Welcome to Fractal Forums

Recently i made a paradigm shift from Particle-wave mechanics (?) to fluid mechanics asthe basis of my model of the observables of natural philosophy.
I have basically finished my review of the foundation of mathematics and found myself questing in the area of Electromagnetism, inspired by Maxwell's (so called) Equations.

The Grassmanns have been my intermittent study for a while, but i keep getting inspired by something they say and get diverted! Nevertheless i have to return to the Ausdehnungslehre at some stage, but for now i want to look at Helmholtz, Lord Kelvin, Maxwell Tait, and Friedmann, and Kolmogrov as an application of Grassmann Hamilton Clifford reference frames, although i have only cursorily approached Clifford's work.

There is a problem in Turbulence i want to look at by means of fractal generation apps, as i do not believe you need a Cray supercomputer to model boundary layer fluid dynamics. A straightforward quaternoin based fractal generator should give some insight.

My initial proposition is to make lines of vorticity fundmental. Strreamlines will be constituted from lines of vorticity.
(http://nocache-nocookies.digitalgott.com/gallery/13/410_06_04_13_3_15_16.png)

Fluids are counter intuitive, which is what i have learned to love. These lines of vorticity are the radii of spheres about any distinguished centres. Thus a stream line is a line of these vorticity bubbles, where the radii that follow a "stream" are distinguished.

Because of printing/plotting constraints i restrict the lines of vorticity to orthogonal axes plus a resultant axis of rotation( hence quaternions).

The boundary limit constraints act as a conjugating region. a focal region for study.

i have trialed a intuitive notion of a streamline equation/profile for iterative generation, but i write for Terry Gintz Quasz, so i will have to modify it for anyone else.
This particular trial image restricts the lines of voricity to a plane. i have a 3d version i will load tomorrow.
(http://nocache-nocookies.digitalgott.com/gallery/13/410_10_04_13_6_18_29.png)

The first principle is that the lines of vorticity are a fractal distribution at all scales and the quantity of vorticity when defined, will be assumed to take a Gauss Maxwell Boltzman distribution in an observable space in which the factors of Lubricity/viscosity,densityand pressure are attributed.

The second principle is that these magnitudes are functions of vorticity. Thus the lubricity/viscosity measure and the density measure to be defined will take vorticity and the distribution of vorticity as arguments, and the regional pressure  will take vorticity, viscosity/lubricity and density as arguments.

i will define a centre as an arbitrary region in space that i am aware of.
Between 2 distinguished centres i will construct a spherical spaciometry by which i will define a metron as a fixed displacement(the 2 centres) a plane of dual centres as points and a line of dual points as a straight line, This arbitrary straight line will be my fundamental axis.

Rotation around the centres will be used to define a conical rotation around the fundamental axis. This conical rotation is defining the concept of vorticity.

The concept of rotation will be more general than movement on a spherical or conical surface that is "circular" any closed or open movement on a conical or spherical surface that returns to its point of origin  is a rotation. In particular if an elastic straight line is attached to the moving region and its origin of motion, that line will sweep out a surface that is conelike however warped.

The use of an axis to define vorticity means we can relate rotational motions to positional displacements. When this is done it should be clear that rotational motion does not resolve into motion around and axis. it resolves into oscillatory motions on arcs in general relative to a given axis or surface.

We may resolve vorticity into a number of ways, but the praxis is to use whatever makes it easier to analyse and describe the motion.
For example, for the mandelbulb we regularly switch between cartesian and polar representations.

To include vorticity then i need to add a dynamic addition to the the positional coordinates,

It is traditional in fluid mechanics to define what a fluid is.
A fluid is not a continuum.
It is a region of space, usually conjugated, which consists in fractally distributed regions at all scales.
With regard to motion these regions are free to move rotationally relative to any other region  but constraints on those relative rotational motions will define the phases of matter.
 a solid is a phase in which the regions have a fixed relative displacement with no freedom of rotation
A liquid is a phase in which the regions have a fixed relative displacement but are free to rotate around any region relatively, according to the constraint
Any other phase of matter has no fixed upper displacement but may have a fixed lower displacement relative to any other region and is free to rotate relative to any other region
It is assumed that each fractal region has its own local reference frame relative to which the region may rotate but affected by the constraints relative to other region.
I will modify the word fixed to a constrained displacement with an upper and lower bound dependent on the definition of substance as a perception of these observable behaviours of fluids.

Their are other attributes which also define this motion constraint but these are substantive. I refer to density, viscosity, lubricity and elasticity. Permitivity or permeability may be defined in terms of these 3, and diffusion as a state density distribution of distinguished fractal regions/

The motive principles behind these descriptions are essentially newtonian unless specifically designated.

Substances, when defined will include electric and magnetic as essential material substances found universally in an assumed structure i have recently decided upon.

s=imaj(z)
z=x#+(x#+y#+s+x#*y#*sin(x#))*i+ (x#+y#+s+x#*x#*cos(x#))*j+(x#+y#+s+x#*s*sin(x#)*cos(x#))*k+0.1*c

The trial vorticity streamline equation for Quasz.
This is not a movie but a snapshot of the space distribution after so many iterations. As with all fractal images it is a "negative" sculpture. What is actually moving under the iteration is "cut away" by the boundary condition and highlighted by the colouring algorithm within certain bounds.
The colouring algorithm is fundamental to grasping what may be happening under the iterations. Stepping through the iterations will give some insight.
The movement of the points is not natural in this test equation, it is highly specified. This is because the movement through the quaternion block is highly specified by the for loops. However, this whole approach tells me something about the relative movement of the points under iteration if the next position is given by the equation.
each point (y,s,w)in the quaternion block{ All( x,y,s,w)} moves by its coordinates plus a wobble at each iteration. The wobble i have kept simple, but of course you can make it exponential or Lissajous etc. every parameer is placed in the equation so that it can be tinkered with. So swop them round set some to constants etc.
i try julia and mandelbrot iterations even though this is not a mandelbrot form yet. The idea is to understand these types of equations to see if a vorticity distribution is apparent.

On the face of it , in julia form c represents a constant velocity profile so hence the factor in front of it. In the mandelbrot type it can be varied as that is the main variable profile descriptor and makes z a recursive function of c,  Z(c,z(c)), while the julia is a recursive function of z, Z(z)

(http://nocache-nocookies.digitalgott.com/gallery/13/410_14_04_13_9_17_27.png)

The colour scheme is not great but i hope you can see the whorls within a vorticity streamline . The equation i used is a polar form of the streamline equation, but i neede to pick the correct colouring algorithm to show all the dispositions in the Quaternion block
[rfun
r=x#+0.01*sin(pi*x#/2)
p=y#+0.01*cos(pi*y#/4)
o=imaj(z)+0.01*sin(pi*imaj(z)/6
rend
z=r+(r*sin(p*1)*cos(o*1))*i+ (r*sin(1*p)*sin(1*o))*j+(r*cos(1*p))*k+4.405*c

Again everything is meant to be varied.

(http://nocache-nocookies.digitalgott.com/gallery/13/410_17_04_13_2_00_43.png)

This image seems to be clearer and gives a depth view. Depending on colouring algorithms vorticity lines seem to make good candidates to replace streamlies.
I have to figure out how to get two vorticity streams in opposite directions from the same block, and then how to collide them!

Just a note to draw attention to the fractal basis of fluid mechanics.
http://my.opera.com/jehovajah/blog/2013/04/23/the-strain-ellipsoid-and-the-theorem-of

(http://nocache-nocookies.digitalgott.com/gallery/13/410_24_04_13_11_45_16.png)
This equation is again for all variations. Will it fit as a model for any observed flow marks on sediments?
[rfun
s=imaj(z)
rend
z=x#+e^(sin(x#))*cos(y#)*cos(s)*i+e^(sin(y#))*cos(y#)*sin(s)*j+e^(sin(x#))*sin(y#)*k+0.0*c

Again, there are traces of the Kelvin -Helmholtz patterns. and a deep zoom shows the pattern reforming, which is characteristic of turbulence.

These highly structured freeze frames are interesting but clearly not instructive yet.

Hi Guys, maybe i can put some wind into the brainstorm.
i am also try to develop different aproaches to fluid dynamics, due i hav to cope lots of time with the current stage of technology
which i find also laid back and not efficient.

neveerthe less there is the problem in numeric solving, that the solving influences it'self. a bit like the race of achilles and the turtle.
startvalue influences endvalue for the next iteration asf. so in critical situations you get the classical divergences like in the logistic equation resulting in the
mantelbrot fractal, as an mathematical feedback.

i am not so familiar with the electrodynamics, but i know the principles are similar like in fluid dynamics. so therefor i maybe cannot discuss in detail
with the maxwell and blotzmann equations at the moment. but that doesn't mean that my ideas are less valid.

i recently was calculating a bit with bernoulli and classical mechanic laws, where i discovered, that actualli bernoulli doesn't explain the energy conservation right in the venturi effect.
it's a result that is actually physically incompatible with the newtowns laws of motions. which would not allow several conditions which is explained in the bernoulli solution how to solve
the venturi effect.

this lead to the wrong conclusion of a totally irrational potential theory which doesn't have inertial effects included and states, that an ideal fluid doesn't cause any resistance to
a body. on which again are all analytical theroies built on in aerodynamics.

This is how the venturi Effect should Look like if Bernoulli is right:
(http://upload.wikimedia.org/wikipedia/commons/1/16/Venturi.gif)

This is how venturi looks really:
http://www.youtube.com/v/KS-GSLrkf30?hl=de_DE&version=3

I Opened a Thread recently discussing the CORRECT explaination of enery conservation in the Venturi Effect here:
http://www.fractalforums.com/let%27s-collaborate-on-something!/developing-fractal-algorithm-for-fluid-dynamics/ (http://www.fractalforums.com/let%27s-collaborate-on-something!/developing-fractal-algorithm-for-fluid-dynamics/)

the same thing explained in 15 Minutes:
http://www.youtube.com/v/lBdpZlLybcY?hl=de_DE&version=3

i attach a file for PBH. I want to use it later , but it is too large so i have submitted text only.
http://mmacklin.com/pbf_sig_preprint.pdf
This is the link.

I have some corrections to make to the posted formulae.
Note to myself these formulae are in the so called Eulerian frame of reference. I will have to figure out some in the Lagrangian frame of reference using iteration steps. These may then be animated.

(http://nocache-nocookies.digitalgott.com/gallery/13/410_01_05_13_1_24_43.png)
[rfun
s=imaj(z)
p=sin(x#)
q=sin(y#)
r=e^p
t=e^q
rend
z=x#+r*cos(y#)*cos(s)*i+t*cos(y#)*sin(s)*j+r*sin(y#)*k+0.0*c

This reformulation removes a procedural error on my part and compensates for the increase in processing time.
The following is the corrected strain ellipse envelope , but quasz shows no solid platform is produced except in this one special arrangement!
[rfun
s=imaj(z)
p=1/sin(x#*pi/y#)
q=1/sin(1#*pi/y#)
r=e^p
t=e^q
rend
z=x#+r*cos(y#)*cos(s)*i+t*cos(y#)*sin(s)*j+r*sin(y#)*k+1.0*c

I think Matts methods here
www.fractalforums.com/theory/an-old-formula-revised/90/
May be useful in getting the streamlines to move relative to each other.

http://my.opera.com/jehovajah/blog/2013/05/01/standing-waves-and-the-strain-ellipsoid
There is an unfamiliar concept of wave here, not based on the unit circle, but on the strain ellipse. It reminds me of De Brogelie waves and Milo Wolff's standing wave theory,
http://www.quantummatter.com/

a local reference frame is necessary to describe deformation of an object posseesing that local frame. While the shape of an object can be described in an external reference frame it makes more sense proprioceptively to describe it against a local one, as this seems to be the way the brain processes distinguished objects.

Because of his tendency, i can observe that certain seemingly definite objects are in fact artifacts of this brain processing Tendency, especially in dynamic situations. The tendency seems to be to boundarise a form, however indefinitely, relative to a polar coordinate referene frame. The geometric centre of the "image" is not usually emphasised, but is known proprioceptively. Sometomes this corresponds with a centre of symmetry in the form, other times  centre of dynamic dynamic symmetry, such as a barycentric centre. This determination, because of the iteration process in perception, only needs to be a rough guess, as within nanoseconds the true centre is converged upon as the image dynamics stabilise.

The strain ellipsoid provides a boundarisation surface that is dynamic but it does not interact with its suroundings. This requires the envelope of the strain eelipsoids as they progress through vorticity. To this may be added any translation caused by normal and vorticity stream pressures.

Since the strain eelipsoid contains a measure of vorticity the question is do we need vorticity streams? I think so because of the situation of being scale free or fractal as a requirement i have imposed.

for vectors a,b,c

if c ~ a+b (That is c is a label to a+b because they both have common beginning and end points)
Then the strain deformation vector ∂c

∂c =mod(exp(1/[a x b]))c
 
where [a x b] is the cross product of the two vectors or the wedge product, that is the area of the paralelogram formed by the 2 vectors.

This defines the strain behaviour between 2 points identified by c, and by a + b.

This can be written in terms of unit vectors, which make plain that a,b,c are fixed but ∂c is ≠ c
As the unit vectors are varied ∂c varies, so  dynamic , iterative function form can be written to model strain behaviour.

Let dc be a vector type
dc :=b means set dx as b and treat as a vector.
Then iterate
dc:=a+dc
dc:=a+mod(exp(1/[a x dc]))dc
Finally ∂c:=dc

It is important to note that this is a wave function of a different form to the sine one and produces sanding wave patterns relative to the initial point and end point of c.http://www.math.utah.edu/~gustafso/s2012/2270/web-projects/christensen-HistoryLinearAlgebra.pdf

This is an accessible survey of the history of Matrix algebra.


However it is noteable that the fundamental contribution of Hermann Grassman (1844)(and in part Robrt Grassmann(1862)ff) has not been noted, nor indeed the fundamental contribution of Sir William Rowan Hamilton(1834.1851). It has been claimed that Grassmann's work was obscure, and difficult to read, but i have not found it so even in the original german. It is iintoxicating and innovative, as i am sure Peano, Whitehead and others will attest.. Google "jehovajah Grassmann" if you want to know more.

i posted this to help with the matrix notion used in the ellipsoid paper.

I have a link here to the concept of a potential flow equation in fluid dynamics.
http://en.wikipedia.org/wiki/Potential_flow.
Why describe it as a potential flow I am not yet clear on, but the gradient of a scalar field showing streamline speed as opposed to velocity is just an equation which if graphed shows the tangent to a curve or line exhibiting the relationship of speed at certain distances..
Say the gradient is everywhere one, then as one moves in the direction of the parameter increasing the speed increases. A vector determines the flow direction.
http://en.wikipedia.org/wiki/Velocity_field
http://en.wikipedia.org/wiki/Velocity_potential
http://en.wikipedia.org/wiki/Conservative_vector_field#Irrotational_vector_fields
http://en.wikipedia.org/wiki/Hamiltonian_fluid_mechanics

In Wave mechanics there are several types of wave, but most consider the typical wave to be sinusoidal. This is not the case however. A typical wave in natural surroundings would be trochoidal, only approximating the sine form at low amplitudes.

The use of the sine function to generate and control oscillation has to be released from its usual diagrammatic constraints, For example it is usual to depict a sound wave using a transverse Fourier wave, with a warning that the oscillation is in fact longitudinal! Although it may seem a trite thing, this usual depiction, for want of rotating through π/2 radians conveys a graphical conundrum! The longitudinal wave dissipates as it travels forward. It alsos propogates in all directions in the medium. Because of these vector motives there is a natural vorticity in such wave propagation's which are completely obscured.

The unsatisfactory explanation of the mathematical description of vortex rings is down to this convention of showing longitudinal waves as transvers analogues,

Now, considereing a transverse wave motion in a medium requires the motive of propagation. A sine wave does not propagate  in space. The graph models the relative displacements of centrifugal and centripetal motives. The change in celerity or its effect velocity is crucial to the model, but it models local rotation only. Thus for a sine wave propagation there must be a net celerity  or motive that by definition is longitudinal, or in a medium radial.

What this means is that vorticity is induced by a combination and possibly a compounding of these transverse and longitudinal oscillations, and spread by the net radial or longitudinal propogation, or conversely compacted by the same.

Do we ever see a purely transverse propagation?

The skipping rope is a fundamental example of wave propogation. The tension in the rope is crucial, because it is this tension that propogates the transverse motion!

If the tension is low, the transverse motion may be damped by the longitudinal motion of the tension dissipating. If the tension is sufficient to reach to the other end, hen a transverse motion may be carried to the end, and return on the returning tension!

Newton's Third Law, although i have explained in my blog that this is an unfolding of the inertial frame concept Newton had in mind. posits this contra action in all interactions where equilibrium is disturbed.. The inertial frame maintains a static or dynamic equilibrium. In the case of a rope, the equilibrium is dynamic, even if the rope itself is static.

It is this insight about the nature of spacematter, a dynamic fluid medium in which static "motion" is a net effect of ever present dynamic ones, that Newton hoped to take forward into the study of fluid mechanics in book 2 of the Principia.

The nature of fluid spacematter was beyond his time and duties burden, and despite fundamental insights and experiments, he left the field to others to develop, secure that his principles and praxis would yield results.This in part was why he contradicted Huygens, apart from religious differences. Huygens offered no convincing empirical data, only some "mathematical and geometrical guesswork! This Hypothetical approach, Newton did not encourage, propounding that in these matters we must proceed with caution from certainty to certainty, deriving hypothesis from empirical data.

Huygens may have protested that he did  derive his solution from the empirical data of the fringe  anaomalies, but in fact, this was not demonstrated by him, just posited with no empirical evidence for the widely held belief of an extant Aether with the attributed properties.

Newton found fluid mechanics confusing/perplexing. Consequently he could say little that was certain about it. In particular he could neither demonstrate or eradicate the prevailing "guess" of DesCartes, that vortices moved the planets. For this reason he used the ill defined term gravity, in contradistinction to levity, to describe a tendency in spacematter to clump together, but as to what it was he made no hypothesis ! He had no frame for it! This from the man whose reference frame, derived from observing rigid matter, elastic matter etc, was so persuasive! Newton assessed that his rigid motion laws were a better match to empirical data than his fluid motion ones, and some have misread this as overturning a vorticular dynamic in spacematter.

In making such a comment i defer to Newton, who fully acknowledge the work of Hooke and others in his insights. It is Hooke's law, as well as the behaviour of pulleys that supported his third law of interaction.

So the propagating force is tension, and because it is opposed, when it meets a boundary the tension "reflects"! We can understand this simply as he net force being opposed by a reactive force which itself must inhere in the rope under tension. Having achieved the require tension to overcome the propagating tension through strain, the rope is now in an unstable deformation. Consequently this unstable deformation becomes a net driving pressure in the opposite direction.

In  a rope with no tension, there is no clear or transmissive boundary condition. consequently "gravitational" pressure counteracts the tension force as it propagates, but provides no unstable deformation in doing so. It acts as a restorative pressure maintaining equilibrium. The dissipation is one of our clues to the transformation of motion into other motions and forms that move away from the source through the inertial background. We tend to call this energy dissipation, which is dissipation through work. It is this work concept that allows us to link heat to mechanical actions, and underpins the mechanical description of electromagnetism, But it took Maxwell and others like Helmholtz and Kelvin to suspect that fluid mechanics would be the best reference frame in which to describe all material interactions.

The longitudinal wave, therefore is the necessary tensile support for the transverse wave. The longitudinal wave must precede the transverse wave.
The longitudinal wave is the propagation mechanism for all reflection in wave mechanics, while the transverse wave is responsible for diffraction and refraction.
While a longitudinal wave may propagate without a transverse motion. a transverse wave must always propagate within or adhering o a longitudinal one. Because a longitudinal wave propagates radially, it also contributes to refraction, but the refractive enhancement of the transverse wave is what allows a prism to spread out the colours of the spectrum.

The fact that the longitudinal wave reflects is evidence of the longitudinal oscillation. Thus as a longitudinal wave propogates it is oscillated by back pressure at each encounter in it advancement through the supporting medium. This back pressure  would be thought to be radial, but in fact it is not. The back pressure is arbitrary, defined by the response of the supporting medium. In addition if the medium has its own dynamics this acts on the longitudinal propogation. At best i would characterise the back pressure as spherical resulting in an imploding but weakening spherical wave front at the same time as an exploding weakening wave front propagates. In such a scenario i would characterise the resultant vorticity as regionalised and even turbuelent within the advancing longitudinal propagation. Add to this any transverse wave motion and i would expect trochoidal filamentation occur at some given radial distance from the source as the longitudinal reflective back pressure dissipates.

In a smoke ring or fluid behaviour around a jet we can see that this filamentation is not necessarily linear, but in fact curls of into mushroom like protrudences depending on the supporting medium's Reynolds number. This number reflects the relative effect of viscosity over inertia. That is to say the internal pressures inherent in and defining a substance which has a separate boundary to other distinguished substances with differing internal pressures. If one substance is placed within another such systems , by boundary layer interactions may generate motion which is inertial within the surrounding medium, if it is taken as a reference frame.

Thus an inertial reference frame is a matter of choice and convenience, but chiefly it encapsulates a region in which all the pressures may be considered and distinguished. I am sympathetic to the thought that vorticity may underpin viscosity/lubricity in a mechanical way similar to gears and bearings.

The longitudinal wave i feel is modeled by the strain ellipsoid, but requires a dissipative factor even as it tends to infinity. In the same way the reflected longitudinal wave would dissipate, but the superposition of components from the radial spherical reflection may be significant in the short distances from source. The quality of this reflection and thus the perceived longitudinal oscillation will depend on the reflective properties of the medium.

In passing, the electromagnetic behaviours seem so superior in this regard, that i would conclude that the basis of their efficacy is a substrate of perfectly tensioned spacematter with ideal reflection properties and very little radial dissipation. That is a focussed signal tends to remain focussed over long distances. This directional quality of strain is inherent in the parallel trammel lines along which it is defined in the strain ellipsoid.

https://www.fractalus.com/kerry/articles/vortical_flow1.pdf
http://www.me.jhu.edu/meneveau/pubs-fractals.html

Some active links from Jules Ruis (thread earlier).
http://en.wikipedia.org/wiki/Kelvin–Helmholtz_instability
http://www.academia.edu/1253068/Electrostatica_y_Magnetostatica_de_cuerpos_elipsoidales_formalismo_del_tensor_depolarizacion

(http://nocache-nocookies.digitalgott.com/gallery/14/410_07_05_13_9_00_46.png)

[rfun
s=imaj(z)
sc=imaj(c)
l=real(c)
m=imaj(c)
x=x#
y=y#
p=1/sin(x*pi/y)
q=1/sin(1*pi/y)
r=e^p
t=e^q
v=l+m+sc
rend
c=l+(v+l*m*sin(l))*i+(v+l^2*cos(l))*j+(v+l*sc*sin()*cos(l))*k
z=x+r*cos(y)*cos(s)*i+t*cos(y)*sin(s)*j+r*sin(y)*k+0.8*c

The full vorticity strain ellipse formula.
My strain ellipsoid has a special formulation to produce visible sculptures, but in general it can be varied. The strain ellipsoid shapes thr vorticity streams.

The combination of the Z and the C Quaternions in Quasz reveal that in this case a heightening of definition results. As these are not animated i have to look at the fluctuation as iterations increase. which i have not done yet.

I feel if the combining of different streams is not just as simple as a vector addition of the 2 stream equations, this format offers me some pathway towards conflicting 2 streams, but i do not know that yet.

Again i have to remind the viewer these ae "negatives of the actual turbulent flow which is sculpted away.

The far view of the strain ellipsoid shows me that the concept is being replicated by the fractal generator, but as i am just fooling around, i have not considered what it is i am creating and how the boundary conditions sculpt it.

The boundary conditions help me to produce a negative sculpture only in the correct fornulation. The surface produced shows me all those quaternion points that do not break the boundary condition.

Straight away  this means if i can run two motion formulae together with different boundary conditions, one inside the other  i may be able to see the combined effect of the motion formulae.

 i like to keep the idea clear and simple, and then explore it through manipulation of the idea. Although this would potentially create a "false boundary in the sculpture, i want to see what that looks like first in a simple way before complicating the boundary condition. i mean the Mandelbrot set boundary arises from the simplest boundary condition!

The other  thing is the strain ellipsoid approach relies on a straight boundary which is "scale free". The advantage of this is that the strain ellipse shape and deformation rate reflects the orientation of this straight surface to the spherical test bubble in a laminar flow. Understanding how the strain ellipse represents this interaction would help to define pressure fields at boundaries.
the advantage of this is that the notion of boundary has nowhere been defined as  rigid, and yet the paradigms are borrowed from rigid dynamics. Technically we can modify the formulae accordingly, if we know what we are doing. However this requires a greater freedom to add detail than i believei have in Quasz, so i may need to do some working round to fit it within my codelimits.
My first idea is to see how the strain ellipse/ellipsoid copes with a parabolic curved surface y=x^2

So my strain ellipse formula is a model of pressure, not of strain. It is proactive not reactive.

Analysing the theory on strain ellipses and hoping to use the strain ellipsoid as a model of pressure lead me to explore how the background stream flow strains an ellipse. It suddenly became clear that my formula is accelerative in an ellipsoidal way. Bingo.

It was a motion formula acting like a source of pressure!  A shaped region of pressure, and if i can a shaped region of reaction with some physics might take me somewhere!
http://my.opera.com/jehovajah/blog/2013/05/05/the-problem-with-force-is-it-is-non

The more i meditate on the ellipse. and ellipsoid the more i see its appllcability.

The strain ellipse comes out of deforming the circle. The theory starts with laminar flow, and the circle at rest in the flow. This is the same as saying that the internal points in the ellipse move with the background.

Thus i can add the background flow equation to the points in the ellipse. Adding them to the ellipse boundary should be sufficient.
Now changing the background flow changes the ellipse accordingly. By working out the position of the elliptical axes and some other orthogonal radials inside the ellipse i can read of a measurement of the train for those radials inside the ellipse.

spaciometrically we can see that there is rotational strain that is twistorque as well as axial strain that is centripetal and centrifugal . From these i can resolve into tangential strain for the selected points on the boundary where the radials fall.

Looking inside the ellipse i can see 4 regions of counter clockwise rotational strain, that is counter twistorques which are trochoidal, that is they exhibit a complex path like a trochoid rather than just rotating relative to the ellipse centre as one would unconsciously imagine. This is because as the point moves the ellipsoid rotates. To be part of an ellipsoid with any internal viscosity each internal point has to move with the appropriate rotation. The back ground motion therefore is not indicative of the motion of an elliptical strain reaction.

The difference is due to the difference between Lagrangian and Eulerian reference frames. Thus the strain on the ellipse boundary hides a simple fact: the boundary is instantaneously dynamic. It draws in from within and without fresh points of material. in a fluid this is possible, in a solid this is overlookd in the Hooke's law based strain equations.

In an elastic deformation the marked poits of the metron stretch apart. Physically this meeans that those fixed points in the laboratory reference frame (eulerian) are instantaneously replaced by new points from WITHIN the material undergoing deformation. We ignore this in simple models. but we do not in fluid models. The venturri tube which Bernoulli experimented on examines just this strain phenomenon under the guise of fluid hydro static pressure.

Applying material conservation laws leads to some explanatory principles and realations, but Bernoulli's Important observation is too simplistic to explain the fractal picture unless it is applied fractally and multi directionally in a fractal generator.

The hydrostatic pressure misleads us. A pressure system is multidirectional and thus the hydrostaic pressure is bur one net pressure reading. We need the stagnation pressure as well for a fluid flow. This is in fact the barest minimum requirements for a reasonable model, but the strain ellipsoid model actually allows us to independently measure 16 points of varying pressure on the eliptical boundary. This pressure varies because of the instantaneous motions of the strain ellipse in an instantaneously varying laminar flow field(such as near a boundary).

So Bernooulli needs to be updated by placing pressure meters at the various points around the ellipse and upstream of the flow as well as downstream.

The conservation law reveals that a cylinder of fluid is deformed longitudinally and cross sectionally. The assumption is that the material points push each other into the respective positions under the influence of an external pressure. However this cannot be the case at the boundary of the pressure systems. For example a point at the boundary has to move at zero velocity. If there is no viscosity these points are lost to the fluid body in motion, the matter is not conserved, except pragmatically.

Secondly the pressure "forces" material points into each other. Do they coalesce or do they separate a space between material points so they can "fit" in?

Logically they must coalesce, so again matter is not conserved. If we say they create "space" in a continuous or contiguous medium we have to acknowledge that work is being done in a pressure background! Thus to conserve matter we have to do work. Where does the energy to do this work come from?

http://www.youtube.com/watch?v=QWEq3xifCDw&feature=youtube_gdata_player

In our strain models both rigid and fluid we have not accounted for energy or work done in conserving matter. In our Thermodynamics we hav not accounted for work done in conserving matter. In our electrodynamics we have not accounted for work done in conserving matter.

We can derive Bernoulli like principles from the point of view of work done in going from one state or configuration of matter to another. The terms are recogniseable as so called energy terms. What is not realised is that these terms represent the work done to conserve matter as well as to move it and transform it.

We have a conservation rule for energy in certain mechanical systems. It is an extremely useful principle but we are not sure if energy is consrved in all interactions. We certainly know it transforms the matter involved in the interaction as just described, translating it, conserving it , rotating it, but we have to consider the triboelectric and tribomagnetic effects of "collisions" with pressure fields . in fact we have to consider whether consrvation laws apply to these transformations of form, that is into electric and magnetic and thermal pressure systems. In fact is their a conservation of "pressure" rule!

Since the Dirac Fiasco, Physics has lived with this notion that energy can be created and destroyed. This was dreamed up to avoid having negative energy, a concept that goes against the laws of thermodynamics! But since , i sayy there is no real justification empirically for excluding negative energy, maybe we ought to revise the Thermodynamical laws, and see them for what they are part of a greater role for some metaphysical notion called Energy which can transform translate and conserve matter, or rather spacematter, and it can do this dynamically in tatic or dynamic equilibrium statusee as well as in explosive dynamic statuses, but it does so because their are contra forms of Energy just as there are contra forms of everything else elemental to our reality experience.

So as you can see the strain ellipse. modified by energy or work considerations may be a way forward., providing we break free from the notion that laminar flowconserves matter within a boundary, or that Hookes law conserves matter within a boundary. Material points in a dynamic situation rotate through and conservation boundary on a trochoidal path, eventually failing because no new internal material points can rotate to the boundary and external material points then rotate into the material. This is the failure of the conservation of matter work done by energy in that material with that viscosity, The work done by the energy from that failure point onwards travels ino the breaking of electromagnetic and nuclear bonds, as we understand them, and into the radiation of "heat" and the rotational motion of the material points exhibits as sound and heat vibrations as well as light rotational disturbances in spacematter(Aether or Ether or a Lagrangian or Hamiltonian Description of material point behaviour).

Another important aspect of modeling fluid dynamics is the model of terminal velocity.
Using the strain ellipse methods in this situation may be revealing, because terminal velocity clearly gives us a mechanism for generating the velocity gradient field close to a moving object.

This velocity gradient field is again based on a potential flow equation which may be a better model for strain ellipse method than the ubiquitous shear differential.

Thinking about the strain ellipsoid made me realise that one solution for the flow(i think laminar) is 8 contra "eye like " forms facing each other contiguously with antysymmetric rotational flow patterns, all shaped in the ellipsoidal boundary.

 The image I have in my head is like an ellipse with each octant around 3 d axes having a flow that ranges from hyperbolic to circular as the test point moves out from the centre. These are shaped within the ellipsoid boundary. 2 of this type of strain ellipsoid form a stable orthogonal pattern that mimics " N–S" polarity mechanically for the whole system of 16 "eye-lets"

While this is not a definitive solution , being one of many, i think it is characteristic of a great many strain situations including electromagnetic. in fact i might consider a work up that replace the dipole with a strain ellipse!

I am currently finishing off some other research while assimilating universal hyperbolic geometry.

I intuit that this is in fact the natural geometry for fluid dynamics and may provide a natural insight into the pressure ellipsoid and the strain ellipsoid for a general fluid. The natural vorticity in these forms may be sufficient to describe the centripetal centrifugal tangential frame devised by Apollonius, and adopted by Newton as the natural descriptive geometry for the action of motive as an acceleration of a body at all scales from the point to the distinguished form,

Within this geometry vectors etc are derived naturally as projected objects/entities, and boundaries arise as resultants of complex fractal iterations of projective processes. That mens they are ephemeral yet clearly experienceable, suitably described by the notion of instantaneous formations.

For me, the Elliptical error is explained best by iterative processes in universal hyperbolic geometry.

http://my.opera.com/jehovajah/blog/2012/12/10/the-elliptical-error

Just back off holiday. Visited the Alhambra in Spain. Fractal Mosaics or what!

Anyway watching the Actual waves in the Ocean confirmed that the sine function as a wave model is totally misleading. Actual waves are trochoidal in 3d, that is they form ellipsoidal vortices that feed into each other , but whose centres do not move. The matter of which they consist rolls around these centres transferring not matter but motive, Newtonian source of acceleration!

The structure of these Actual waves is also interesting. The waves not only are ellipsoidal or trochoidal in one axis but in all axes. Thus the material rotates in a trochoidal pattern around the centres . Those who have followed my research know of the work of Lazarus Plath, a pseudonym for a creative programmer. Find him on YouTube under qqazxxsw.

http://www.youtube.com/watch?v=rz8A5l_yn34&feature=youtube_gdata_player

The Trochoids his programmes can create are deeply informative, but based on the circle. Although he provides real time variation this is only sinusoidal. However the number of trochoids that can be " layered" is phenomenal, so this is not such a limitation as at first seems.

The trochoidal wave forms in the ocean demonstrate a surface feature in which they appear to be rolling " forward"  toward the beach only. But in fact they also roll " sideways" along the beach at the same time. Thus the wave is not a parallel phenomenon but a radiative one, and the radiation is not concentric either!

These standard concepts are in fact explanatory models that mislead the student. The behaviour of actual waves in the sea are as interlocking, fractal vorticular regions. That is vorticular shells radiate out from any " motive" source in a liquid substance. The ripple effect is quite noticeable but it's fractal nature is not. It is only in the coronal display of the initial " splash down" that this ellipsoidal/ trochoidal behaviour is distinct enough to overcome the surface " tension" effects of a liquid.

http://www.youtube.com/watch?v=VORN5SHiT7A&feature=youtube_gdata_player

The surface structure of the liquid is also fundamentally different. When an actual wave " breaks " on the shore the regional ellipsoid/ trochoid gains kinetic energy features that spread it out rapidly into a foam on a sandy beach or a highly projected film on a smooth surface.

This film returns into the back wash depeni g on the "gravitational" effect of its potential energy. As it returns it is actually lifted into a film of returning liquid which rides on top of the advancing ellipsoidal waves. Thus a thin layer of returning liquid is shaping the rolling advancing " wave" front. This thin layer returns into the advancing ellipsoidal spread and appears to do so smoothly. But this depends on the varying energy  of the advancing wave front. A test floater reveal the vorticity in this layer which generally follows a largely predictable path, but which also shows extreme and unexpected variety.

It was noticeable that for certain conditions this film provided a current that took the floater out into the ocean returning it to shore unexpectedly, sometimes to one side sometimes to the other side of its entry point. There was a region where the tester was never taken out to sea but merely washed in and out slowly moving along the shore line.

This set of observations was applicable to a liquid, but in a gas, the centres of the ellipsoids or trochoids themselves move arbitrarily.

The use of a background flow in which these centres are embedded is the classic description of fluid behaviours under " laminar " flow. However laminar flow is a special boundary case for objects moving in a fluid. The general motion is left uninvestigated, although interest in general turbulent regions within and outside of boundary flow is of great interest.

http://www.youtube.com/watch?v=7aIMapx2pag&feature=youtube_gdata_player

The fractal model provides the best in class approach to modelling these fluids, but I believe that the trochoidal formulation of the flow will provide the most general apprehension of what is the potential structure of kinetic and potential energies or Newtonian Motives in fluid mechanics.

http://www.fractalforums.com/gallery/hyperbolic-fractal-infinite-trip-animated-gif/msg63722/

Another great example by Bib

I have 2 resources on Newtonian Fluid Mechanics.

http://my.opera.com/jehovajah/blog/2013/07/05/newtonian-fluid-motive-as-a-newyonian-active-principle

http://my.opera.com/jehovajah/blog/newtons-principia-on-fluid-mechanics

The main point being that Newton retreated back to his model in book 1 of the Principia only after considerable effort to crack the more complex fluid mechanics, in which he made not a little headway.

The fractal complexity of the situation is not what defeated him. He had the correct tools and intuitions, honed by years of meditative study of the Ancient Mechanical Philosophies. He also used uniformity as his gateway into the fractal world of nature. While I prefer to approach fractal nature from a fractal Mosaic, essentially newtons methods of fluents and Fluxions is the approach, both rhetorically and algebraically, and spaciometric ally.

The notion of infinitesimals current in Newtons day, backed by growing microscopic evidence of corpuscular forms, notably Hookes and Loewenbrooks treatises encouraged Newyon in hi application of the methods of exhaustion from the Greeks. Where he differed from the European school including Leibniz scholars, was in the rhetoric on limits. His limits never vanished. They were in fact ultimate ratios or proportions of evanescent or nascent quantities, which themselves were formed out of magnitudes by some boundary process.

Newton therefore was always pragmatic and perisos, that is approximate, rather than Artios which is precisely accurate!

What defeated Newton was the degrees of freedom in 3d space. There are 6 degrees of freedom that are orthogonal, and innumerable axes of rotation. In the planar case the 4 degrees of freedom are complemented by only one non planar axis of rotation. Thus , given a rigid body, reducing it to the planar case captures every possibility for that plane using the rectilineal axes ans the circular adjunct to them. This reference frame , in the plane gives every point 6 degrees of freedom as a basis " vector" .

Generally we split this basis into overlapping Cartesian and Polar coordinate frames in the plane.

The important notion of a spanning basis was actually first expressed by Hermann Grassmann, but it was clearly intuited by Newton, and forms part of the Toolbox Apollonius used in his work on Conics. Possibly Euclid's Conics carry the same notions. This reference frame in the plane is in fact the reference frame for universal hyperbolic geometry. I would strongly recommend viewing Norman's lectures on the subject.

So it is clear to me that Newton's general" geometry" or rather Spaciometry was that of Apollonius, and so was a hyperbolic geometry. The fuss created by Gauss, through Riemann was more an attempt to garner the intellectual high ground on the topic to himself. He did little to encourage Bolyai, and he attempted to suborn Lobachewsky by learning Russian so as to write influential commentary on Lobachewsky's work.

It is ironic, therefore that the gift he turned away, when dismissing Grassmann's work, was the very thing he was aspiring to: a great work to maintain Prussia's intellectual standing and to enhance it! And delivered by Gauss as a gift to the Prussian nation and the Emperor.

Newton's attempts to tame the surging tides of fluids by equating the fluids to infinitesimal solid cylinders go infinite length fails precisely there! The cylinders of fluids are by no means rigid. Nor is there any reason to suppose that they form independent boundaries! Thus all 6 degrees of freedom have to be used to describe the motion of a point in these cylinders and in addition an innumerable number of spherical shells representing an innumerable number of axes of rotations.

Bearing this in mind it is no wonder his work on fluid vortices failed to be accurate enough!

Universal hyperbolic geometry now offers a way to project these degrees of freedom onto suitable planes, and then to algebraically return to the 3 dimensional situation.

In addition, the advent of computing power makes this a much more manageable but still overwhelming task.. Computational Fluid Mechanics attempts to do this task, but I believe using a less elegant approach than UHG.

We shall see.

In addition, the Newtonian notion of motive has clarified for me the structural dynamics of vortices in the UHG and leads me to posit a 3 region structure to a fluid element, which will not necessarily be a cube, but more generally a bubble with an interior spiral, a bubble skin with a trochoidal dynamic and an outer region where high energy interactions take place with the bubble skin driving the internal dynamics of both skin and bubble interior, or transmitting the internal dynamics of both interior and bubble skin.

This skin interaction is studied under surface tension, and that needs to be fully exposited in its transmissible nd receptive roles.

Finally, while this naturally applies to liquids, fluids in fact are a more nebulous aspect of SpaceMatter. Consequently the principles discovered in the liquid case will also have analogous implications for the more ephemeral and nebulous gases, plasmas and electric and magnetic "fluids".

I would also add that in none of these concepts will mass be defined by gravity, nor will gravity be defined by mass. The notions of Mass will be entirely Newtonian, and defined by the action of a body under the influence of motive. Consequently, as we shall see Gravity will be the resultant action of a number of motives which will all be physical in the classical sense of Phusis. That means they will all have opposing motives!

(http://nocache-nocookies.digitalgott.com/gallery/14/410_17_07_13_8_35_41.png)

This is a trochoidal description of a fluid . In its simplest form it shows many interesting structures, which are constructs. They construct the spatial regions where strain is in equilibrium and thus potentially at its greatest. The dark areas show where stress has broken the equilibrium in the many directional attitude of pressure.

the effect of a global pressure may be evident but it is very slight, so no clear vorticity streans are evident.

The next stage is to explore many motive laws to see what boundary conditions they generate!

Note the difference. the boundary conditions within the fluid are generated by the motive laws. We only observe them when they achieve some kind of equilibrium, otherwise they are beyond our senses.

I have some interesting structural results , but the margin here is too small to write them in.

(http://nocache-nocookies.digitalgott.com/gallery/14/410_27_07_13_11_25_35.png)

This is a part of a much larger sculpture designed to explore diffraction and interference patterns in a fluid medium. The image shows a region where vortex shedding is evident .

At the same time, this vortex shedding is interpretable as interference patterning during diffraction.

Fluid dynamics really has it all covered!

Best of all it is Fractal.
http://jehovajah.wordpress.com/jehovajah/blog/2013/07/27/single-slit-interference

I posit that trochoids of the sphere are general solutions to the Navier Stokes and Euler equations for fluid mechanics

http://jehovajah.wordpress.com/jehovajah/blog/?id=60182732

But I have much to do to recommend the idea.

A solid has 6 degrees of freedom but a limitless set of axes of rotation.

A fluid has n degrees of freedom as well and a limitless set of axes of rotation. It is the near solid or fixed relationship between these degrees of freedom in rotation that is usually studied. The surprising fact is that rotating frames of reference self organise!
http://www.youtube.com/watch?v=Ans3tnvMyTk&list=FL-UDZA0WQ_UwF1mpBTSjceA

So now i am pretty much comfortable with the notion of Aether and Ether.

Philosophically it is unavoidable, the science that denies it is Luddite. The fact is the mathematics is derived based on its assumed or implied existence, and it is merely blinkered thinking to insist that the mathematics is sufficient to describe what in fact we require philosophy to do.

Somewhere in the early 20th century huge propaganda machines warped our natural philosophical understanding of the experience of reality, for mainly national security reasons. Then scientists found that this se of affairs was a cash cow for research funding, and no one really wanted to change it! do not rock the boat!

The Newtonian fluid motive is my starting point. To it i must add a notion due to Dirac and Newton, the notion of an Impulsive force. The Dirac delta function is the mathematical model of an impulse, and what we have to observe is that an impulse is an instantaneous force effect. the notion of instantaneity does not rely on time, In fact time is problematic. Instantaneity depend on differentials of motion, From the differentials of motion we necessarily must expect a motion transmission .

Because there is no real notion of time , we must expect that motion is fundamental to the reality in which we have our experiences. Thus if i apprehend a motion, i know from experience that motion will continue in some fashion. The fashion of this motion is varied and complx and fractal, but we can "rationalise " it. When we do the ratios of experiences of motion we define this ratio experience as time!

Tyme, the old word from which time is derived meant the records of positions in space. The modern notion of time retains that idea fundamentally. Time is the position of some hand on a clock face, the position of some digital represetation in a sequence, the postion of some pensulum in som sequence of periodic motions etc.

The point is that time represents the position of some specified indicator in its inherent motion. Instantaneous therefore refers to that position in its motion if we could take a picture of it on a very fast film, in a movie. Each frame would capture an " instantaneous position".
The fact that some logicians might confuse themselves by discounting the term instantaneous, or confusing motion with the existence of time need not concern us. Logically motion must be fundamental, Without it nothing would move!

So now the instantaneous motion or displacement we have a tool to measure, say Hooke's spring force measure, must lead us to expect the motion to continue. This motion continuing is what we call strain transmission. What Newton attempted to get us to focus on was not continupus motion, but continuous rest, or uniform motion. These phenomenon reveal that forces are in equal and opposite action, in a space or sphere of equilibrium, called inertia, so that motion is kept in a dynamic equilibrium, or returned to a dynamic equilibrium by these inertial forces.

The consequence of this is that motion in space is not only endemic, but it is in a kind of dance that attempts to restore a notion of equilibrium, or inertia. Inertia in this sense is a local phenomenon, and Newton concluded it was relaed to density in space. His vision of space as a unierse was that it was not uniform in density. What he could not grasp was how that density varied. In particular, in a rarefied atmosphere how could force be transmitted?

We must be aware of the great sophistication of Newton and La Sage. The notion of a wind was not inimical to their thinking. For a great while Newton, in correspondence with La Sage felt that a corpuscular wind might be the transmitter of force, much like the way the wind blows the sale of a ship. However, this explanation came very close but failed to give a consistent explanation. Newtons own work on Fluyid mechanics gave him the same hopelessly complex situation which he could not resolve.

The resolution was universally thought of as being the field concept of Faraday, and Maxwell. However this was not the case! The real solution is the correct notion of motion and the Diract delta function. 

http://www.youtube.com/watch?v=6MstJEAlNFs&feature=youtube_gdata_player

These are the equations I will modify to describe fluid dynamics with shock waves described as Dirac delta functions.

The general pressure bubbles will be modelled by trochoidal type equations of these forms.

I have been busy grasping at straws.

Fluid mechanics is such a deep paradigm that i have too many things to declutter from to grasp its essential simplicity.
Initially fluid mechanics was a means to an end: to understand electrostatics and magnetostatic in terms of the Shunya field, the field whose spaciometry is the vortex and whose trigonometry is the roots of unity and the zeroes of space.

``it has taken me a while to be able to formulate it in language in this fashion, and it is poetry that is more meaningful to me than the reader, i suspect.

So again i recommend the amazing app by Lazarus Plath here
http://home.comcast.net/~trochoid/TroWithMesh.html
for you to explore in your browser running java script.

Once i thought i understood more than his ineffable explanations at youtube qqazxxws, but i realised pretty early on i did not have the insight that he intuits in a way he finds hard to explain.

Play around with clicking and dragging. click on the numeral boxes and drag. click under or over the numerals. Click the tiny boxes in the top left. Explore.

this app is based on what i call Twistors. only radius and rotational frequency are used. everything is displayed by radius and relative rotational frequency.
Meditate on that, and how 2 dimensions rotated relative to each other in the screen plane ostensibly can develop a rich affine geometry/ spaciometry.

Think as Grassamnn thought, particularly about Phorometry.

This is why Newton could only go so far in his fluid dynamics, and why he came closest to understanding action at a distance.

Claes Johnson on fluid dynamics

http://www.youtube.com/watch?v=t7e_6bkUFzE

And on D'Alembert paradox.

http://www.youtube.com/watch?v=3u3OzSlEvdk
Johnson and Hoffman solved the paradox but did anyone care?

This article on resonance is particularly interesting for fluid mechanics.
http://en.wikipedia.org/wiki/Self-resonant_frequency

It is known that Freak waves appear in the open ocean unexpectedly. What is not known generally is that marine scientists used to cow tow to a theoretical model which said they were impossible! Thus resonance in fluid systems was denied by ignorance ?
http://www.ivorcatt.co.uk/x3cj.htm
This reticle by Ivor Catt shows how this ignorance extends into the " electric" realm, in terms of the notion of self resonance.

Self resonance seems to be an oxymoron. An object that vibrates clearly vibrates with itself! Resonance is where another object vibrates in sympathy with another or when a vibrating object shows marked and distinct amplitudes in its vibration.

In the mantra of sine waves this is when waves constructively or destructively interfere. This happens because " waves" pass through each other and interfere with each others wave motion.

Thus we can refer to interference patterning when we are discussing Resonance, but we must not confuse the sine wave model with physical wave processes.

Self resonance can only refer to physical vibrations within a bound form reflectively interfering with each other. Unless reflection occurs there can be no self resonance.

Ivor's articles states that a capacitor has no self resonant frequency for in series capacitance.  If there is no undulation or vibration then there can be no interference pattern. However. A reflective plane wave must aggregate additively in this case, so I would expect the capacitance to increase over time.

Thus the charging of a capacitor is the interference behaviour of a plane or step wave in this case.

Ivor mentions that self resonance exists in an inductor. This means tht magnetic flux within an inductor oscillates or undulates. The variation in magnetic flux kin an inductor may itself be due to the variation in the energy pumped around an inductor, ie an AC induced melectromagnetism. In this case , there is no reflection, but hysteresis provides the variation in the magnetic flux behaviour, and so the inductor exhibits resonance , but not self resonance in terms of reflection. This would be induced or driven resonance.

As far as physical waves go, I have previously mentioned the strain ellipsoid and how that gave me a notion of a pressure bubble, I also mentioned my experience of physical waves on the beach and their cylindrical rolling action, and how they seem to roll onto the beach at an angle while collapsing into foam.

Looking out to sea I noticed the characteristic undulatory appearance and dynamics, and of course an extended ellipsoid fits the image better than a extended cylinder. The dynamics is like beer barrels rolling over each other! Stacked logs come to mind, but they are too cylindrical.

Smooth pebbles might be another physical sample especially as they are tipped out of a dumper lorry. Any rockslide has the same characteristics and in fact scientists have expressed surprise at how fluid the motion of rolling rocks appears when seen on films of landslides or rock avalanches. Snow avalanches, once they begin to break up into rolling chunks have the same motions.

However the ellipsoid is just one physical form that fits the bill because we have studied it very well. The other Shunyasutras I will refer to as trochoidal forms and these have a sea shell like form or appearance. A special such form involves trochoidal space curves called Loxodromes and these are particularly significant for descriptions of electro Thermo magneto complexes as wave foms.

In this regard the simple helical space curve considered as a wave rolling perpendicular to its central axis may be a sufficient model for an extremely long ellipsoidal or loxodromic rolling physical wave.

The sine wave model has really obscured the physical wave models, and in particular, the propagation of energy in a physical wave.

For instance: the wave that attaches itself to a conductor is such a long rolling wave. It rolls onto the conductor from the surrounding "dielectric" by means of magnetic coupling. The energy transfer along the conductor is in the space around the conductor and it progresses along the surface of the conductor as a wave coupling action. Think of a sea wave rolling onto the beach . The wave engages the shore line as it is , not as a straight line, consequently the foam takes on n undulatory form relative to the shore line. Similarly, as the approaching wave from the dielectric is magnetically coupled to the conductor a sharp rise time is seen as it progressively attaches. This rise time travels at the speed of light as a pulse front along a magnetic ' shore line'.

The discussion of how a cell battery or a capacitor battery achieves or is involved in this I will leave to another time, but it involves twistors and Twistorque.

The secret of flight is such a simple breakthrough it deserves pondering

http://secretofflight.wordpress.com/

Essentially density is the key.

The concept of density enwraps gravity and levity into one misunderstood concept! We measure gravity by levity and levity by gravity, and say that the denser object is a multiple of the density of the lighter or less dense object. And we imagine force moments rotating against each other. The levitating force in the beam is ignored, the levitating force in the environment is ignored. It is only when a balloon lifts that we recognise it. Then we mis ascribe it. Density pushes against its environmental density. It does so through surface area. Density is another example of Newtonian fluid motive or energy. And the first transformation of energy is into density force within a volume with a surface area.. As that surface area increases that density force spreads out to act on more of its environment

The greater surface a density force acts through the more levity it exhibits .

When the environment acts on a surface the density force is crucial to levity. By compressing the environmental density force into a smaller volume levity is induced in the wing which being rigid maintains it's density force

The environment above an aerofoil does not compress, instead it works to move out of the way . Consequently it does not impinge its density force on the aerofoil.the net density force of the aerofoil helps spread the environment out above the wing. Finally, at the trailing edge the disturbances generate vortex columns in the environment. These vortices have no net density force because it is applied in a spinning vortex. This allows increasing environmental energy to be stored in the vortices but means the leading edge of the aerofoil is applying density force to generate work on the environment that results in the vortices, and the compressed environment is reacting with density impulse force, creating levity and drag.

Drag is levity applied in a direction other than up.

Fluid mechanics from an artists point of view!


Hermann

Thanks Hermann! Some great impressions!

My research into barycentres has thrown up some new insights.

While the aether was a common religious concept borrowed by theosophers in their philosophical attempts to describe the workings of the material world, interdpfaced with a spiritual world. Philosophers were not do keen. In fact they often avoided using the term, which was becoming a catch all.

Descartes popularised its fundamental mechanical use as a medium of extension within the spirit enfused reality ordained by god. He ran into trouble with those who believed in transsubstantuation, because god to them could not in any way be physical, or have physical properties!

Leaving their ideas to one side we find the aether taking on a kind of strain or active medium role. Although the term Plenum is not strictly the same as the aether, by the time of Grimaldi and Huygens, the plenum was generally confused with any number of ideas of an aether.

The more mathematical and geometrical philosophers tended to keep clear of too close an association with any particular concept and instead focused on mathematical or geometrical laws or uses of resistance go motion.

The mechanics of Archimedes and Timaeus had passed down through the Arabic scholars to the west. By that time the geometry of the mechanical systems had been extracted and in this guise geometrically examined and improved.  The Geometers of the west tended to despise mechanics, Leonard's Da Vinci bing a noteable exception. Consequently many confused geometry with Euclidesn Pythagorean teachings, which were also extracted from the mechanics of the ancient civilisations.

It should therefore be no surprise that geometrical forms measure mechanical and physical motions, energies and forces, What was difficult and not easily studied was the dynamics of fluids.

It really did have to wait until Archimedes to make that Eureka discovery, that volume and density of liquids were related to flotation in a measurable way.

Newton's notion of density girs straight back to Archimedes formulation.

In his work on fluids Newton as was common characterised it as a resistive medium, rather than an elastic one. Elasticity was the boundary between a fluid and a corpuscular body! Thus fluids acted on corpuscles by laws of resistance. In that regard lubricity became a measure of hw slippery a fluid was.

Because of the clear boundary of geometrical forms, Newton's Anlysis was always based on uniform and smooth motions and deformations. He had no way of dealing with chaotic or rough systems, even with his Fluxions. Thus his analysis of fluids was doomed to result in less accurate results than his point mass analysis.

The definition of a fluid is crucial. Considering it to be a resistive medium eliminates many of the contradictory behaviours that then arise!

For me

Space is a fluid that consists in contiguous regions of varying density, which are fractally distributed dynamically.

The contiguity of spatial densities is crucial in this dynamic setting. There is for me no container space, just varying density space. Denser space pushes through less dense space, but also denser space can fragment by rotational action in less dense space., and this is directly related to the barycentres of a rotating element and the viscosity within that dense region.

The contiguity of space means that this rotation and change is always transmitted my strain propagations throughout space, but at the same time moving Fractl " masses" also displace the status at the time.

There are 2 gyres of note , the third being a fixed radial gyre. The first gyre expands, while the second contracts. The notion of clock or anticlockwise is relativistic, although if a gyre actually stops spinning and speeds up in the opposite direction that I would call non relatistic variation.

The behaviour of these gyres  in combinations would constitute a proper area of research in fluid dynamics.

The following principle suggests itself:
In any region of uniform density a uniform viscosity exists which transmits strain through that density in a time and material dependent manner. Furthermore, at the boundary of such a region of uniform density reflection and refraction and diffraction of strain occurs according to Snells Laws.

The deformation in the the region is a density deformation and is spherical in behaviour.

In addition a general principal of Newyonial fluid motive is that it is equivalent to energy and transforms into different firms of energy the chief among which is usually called Force or Pressure. This motive promotes rotational pressure (force) as its first transformation .

The barycentres of these rotational forms play a significant role in the geometry of the dynamic equilibria that result in terms of elasticity and viscosity.

(http://www.wackerart.de/Foto/munic/gradient-2.JPG)

I think this is incorrect. You only need 3 axis of rotation in a 3D space to archieve all possible rotations of a solid, in the same manner as translation based on 3 axis is enough for translation in every possible direction.

Sources
https://www.youtube.com/watch?v=vOFM8eG8kVc
https://en.wikipedia.org/wiki/Degrees_of_freedom_%28mechanics%29
https://en.wikipedia.org/wiki/Euler_angles

Really, rotations are not along axes but rather across planes. I mean obviously you can represent a rotation of a plane with a rotation around the normal of the plane, but if you go beyond 3D, thinking of rotations with planes is more intuitive than thinking of it along axes.

As such, in 3D, you have 3 planes that are normal to each other and can be freely rotated. - however, you can choose the precise orientation of your planes arbitrarily, as long as they all are normal to each other.

So it's a matter of view. Or rather a matter of what you are really saying.
If you are talking about how you represent rotations, you only ever need 3 planes.
If you are talking about what you can choose, you have infinitely many options with only one requirement.

Aha, but if we were talking about the options that you can chose, the solid has infinite freedom both in translation and rotation. My point was that there is no fundamental difference between translation and rotation if we are talking about the degrees of freedom.

Fabulous image of a weir, Hermann. Illustrates how contiguous regions of different densities just morph visually into one another!

Rotation and Translation are, in fact, so similar that you can introduce a point at infinity in order to make translations be treated as rotations around that point, in which case you end up with six degrees of rotation, three of them in the usual sense, and three around the point at infinity, all six such that that point does not change.
That's what happens in projective and also conformal models of space.

Thanks youhn, and Kram 1032.

I happen to agree with both of you!

Yes degrees of freedom is too loose a term to describe translations alone , and yes axes of rotation is an idea from my kindergarten days!  However , using only 3 axes of rotation as a basis set can describe any rotation providing you remember to use half angles!

I find the bivector notion of rotation interesting. It is natural to observe complex rotation without being ble to identify an axis or an equivalent axis of rotation. One of the tools we use to specify rotation requires an axes but another tool does not. Often I will revert back unwittingly to sn old way of thinking without realising.

The rigidity of a solid is locally well captured by the 6 degrees of freedom in a 3 orthogonal axis system. This can be reduced to a quadrant of the 6 degrees system, if preferred.

Rotation using this axial system is more complex to describe, whereas using the bivectors of a parallelapioed that contains the object nicely describes local rotations.

The more complex rotations require nested sequences. It is only since my research into The Barycentric calculus that I have realised a more flexible reference frame has always existed. This bears directly onto the Grassmann method of analysis and through that onto the Clifford and Geometric Algebras.

Yes I have still got lts to clear out and relearn, but it is great fun doing it.

I tend to use your last point a lot Kram1032. But I have had to take instruction from Nran on projective geometry to make sure I am not making any naive assumptions.

For many motions I use a local reference frame, but for a general discussion I use a projective reference frame with homogenous coordinates.

I can recommend Normans universal hyperbolic Geometry course for clarity on these points at infinity.

You do not need any kind of nested sequence for "more complex rotations" - all rotations lie in the same space.
You do need up to three rotations to describe arbitrary rotations in coordinate system.
However, this is only a constraint if you actually fix coordinates.

If you take a coordinate-free approach to rotation (which Geometric Algebra and especially Conformal Geometric Algebra lets you do without a problem), all rotations you can possibly imagine require only a single step.
Quaternion rotation also does this, and in fact, what you do in Geometric Algebra to rotate planes is the exact same thing as what you do with quaternions. It's just that the framework of Geometric Algebra is so much clearer and, in a sense, more polished than what quaternions do, that it seems like an a lot easier operation.

Quaternions seem to come out of the blue and they are messy and confusing, them being 4D and all that.
Inside Geometric Algebra, this makes a lot more sense. You have (for a 3D space) one scalar dimension, three locational dimensions (corresponding to translation and vectors), three rotational dimensions (corresponding to three planes, the bi-vectors and rotations) and one "volume dimension" which is, in a sense, degenerate and acts almost like a scalar. (It is, in fact, a pseudo-scalar.) That last dimension, besides closing the algebra so it all becomes workable, really only is there to change operations into their dual forms. (For instance, if you multiply a bivector that describes a rotation by this trivector/pseudo-scalar, you'll end up with an axial vector that describes the exact same rotation, and vice versa).

Quaternions turn out to be precisely the scalar plus the three rotational dimensions. That's why quaternions are four-dimensional. And that's also why all the action of quaternions typically lies in their three imaginary parts. - Those are precisely the three bivectors.
The reason why they are confusing really only is because they are, in a sense, incomplete. They only focus on rotation and completely leave out translation. Upgrading them to dual quaternions - which is possible and gives that missing power to the algebra - still is confusing, but if you approach this problem from a perspective of Geometric Algebra, it suddenly becomes a lot more obvious.

Well, excuse me for still being in kindergarten :-)

Since I'm an engineer, my perspective is mostly the physical 3D world (opposed to the much wider and more general mathematical world). Fractals are to me a subject that attracts because of it's (visual) beauty. Add a little curiosity and I sometimes dive into the math, but mostly not very deep. So I've heard about rotation around planes from the documentation of gnofract4D.

But in practice ... what would higher dimensional rotation mean for (an engineer like) me?

Well, the way to turn rotations and translations into one and the same concept is to embed your 3D space into a higher dimensional space that adds one extra dimension.

The reasons for this are that certain very practical, not at all unusual transforms become much easier that way. They obtain a much more natural description that is easier to work with.
In particular, you'll get screw-transforms.

Another reason is to stop the center point from being a "special point". In nature, in the real world, there is no inherent difference between any two given points in space. All difference simply comes from what particles are in its direct neighborhood.
In a naive mathematical model of space, you get some kind of singularity in the origin of your coordinate system. This singularity is entirely artificial and you can get rid of it in a particular point in space, simply by shifting your coordinate system around. However, naively you can't get rid of it altogether.
To do that, you need to add another dimension. If you do that, you get homogeneous coordinates.

If you do both those things, you end up with a conformal model of space which is as well-behaved as it could be.
It has lots of advantages:

• The Origin is no longer a special point of your space
• You can do all screw transforms in a very direct, simple way, since translation and rotation ends up being the same thing
• Additionally, points (not vectors), spheres and planes become one and the same object for, respectively, 0, finite and infinite radius, which is very natural and easy to deal with.
• Conversely, circles and lines also become one and the same, and a circle of radius 0 becomes something like an infinitesimal vector - a point with direction, which is a great notion for a normal vector.
• Things that actually should be different quantities, like points and vectors, are usually mingled into a single concept which can cause a lot of confusion and head-scratching. In a conformal model of space you get notions that truly separate the two.
• And on top of all that, in a conformal model you actually get rid of the need of a coordinate system. - All your calculations can be done algebraically without ever referring to coordinates, which is a really natural notion.
  Coordinates are just a mess to deal with, they add a lot of complexity and they are entirely a mathematical construct with no physical relevance what so ever.
  In fact, many misconceptions behind the theories of relativity, and even behind some Newtonian physics stem from failing to see the difference between coordinates and the actual space. So if you can do all your transformations on a space, without ever referring to coordinates, you avoid all these problems. The structure of problems just becomes a lot easier that way.

Other than that, a 4D Minovski space obviously is good for special and general relativistic concepts and you can also go for a conformal description of that space which gives you a 6D working space of which only 4 Dimensions have direct physical relevance.

It might be surprising, but adding in those two extra dimensions actually makes things easier, combining notions that shouldn't be separate at all, separating notions that shouldn't be confused and making transformations easier that are quite relevant in the physical world.

If you, however, are talking about higher dimensional spaces in which each of the dimensions is supposed to have physical relevance, then there might not be much to that in practice.

Unless, that is, you count statistical data as physically relevant: Such data can have a virtually infinite number of dimensions. And often it's necessary to transform data sets to be "shaped better", such that algorithms which, for instance, search for optima or structures in your data behave better.

Beyond that, even for naive 3D Space, without a homogeneous, projective or conformal extension, thinking of rotations to act in planes has certain advantages.

Note, however, that the conformal model is actually closer to the physical reality than the naive form. - At the very least you need a homogeneous model.
The projective part that deals with a point at infinity you might argue against by saying that in reality there is no such thing as a point that is infinitely distant. (I personally wouldn't argue that) However, even then, that addition is one of convenience. It just makes things so much simpler to treat translation and rotation; circles and lines; planes and spheres as one.
And by combining the two you end up with a system you can manipulate without ever referring to coordinates, which definitely is very natural and much closer to reality.

Reading this and your reply to Youhn certainly whets the appetite. But I can't say I understand how you would model an n body rotational problem off what you have just written. By this I mean a solar system rotational description with planets having attendant moons.

For this thread I am particularly interested in vortex shedding and compaction.

For this isssue their exists an excellent book: "Quaternions and Rotaion Sequences" from Jack B. Kuipers very easy to read and understand. Even if you have only highschool maths.
That is the way I all maths books should be written. An example of very good didactic.

By the way with quaternions it is possible to make fantastic fractals:

(http://nocache-nocookies.digitalgott.com/gallery/2/1307_04_04_10_3_27_56.jpeg)

I came to quaternions through the art work not through the mathematics as first step.
If one looks at the picture it looks like mixing dough. Has something to do with rotation and stretchy dough in a mixer.

Hermann

@hermann
I also first heard of quaternions through their use in generating fractals. I'm pretty sure that's the case for most of us here.

And I didn't mean that it's impossible to present quaternions in an easy way, but rather that they are inherently more out-of-the-blue than how they result as a subset of geometric algebra.

@jehovajah
Vortices aren't just rotations. They are indeed much more complicated. Rotation alone doesn't generate a vortex. You'll need some drag forces and friction for that. Friction complicates all matters.

Conformal geometry in Wikipedia.
http://en.wikipedia.org/wiki/Conformal_geometry (http://en.wikipedia.org/wiki/Conformal_geometry)

This isn't the conformal model of euclidean space, though it is related.
That's more like it:
http://en.wikipedia.org/wiki/Conformal_geometric_algebra
though it's not exactly one of the best wiki-articles out there.

Wiki sucks for understanding, though it is great for reminders, facts and the links in the categorized web-of-knowledge.

For understanding I need to play around with the concepts and read it in a more story-fashioned way. The cold and bare wiki language doesn't really enlighten me.

Thanks for the candles everyone ... i'm out to find more fire the next days/weeks.

Deep within our thought world rotations are embedded as entities that exist spatially. It has taken me a while to apprehend that ineal motion is in fact singularly rare! Vortices do not need to be generated in my opinion, because they exist at all scales. What I am slowly reaching for in this thread is a method of synthesis and analysis that is founded on this Förderung, that is the idea I am promoting.

Claes Johnson shows numerically that vortices form part of the solution for explaining flight. The drag on the wing , the frictional forces of the boundary layer attached to the wing create an instability that results in vortices streaming from the trailing wing edge.

In fluid dynamics these are traditionally called turbulence, but Claes has shown they are initially well formed vortex plumes.

Drag does not create vortices per se, because the idea by Prandtll that boundary layers separate at the crest of the curve for an aerofoil is not physical. Instead the boundary layer separates as it must at the trailing edge creating high potential instabilities that spin out as vortices, and consequently have a low or mean zero potential!

Also, the aerofoil shape creates a venturri effect in which the lower pressure sucks the wing upwards

At supersonic speed however this description alters. The high compression of the boundary layer creates a " solid " like fluid mass which can no longer follow the conyours of the wing . The separation of this boundary layer produces profound instability as a shock wave of sound, pressure and heat and electromagnetic properties promulgates through the surrounding medium. The vortices are do powerful that they form a cone, and the aerofoil is heated and stressed by their powerful dynamic . Drag and lift are compromised as the plane is held aloft by powerful impulse forces generated by vorticular variation. Sometimes thes can reverse the control surfaces generating gravity instead of levity!

These are the kinds of fractal dynamics I want to understand better.

http://claesjohnson.blogspot.co.uk/2012/11/lifting-line-theory-illposed.html

http://secretofflight.wordpress.com/

http://www.youtube.com/watch?v=5WKU7gG_ApU

These kinds of vortex plumes are common in fluid dynamics, but we tend to ignore them in our thinking as Claes points out.

They have a Barycentric connection which Helmholtz seems to have left out of his theorems, as did Kelvin.

Defining Torque as vorticity obscures rotation, because rotation has many axes or barycentres and the vortex Barycentre is the systems rotational one in hich centrifugal and centripetal acceleration balance.

If centrifugal does not balance centripetal acceleration then body deformational forces arise and the shape is deformed, resulting in a new Barycentre and new tangential contact points.

Vortex rotation starts when centripetal and centrifugal body forces balance around the Barycentre which becomes a vortex Barycentre.

Current definitions of torque should not be.confused with rotational vorticity. However torque explains shear deformations of a body and when a body will break into several masses that spiral around the vortex Barycentre.

The notion of rotation as vorticity and the notion of Barycentre allows fluid dynamics to describe fluid element ensembles using a youngs modulus approach or a spring mesh with a damper and a youngs modulus distribution ,

I can now model a fluid dynamic not by Kelvins kinetic theory but by a Barycentric mesh kinetic energy model! Instead of Root mean collision paths we can use Barycentric and tangential contact point analysis to. Odell the fluid element deformation and include spring and damping forces with modulii reflecting the material nature of the fluid.

These spring and damping forces model the electro Thermo and magneto interactions that are evident in matter.

While we cannot leave out collisions ltogether, the nature of most of these are elastic near misses modelled by electro Thermo magneto  considerations, or simply gyroscopic rotations of point mass or Barycentric centres fractally distributed in the volume. Some direct collisions of barycentres must be figured in.

If youngs modulus for a material is exceeded then cracks and complete bodily separation has also to be figured in.

For a gas it is usually assumed that this is the case, but pressure behaviour in the kinetic theory has little to say on this matter.

Just a small addition to this topic.

Benefits of turbulence small-scale-vortices on the slightly bigger scale properties: drag-reduction and self-cleaning

https://www.youtube.com/watch?v=9nxRO1EhmmM


Additional sources:
http://rsta.royalsocietypublishing.org/content/368/1929/4775.full.pdf
https://www.beilstein-journals.org/bjnano/content/pdf/2190-4286-2-9.pdf

Turbulence is not equal to chaos or badly-shaped vortices. It has to do with things happening at smaller scales than the scale which is used to evaluate the fluid system. In computational fluid dynamics it takes to much computing power to model all those small vortices, which of course eat up a little of the total energy and convert it to heat. Instead of modelling these all smaller-scaled and transient vortices, the loss of energy is accounted for in a turbulence-model. Most of these are both space-averaged and time-averaged. This makes it possible to carry out steady-state fluid dynamic simulations. To obtain a more physical faithfull results, a very fine mesh, no turbulence models at all (just the navier-stokes equations) and a transient approach is needed.

Source:
http://www.cfd-online.com/Wiki/Turbulence_modeling
http://www.innovative-cfd.com/turbulence-model.html

Thanks Youhn.
The initial start of the plume I think is a vortex, but very quickly it becomes a complex structure I thought might be turbulent because of all the vortices implied . Frankly at this stage I do not know how to describe it, but claes computation gives well defined vortex plumes , which may be physically observable. The examples I had in mind always described it as turbulent, but then they admit it was too complex to describe it by any other name. The fractal structure was observed however.

The importance of the Barycentre AND the tangential contact points in a description of space behaviour under strain reminds me of the strain Ellipsoid!

A search is intriguing as it hints at this being used in slowly deforming rock , but as yet I have not found a good enough reference to mine!

However this link
http://www.researchgate.net/publication/22681376_Perceiving_the_centroid_of_configurations_on_a_rolling_wheel/file/e0b4951a510855345d.pdf
Shows how our perception is naturally attuned to these kind of relationships..

For me the use of the Barycentric calculus of Grassmann epithet his tangential contact points could be linked to the ideas of the strain Ellipsoid. This then gives fluids a representation by this structure which simplifies regional computation of fluid behaviours, potentially.

http://journals.ametsoc.org/doi/full/10.1175/1520-0469(1998)055%3C3358:SALRNF%3E2.0.CO;2

A discussion in the   Second topic heading  refers.

Hallo Youhn,

When we speak of rotation we always have a rotation axis. You can think of this rotation axis as a vector. (Right hand rule)
In three dimensional space this vector has three components in higher dimensions it has more components.
One component for each dimension

Here is a short over view on the rotation of an vector (not the vector of the rotation axis)

A three dimensional vector can be written in the following form:



A four dimensional vector can be written in the following form:



No problem to write down even higher dimensions:



In mathematics you can perform rotations with the help of special matrixes called rotation matrixes.
Let be the rotation matrix, be the original vector and the rotated vector:



In this form a rotation can be seen as a simple matrix multiplication. (No archmetrics with sin, tan, cos etc.)
Even in higher dimensions!!!
That's the trick.

In three dimensions you have a 3x3 matrix for rotations. In four dimensions you have a 4x4 matrix, in higher dimensions a nxn matrix.
This kind of matrix operations (rotations) are called ortogonal transformations in mathematics. If you allow complex numbers as components of the vector and the matrix this is called a unitary transformation.

It is also possible to derive the rotation axsis from a rotation matrix and vise versa. The vector on the rotation achses is allso called the "Eigenvector" of this rotation. A more abstract formulation of Eigenvector is used in Quantenmechanics.

This lecture on stress strain fields relates to the topic.
http://www.youtube.com/watch?v=r8KzP7G7Uks

In this case density of materials allows stresses to act more slowly in the materials showing the strain fields. For less dense materials stresses act more quickly and so the strain fields propagate more quickly to catastrophic collapse or permanent , liquid deformation.  In fluids the strain field propagates almost instantaneously, catastrophically collapses almost as quickly and fragments are dispersed by the environmental stressors..

Within the body of any fluid the environmental stressors are likely to be hydrostatically arranged ( in a gravitational field) or symmetrically arranged by electro Thermo magneto complex energy rotations. Thus stress introduced internally should create strain propagation that leads to catastrophic collapse , but those products are held together and are  dispersed by the environmental stressors or flow currents. This would be the basis of vortex shedding?

What do you mean when you write about "stresses act more slowly" ... ?

What are strain fields? When you say liquid deformation, do you mean plastic deformation (as in opposite to elastic deformation) ... ? More keywords could be Young's modulus and creep (which adds some viscosity to solids like PTFE materials show very much, but also all kinds of steel at elevated temperatures).

This sounds like the reverse of the statement above, which was about dense (solid?) bodies. If I understand correctly, then I agree. Forces acting upon fluids generate almost instantenious plastic (non-elastic) deformation, while dissipating all internal stresses rather quick. Forces acting upon solids lead to very slow (almost non-existing) plastic deformation (called creep), while elastic deformation happens almost instantenious together with the internal stresses.

What do you mean by "hydrostatically arranged" ? Same question for the symmetrically arrangement.

Yes, internal stresses lead to strain (propagation). But what do you mean by the catastrophic collapse? Collapse of the fluid? The shape of the fluid? The relation between stress and strain ... ?  :hmh:

I'll scan the lecture later this evening. Might answer some ofthe questions.

I think you mentioned density as a key word, but probably in another topic. But since the whole universe can be seen as a fluid on the bigger scales of time;

Titel: If the Universe is expanding, why are galaxies still merging?
source: https://medium.com/starts-with-a-bang/26c0f36ddc89

The Greek already knew; Panta Rhei.

Youhn I apologise for my lazy editing and correction policy! Typos abound and I only correct what I spot or if I am reviewing a post to answer a question.

I am not a materials engineer and barely study above A level physics if I can help it! Youngs modulus I understand , plastic deformation in rigid materials I understand intuitively from pictures and graphs.

In a solid the elastic limit is reached and breached when the rigid material does not return to its dimensions. The plastic deformation of a solid or creep is like bonds slowly snapping gradually exposing the undersurface material. Slow snapping is where a crack appears in the surface and propagates inwards to the centre. Meanwhile the exposed faces deform forming a kind of bottle neck. I think I used liquid deformation to mean plastic deformation, and I apologise for any confusion.

Again catastrophic collapse is my term to describe when the material separates into distinct fragments with their own surface tensions. When this occurs they are surrounded by the environmental material
And interact with it as a fragment.

Strain fields are my terminology for fluctuating regions of strain. The fluctuation, from one point of view may look like progressive density variations. From another it may be a standing density variation, a region suddenly increases or decreases in density.

Density is a fundamental Archimedian concept measured by displacement of a fluid which is weighed and the proportion used as a definition of density. Because of this I used hydrostatic to describe one kind of density variation. This is akin to gravitational compaction of matter.

The propagation of strain fields creates tensile stress variations. Sometimes these stress variations are slow especially larger amplitude strain , giving rise to large stresses in that region . But some strain propagation is associated with fast minuscule amplitude propagation. The stresses that these strain fields initiate are minuscule. These fast propagations are typically in denser materials.
http://youtube.com/watch?v=-wgci_qDAy8

Certain Hugh speed events reveal the fundamental fluidity of matter.

http://youtube.com/watch?v=QfDoQwIAaXg

What is still harder to see is the fluidity of light, but at a trillion frames per second even that is revealed.

The notion of continuity has to be changed. Continuity has to be a signal that we have reached the limit of our ability to distinguish!

For example, much of our mathematical modelling provides us only with frames showing actual positions as calculated or differences as calculated. This means, for a rotating object or physical region it would appear that it rotated along a continuous arc without physically passing through the space in between calculated frames.

It is only when we run the frames together that we notice in the "film" effects that we unconsciously " fill" in the causal pathway for. This causal pathway is what we experience as physicality.

Our formalisms or mathematical models can not tell us the full physical picture. We have to intuit that ourselves. In so doing the notion of cause arises in the sense of continuity. However, we can also consider cause in the sense of contiguity! Thus we experience the notion of first or most primitive cause.

First cause or most primitive cause occurs as an experience where the observed or calculated behaviour has no continuous physical interpretation. Thus my knowledge of an event is initial and I assume that all motion is prior to this initial knowledge state. Then as my knowledge tracks the changes in each frame I intuit a physical continuous cause.for motion( say, Newtonian motive). In this intuitive ascription, concepts like density and reaction and interaction are appended.

Now when a motion becomes apparent, at some stage in the middle of the " film" sequence, it will have no prior frames to intuit its cause in a continuous model. This is when I intuitively shift to a contiguous model of cause, and intuit a first cause at the frame or between the frames where the change is first noticed..

Contiguous causality really allows for the mysterious and mythical explanation of causality to have its justifiable place.  Between empirical observations and intuited continuous causality we must allow for contiguous causality, and that means unobserved causes which reveal themselves only by a change in observed motions.

No matter how rigorous we become continuity and contiguity support both myth and fact making in science, so called.

Contiguous causality inspires us to search more carefully and to have a broader view of causality. So for example a change in motive behaviour may accompany a change in local density, and this may be sufficient to simply explain observations. But as Maxwell Observed, we cannot rule out the little daemon in the fabric of space!

It is therefore a personal choice, outside of empirical observation and deduction and induction whether whether you rule out or rule in some none observable reality , apprehended o ly by its effects on what is observed. But certainly, to develop a mythical world reality on the basis of empirical data is not supportable by empirical data. Empirical data used in this way serves only as a metaphorical or analogous description. Thus if one accepts this implied causality one is free to creatively mythologise it, but with no moral authority or empirical one.

That being said, one being free to believe as one wishes does impact on what one ultimately observes, and recognising that makes any notion of ultimate truth a miasma of coercive opinions without ultimate justification.

To claim the scientific method as the source of ultimate truth is as misleading for scientists as it is for those wo support their religion by it. Absolute anything is absolutely misleading! Lol!

It is possible to live approximately, always learning, always skeptical, but always striving to improve to the benefit of the majority who want it, or a minority who do not reject the change.

So back to the fluid nature of physicality: contiguous causality is associated with rotational motion and rotational motive/ energy., almost by definition

I had a singular question in my mind!
How could normal forces contribute to tangential force in a viscous medium?
I was watching the following video in which the viscous properties of water are apparent.

http://youtube.com/watch?v=i8nZ7B1h5Mw

It is clear that a distribution of velocities and thus forces shape the evolving structure, but always a core flows into a rotating ball that breaks away from the supporting and transmitting " rest" of a plume.
So how does laminar flow become rotational!

The answer is all flow is rotational. Thus when constrained to be laminar by some energy constraint it represents a suppression of a curvature.

Newton recognised this dynamic behaviour by establishing it within his inertial reference frame.

Most confuse Cartesian coordinates with the inertial reference frame. In fact the advanced reference frames respect curvilinear coordinate systems in dynamic rotational motion. This sounds and is complex, but Newton simplified it to a combination of Cartesian and polar coordinates combined! In particular his centripetal and centrifugal axes are radial And are combined with a tangential axes and a circular arc axes curving from the point of tangency.

Thus any tangential force also has a curving arc force associated with the point of tangency.

When a tangential force contacts a resistive surface, 2 forces are generated that are equally opposed: an anti tangent and the anti tangent arc! This arc is responsible for the notion that normal forces can contribute to tangential forces! By altering the centripetal centrifugal arc or polar reference frame a tangential force is generated instantaneously. In fact it is not a " tangential" force but a tendency for the centre to move!

http://youtube.com/watch?v=51-6QCJTAjU

This tendency is also measured by viscosity and the Reynolds number. It is a measure of how likely a centre of rotation is to move in a direction or a preferred direction.

Viscosity is a measure of how proportionally connected normal and tangential forces on a surface are. The Reynolds number therefor measures this tangential point arc force and thus is a measure of force curvature at the point of tangency . It is clearly related to the substance surface boundary as well as the fluid substance the surface is moving through.

Instead of viewing the normal and tangent axes as the reference frame for curvature it is better to view them as resolved axes of curvature, that is generated by the curvature! In this regard an orthogonal reference frame is generated by a curved surface using tangential line segments. Obtaining any 2 non parallel tangents to a point on a curved surface allows a normal to be generated.

However the movement of the surface is not describable by a single normal or tangent. It is better described by spherically rotating elements.

ThE Reynolds number gives a kind of average curvature for these spherical elements, but what an instantaneous surface looks like  is an aggregate of these curved elements, their curves of intersection and their interaction with light spherical rotations.
Normals and tangents enable a surface plot of triangles and parallelograms to give a rough idea of the spherical elements aggregate surface structure.

This is a related thread.

http://www.fractalforums.com/new-theories-and-research/revisiting-the-riemann-sphere-(again)/

From Roger Bagula's fractal news email.

http://www.sciencedaily.com/releases/2014/03/140321095023.htm?utm_source=feedburner&utm_medium=feed&utm_campaign=Feed%3A+sciencedaily+%28Latest+Science+News+--+ScienceDaily%29

And Escher patterning

http://phys.org/news/2011-06-exploring-tessellations-escher.html

Rather than make analogies to liquids, I am making the not unreasonable assumption that liquids and gases( classical fluids) are analogous to the electro Thermo magneto complexes which form fractally distributed regions described by us subjectively( or by me at any rate) as SpaceMatter or plasma, and universally as simply space.

Thus the viscous properties of space exhibit themselves to our perception as overall inertial frame effects. The distinction between inertia nd viscosity disappears at the universal level. Inertia is a local phenomenon of viscosity of space.

Viscosity is of empirical experience, non uniform. This aria tion in viscosity also is apprehended as density. Again density is a local variation in viscosity.

The aspects of space described by the Newtonian Fluid motive, roughly equivalent to modern notions of energy, are the behaviour of this viscous medium in mediating acceleration within itself and it's fractal regional parts. Rotation is fundamentally necessary to provide sufficient ground for regionality and thus variation.

I see that the viscosity of a region , under rotation, creates a contiguous divide, in certain cases the divide may store a returning motive called relaxation motive . This relaxation motive is also called surface tension.
Density variation due to viscosity reveals itself in boundary construction. Boundaries do not form static ally, they represent a dynamic interface where rotational motives creat active and reactive forces through resolvable accelerations. Newtons inertial reference frame concepts enable us to resolve these accelerations into normal and tangential accelerations, however trochoidal the accelerations actually are.

Why resolve into these 2 orientations?

Well in our simplicity we only want to know if 2 identified regions are going to remain together or come apart, And if they are rotating! A net zero acceleration means the regions will maintain relative positions. A net zero rotational acceleration means regions will retain relative orientation, any thing else is likely to be too complicated to explain.

Gravitational behaviours is one of those non zero situations. While we focus on gravity we completely ignore levity which also occurs and is not uncommon in space. However our notions of density obscure these distinct force situations, and are compounded by our willingness to generalise local knowledge to a universal scale!

The viscosity of space and of plasmas seen astronomically reveals a fractal distribution at all scales of this variable property. The attendant energetic properties of a viscous medium can be gathered under a tripartite description electro Thermo magneto complex. Of these Thermo magneto is the scientific ground based on the work of Gilbert who examined more closely and more empirically the notions of the ancient observers and philosophers.

We may safely remark that the amazing viscous behaviour around magnets has been in want of anaccommodating notion which links it naturally to chemical and mechanical behaviours through the empirically evident vortex phenomena. That electric magnetism was allowed to obscure the philosophy is just one of those all too common mis steps men have made in the furtherance of their egos, statuses and social reputations, and latterly financial gain.

While the study of this viscosity in space has progressed through the study of its effects in fluids, it is not to be overthrown as an abstracted notion, for all notions are abstracted. Rather it is a pragmatic choice, for the doorway to ideal or formal thinking is in the control of the pragmatists.

It is the pragmatist who surveying the empirical data, elects from the many choices the most practical compromise. This compromise is therefore ideal! It may only ever be approximate to any empirical experience, but it's use is in freeing the artisan to move on to active construction.

Thus as Newton observed, mechanics gives rise to geometry which in turn perfects Mrchanics in an endlessly iterative process.

In this regard fractal geometry serves a fundamental purpose. It teaches those who will learn that complexity can be arrived at from simplicity, and that having pragmatically chosen an ideal form, it is not to remain static, but to be dynamically employed to improve the accuracy of the developing model.

Our forebears spent great amounts of thought and practice in ascertaining these ideal forms. We now must apply them iteratively to create better fractal models of our empirical experience.

What should hearten us to do so is that as we have developed better magnifying tools we have found these ideal forms existing at the very smallest scales. The Socratic platonic game of which is real, the form or the constructed form has its answer: both are real, and we are connected to our universe at levels we do not realise!

The time varying transmission of force through a viscous medium surrounding a rotating body will be proportional to the inverse square of the radial distance. In fact it will be better described as the logarithm of the radial distance  and the arc motion of the interacting bodies, as per Cotes formula.

ix = ln( cosx + isinx)

Where cosx is some function of the radial distance related to the radian measure x or more generally r(x)( cosx+isinx)

This is a bit restrictive but the point is that the inverse square law is an approximation of a more general formulation.

The time rate of transmission of force was ignored previously because the assumption was that space was empty. Thus force had to be transmitted mysteriously and instantly. However we know s
Space is in fact a viscous plasma with variations in viscosity, so the time rate of transmission has to now be included. It is a consequence of spherical geometry that the tangential surface varies as the square of the radius. However this implies an inversion or the force transmission because the spreading strain decreases not increases with surface area. The inverse square law is a rough guide to tangential forces, and by the Reynolds number discussion to centripetal and centrifugal radial forces.

We know from empirical experiments with rotation solids stressed from the central rotation that the transmission forms a Rayleigh like wave and takes time to spiral outwards and then reflect, by relaxation back into the centre. We also know that eventually a region will attain a block rotation if no other external forces are contingent.
The gravitational effect is a state of disequilibrium  while the viscous fluid assymptopically arrives at steady state,

For most viscous mediums there is a rate of force transmission that overcomes the relaxation limit. Particularly in water nd gases this is a very low rate. The result is the separation of the rotating regions into " bubbles". These bubbles develop yjeir own centres of rotation as a result of viscous forces achieving equilibrium. However, this very much depends on the viscosity of the medium and the surrounding media which becomes interstitial on the break up of a viscous region.


Vortex shedding is a commonly seen evnt in flowing water, indicating that rotating regions are dynamic in many ways, and that may not allow bubble formation.

However if bubble formation occurs these interlinked regions will usually have anti spin so that the contact tangent is uniformly directional. This leads to normal forces that do not repel the bubbles. . These structures may form chains similar to gear chains, eventually collapsing as the rotational energy dissipates. Bubbles in the sink ay look static, but the surfaces are highly active, and bubbles coalesce into larger bubbles under these conditions..

The gear chain effect is often associated with antennae and electric current, but in astronomical space or in the atmosphere they may be seen as lightning. As a rule they are short lived at our scale, requiring a conductor to catalyse them, but at huge scales the dynamic rotational energy dominates and these chains can last for aeons!.

The viscosity of space is such a counter intuitive notion that we invent different names for the phenomena. The electro static nomenclature has perhaps done the most to obscure the bubbles of counter rotating space. But recent research findings have supported Ivor Catts long held research findings that there is no static electric charge in a transmission line or it's equivalent a capacitor.  Plasma physics recognises this " double layer" effect in plasma and every planet in the heliosphere has such a surrounding bubble similar to the Van Allen radiation belts.

These are highly energetic active regions as you would expe t from counter rotating space.

I generated this sculpture years ago, but then I only suspected it had something to do with gravity. Now I have shifted paradigms to a fluid mechanical one I am more certain it is a model of electro Thermo magneto complexes as viscous space.

(http://nocache-nocookies.digitalgott.com/gallery/5/410_26_01_11_1_32_17_1.png)

http://claesjohnsonmathscience.wordpress.com/2012/02/22/why-it-is-possible-to-fly-yvfu3xg7d7wt-18/
This important breakthrough has many implications , including in the diffraction propagation of light .
Google "jehovajah vortices " for more details.

http://www.gaalop.de/wp-content/uploads/134-1061-Zamora.pdf
This geometry may be useful to describe general pressure systems in fluids. However, the choice of dot products may need to be made flexible rather than rigid to accommodate different views of a system and discrete behaviours.

http://youtube.com/watch?v=RGevKSlT0yY

fast Fourier Transform by Zedzero on Vimeo. (http://av.vimeo.com/85176/473/12393769.mp4?download=1&token2=1400256134_75c96276bae893d038512c645a34e074&filename=Fast+Fourier+Transform-Mobile.mp4)

http://vimeo.com/zedzero

Understanding that the Fourier transform is based on the general concept of " periodicity" . Now this is a "mash up " concept. The underlying Spaciometry is the circle in the plane. Fourier was interested in describing how " heat" flowed in a conductor and he chose a metal ring!  The temperature at a fixed point was increased and the pattern of temperatures in the ring was modelled at various time intervals.

The parameters are the quantities usually independent , that are measured by some Metron.  Position was therefore measured by polar coordinates r,ø which can be transformed into orthogonal coordinates rcodø,rsinø which might be denoted as trig line segments x,y. Temperature was measured by mercury expansion. Thus this " temperature" is in fact a heat pressure, comparable to atmospheric pressure also measured in height of mercury!

So for each position a mercury height could be plotted. The fourth parameter is again an independent measure, usually the duration of the experiment measured by a metronome.

By recording these data sets in tabular form Fourier presented the scientific world with a technique of statistically managing sets of data. We have to review the work of Gauss and Boltzmann to find similar statistical management. But it was reputedly an idea of Laplace that Fourier developed extensively. The idea is based on the representation of a polynomial form by simpler polynomials that sum to it.

The purpose of representing a polynomial by a sum of polynomials was and is to be able to handle discrete information by a general polynomial that contains it. What this means is best illustrated by the function or polynomial y= x².

Typically we calculate a few discrete values, thrn plot them and join the plots with a smooth , but estimated curve. In one sense we have no way of knowing if that curve is right or rather " true" without calculating the intermediary points. This calculation is called interpolation. Thus it became clear that if one only used straight line equations one could approximate the curve by summing line segments from many lines,

The process clearly is based on difference algorithms and consequently in the limit melds with the calculus methods of Descartes , Leibniz and Newton . Thus the curve can be considered as the envelope "sum" of the tangents at each point.

The question that arose is what was the best fit for any number of discrete points ? Could you recover the " true" polynomial by this approach?  In general the newer is no. But under certain constraints you could converge onto the true curve.
Thus it was shown that you could model any complex curved line by a sum of appropriate line segments, and eventually by a minimal choice of polynomials of a lower degree. Within the given range the convergence could be as close as desired, but outside the range the summed polynomials were seen Togo " crazy" . Extrapolation was definitely not possible!

A Fourier transform was based on Fouriers use of the trig functions of Euler. It was known that these functions could disappear if the amplitude was at the right phase in the rotational cycle. This meant and Fourier demonstrated that over the unit circle any complex form could be modelled by a summation of sine and cosines at different frequencies of rotation. The frequency at which the angle was considered to turn determines the phase of the amplitude .,these would thus cancel or add accordingly creating complex patterns of highs lows and blanks.this was called interference either constructive or destructive. Eventually it came to be called interference by superposition.

Now it was clear that in a ring if this dynamic rotation was occurring, then time would be trapped in the ring, and the angular rotation of these sine and coine waves would also be trapped in the ring by polar coordinates. Thus any amplitude would show up as interesting patterns in the ring. 

Nobody knew or could think how heat was transmitted or conducted, but it seemed it flowed from hot to cold. Fourier wondered if it rotated or oscillated through the medium. If so the ring would show this by a travelling amplitude of oscillation which he could measure by temperature.

Actually his ideas for heat were misconceived, but his analytical method revealed how any complex pattern, conformed to a ring could be precisely modelled by sine and cosine summations.

His heat idea was misconceived because he eliminated the rotation concept, replacing it by the sine or cosine concept. Typically with the thought of his day and even today, the imaginary component of the exponential forms were always thrown away.

Heat flows from a higher temperature to a lower temperature Newton observed. Fourier never explained why, but he did show how a temperature form was represented ar different times in the process.

The reason heat flows is explained by Newtons laws of motion. The heat indeed increases the quantity of motioning a region. However, unlike a blow, the heat Los changes the bulk properties of the material. Thus the increase motion is not uni directional, the expansion is not an elastic deformation, except at the edges and the viscosity of the material leads to a longer relaxation time, a Thermo relaxation called thermal contraction. It takes time! Thus a pressure wave does travel in the ring, but with all these constraints varying it moves so slowly governed by other factors like latent heat and specific heat rather thn bulk material properties of compression.

That is not to say these transmissions do not happen. They do, but typically in a heat conduction experiment the parameters measured are not the acoustic ones!

The Fourier transform is sometimes confusingly presented as converting time into phase. This video shows precisely what is happening.a spatial forms modelled by several parameters only one of which is time.once the spatial form os modelled as an interpolation of some fixed positions and amplitudes at those positions. Then a time argument is devised to add to the spatial form so as to move it along.

Of course spatial form cn alo change with time. In this case the additional time argument is combined ith a time dependent form of the spatial disposition. The form however it is disposed in time cannot exceed its boundaries or the fixed point values by which it is defined. If these values do vary with time a massive compute is entailed and that is why the fast Fourier transform algorithm has been so important to the study of Dynmics.

The wave concept is a misnomer, in my opinion. The rotation or swivelling function is what was meant by undulation, and the progression of a deformation in a medium as a consequence of resident forces in the medium , ie inertial forces , is what deformation propagation is.

Newton in proposing light was a ballistic corpuscle was certain that it's collision with the retina produced the sensation of colour .he did an experiment to confirm this! However later he discovered colour could be described by angle of di
Refraction. This did not alter his opinion that sensation required collision with the retina. There were other details he chose yo overlook because he believed his model was the most straightforward explanation. Without a clear understanding of superposition of deformations in space, Huygens could only suggest that wave amplitudes joined to make a wave front, as circles do in a Moire pattern. By the time of Fresnel, Arago and others had a speed for light , Fourier had a method for describing complex forms that demonstrated superposition, and Young had shown interference patterns in ligh. Newtons ballistic concept had to go. But what replaced it was an arcane question bout form!

It is irrelevant whether a wave is shoed like a corpuscle or like a sea wave! What is and always has been the issue is the existence of a viscous medium for the propagation of light, whether as wave or corpuscle. That medium was called the luminiferous aether and it was thought to exist, until some disputed it. Today it is again believed to exist but it is called quantum energy etc.

Light is a strain that travels in this medium by Newyonial principles of motion and transmits energy throughout this viscous space. My best guess is that the form of this strain is a complex trochoidal rotation in the medium, modelled by Fourier transforms in 3d. The motion is generated by the restoration of equilibrium to this luminiferous aether or quantum energy .

http://youtube.com/watch?v=r4Pc-rGBRJA

http://youtube.com/watch?v=w-XMUQBlcEY

I hope this does not crash anyone's old vomputer.

https://www.youtube.com/watch?v=Tfi8BLca07M
The trochoidal nature of rotation is revealed by these computations. These behaviours are complex enough to explain magnetic behaviours which in turn define electric potential as radial due to contra spinning vortices that pass through each other.

The more I study vortices and fluid mechanics the more I understand that rigid dynamics is misleading . Fluids can abn do pass through each other with minimal mixing.

Also streak lines are different to streamlines as streamlines have a structure while streamlines are instantaneous tangent curves that ignore other forces on a material point that precisely give a streak line its observed dynamic structure.

Yesterday morning I awoke from a disturbed sleep. I had been researching the history of magnetism on YouTube. I had come upon the SLAC channel and found a lecture on the history of magnetism, but also a series of lucid expositions of various other related topics with the " common" touch. As I meditated on magnetism and other issues these lectures slipped easily into the background of my thoughts.

So it was that I awoke with a unified thought. That space is a fundamental fluid plasma which is in dynamic fractal regional and trochoidal relative motion.

This fundamental plasma consists in trochoidal motions at all scales and at all intensities of motion. In my perception I visualised a complex fractal pattern of plumes with associated vortex rings. This was my fundamental Newtonian motive structure as a fundamental primitive environmental plenum in which and of which I am and from which by formal analogies and analogous ratios and proportioning I must synthesise a model of behaviours observed in and of space at all scales.

My issue is epistemological. What can I know of this fundamental plenum which by its nature is in constant flux?

The Pythagorean solution is to define invariants..

The first and most primitive invariant is the inherent Logos response. In this response I interact with spaceas a conscious entity separating itself from all else, while defining itself by all else!. This I formalised into 2 formal and contra sets. The set FS and the contra set notFS. . These sets are formal and dynamic and as the set FS is dynamically defined so is the contra set as everything external to the set FS.

I describe this as a formal conjugation of Shunya, Shunya being the Sanskrit for Everything possible and impossible, probable and improbable, statistical and non statistical, distinguishable and indistinguishable, orderable and non orderable, dynamic and non dynamic. Essentially: Everything!

As a dynamic entity in a dynamic field my first formal invariant is to sing and dance.

The song I sing is a sequenced lyrical response to a regularly changing event. The dance I perform is a metronomic mimicry of the changing event. If I share that event with others it becomes the basis of all communication responses within the logos response. It generates and shapes a common language which distinguishes by sound and by movement a regularly changing event and becomes the progenitor among other spatial progenitors of conceptual order and sequence, both in dynamic relativity and invariant relativity.

As a " concrete" conception, through the syllabric development process it gives rise to a symbolic and speech act representation that is both alphabetic and numeric. At some stage in the history of the development of human writing these 2 aspect of the logos response become separated and distinct. At the same time later generations became confused about the origin of this bicameral logos response.

Sticking with the Pythagoreans I see a development, formal in nature, that captured analogous thinking in and by this bicameral brain, and this developing bicameral mind. It was called Logos and Analogos philosophy.

Later still it was developed and combined into Arithmos philosophy.

The rich vein of formally defined invariants in the Logos response constitutes a forml spatial philosophy which I defined as Spaciometry. Strictly speaking Spaciometry does not formally constitute until the Arithmos philosophy reaches a sufficiently complex form of expression but the intuitive logos response distinctions of sequence and order and distinguished form capture powerful invariant concept defined as " ideals" of pragmatic response requirements, language development requirements and social communication requirements.

You will find more detailed discussions on these topics in the Fractl foundations of mathematics thread.

Thus , returning to the fundamental conjugation of Shunya I can construct only formal models of the set FS. In so doing I may perceive the set notFS as a murky non modelable contra or inverse " reality". The more universal I claim my model to be, the more nonsensical any alternative cotra model appears to be.

It is therefore wise and less arrogant to consider any model as only an approximate or partial or limited model of the set FS. In fact, because the setFS is dynmic, any model defined on it must either be dynamically adjusted or regularly reviewed, redacted or discarded in the light of New data of a sort I will refer to as empirical.

My fundamental forml primitive is therefore a limited model of a complex constructed or structured primitive designed to be the ground of more formal constructions of metrical systems that is constructed measures based on the invariants of the philosophy of the Arithmoi.

Newtons philosophy of quantity marked a distinct break with a qualitative description of magnitudes and behaviours in space.

However his philosophy was firmly based on the Pythagorean principles in the philosophy of the Arithmoi. Using even the simplest expression of this as in Euclids Stoikeia, an introductory course to the philosophy of the Pythagoreans, produced a remarkable set of analogous measures. These measures though well understood were not employed in the service of a quantitative physical theory of the cosmos , that is a more primitive astrological theory of matter, before Newton.

Evenso Newton acknowledges that it was not for want of attempting to do so, for he garnered many of his principles of quantity from the ancient masters of philosophy of natural phenomena. It was chiefly his combination of the principles of Mrchanics with the principles of astrological measures and calculations that founded his philosophy of quantity.

The fact is that the purest ideals of Astrological measures are derived from the philosophy of the Arithmoi. Thehilosophy of the Arithmoi essentially means we can create an analogous mosaic form of any form in our cosmos. Thus every form has an analogous twin which we can explore subjectively by counting and comparing and analysing and synthesising internal relationships and patterns.  It is the invariance of these patterns that we use to found formal systems of measure and logic and comparisons.

At the same time these invariant relationships do not reveal to us the reality of any form, but rather our ability to perceive patterns in a spatial format, as an analogy. Thus they reveal how we may model forms and behaviours analogously without ever knowing the absolute truth of them and their behaviours!

Accepting this limitation was once the humility of philosophers, but later arrogant thinkers declared these analogous forms to be reality , and mans mind as the master of it all!

It will be suitably humbling to read Newtons explanations of his fundamental primitives of quantitative measures, both to learn how he applied the philosophy of the Arithmoi to this end and how we may do so with a greater empirical data set of fundamental behaviours. I am sure we will find that as Ampère and others did that Newton's principles are adequate to the task , but that we must expect to have to demonstrate conformity with the invariants in the philosophy of the Arithmoi.

In addition we now have Grassmanns tools and methods, which while not being different to Newtons own are in fact publicly explained! This was something Newton was loathe to do, and set the general enquirer at a considerable disadvantage as to how he arrived at many of his formulaic expressions for calculating quantities and measures.

Finally newtons first approximation was to the motion of bodies. Ours must be to the motion of fluids and for that we require a whole new conception of environmental structure and primitive motions. I believe that density and pressure will be the fundamental concepts of vis and vis viva, that is newtons fluid motive?

This is a very good series on fluid mechanics giving a sound overview. However the historical development sees the application of solid deformation mechanics to fluid space. The concepts of classical hydraulics were the basis built by engineers that led to computational fluid dynamics.

https://www.youtube.com/watch?v=fa0zHI6nLUo

For the first time I have heard an explanation of Newtons  differential equation for Lubricity! It has been worked over through the years to establish a definition for viscosity and a definition for the rate of shear strain.

I am grateful for prof Soms explanation because it shows a fundamental mistake and how it has been redacted to make it " work".

Shear stress is dimensionally [L]/[ T]² the velocity gradient has dimensions [L]/[T][L]

This means that the viscosity is not a dimensionl constant.

Because of this mismatch the early equations tend to be littered with empirically derived constants and indices. The formulae thus have the feel of being squeezed into conformity with empirical data rather than applying philosphical ideas to pragmatic measure construction.

The issue can be resolved by using a stress that depends on time and space distribution . Then this shear strain concept will be defined by a partial differential relationship.

The analysis of the interaction between layers in the 1 dimensional case then can be seen to be misleading as the interaction is not one dimensional but 2 dimensional. The one dimensional velocity gradient is a diagramatic nonsense, as it requires 2 dimensions to represent it.

Further the analysis does not account for lubricity between the laminar flows. Thus one layer acting on another has to slip past without resistance or friction, otherwise we can expect oscillations to develop between the laminates, that is rotation that Makes the model inns curate ans t.he laminar flow should show interleaved streak lines consisting of rotating fluid regions, and laminar streaklines. Laminar flows should be filamental.

The analyis is in practice used for boundary layer calculations. In that case it would be more appropriate to use a terminal velocity analysis. In fact as a first order approximation to a terminal velocity solution this standard introduction is good empirically. The compressive resistance of a fluid is ignored in this analysis and so the dynamics cannot hope to be acceptably captured.

The analogy is comparable precisely to Hookes law of spring force and Newtons law of force. However it is best to avoid the measure force as a symbol of causation and focus on pressure as the fundamental symbol of causation of dynamics

While researching the flux moment of a rectangle, newtons fundamental definition of a flux in 2 directions at once, and from which he derives his rules of Fluxions and fluents without recourse to limits, I came across Newtons own words regarding the resistivity of fluids.

It is to be noted that the various laws are mathematical models in an attempt to understand what may best describe the observed behaviour. Newton in fact is so bold as to say they are non physical but purely mathematical!

The result of this experimentation lead him unsuccessfully to build a model of fluid behaviours. While his model came close it was nevertheless untrustworthy and misleading. In abandoning it he remarked that future generations might build a better model, not as is commonly taught that vortices d o not exist in space as DesCartes conjectured!

However DesCartes in Newtons School was hardly respected due to his arrogant writings. This should not obscure our view of his general genius in these matters of  philosophy and geometry. In fact reading his own words it is questionable if his arrogance was any greater than any other self publicist who must survive on patronage! The Puritan streak in the British character would have been highly offended I am sure, but landed gentry or Aristocracy could afford to adopt such effacing attitudes where feudal peasantry could not! In France all were peasants to the king!

So now I make progress by regressing.

It so happens that I have been in discussion with Barau A Tour on the Magnetic Universe orim about the fundamentals of Mechnics, and defending Newtons approach from misinformation. However I know my in ledge of Newtons Astrological Principles and the system of Worlds is very Sketchy. In particular it is speculative with regard to several early notions I held by my own translation of parts of a Latin version.

I m no Latin scholar and so only translate bits and bobs as I have interest. There are however some very good Latin translators who have translated the Principles into English. I avoided those until I had formed an independent view. Substantially my view is that most translators and scientists misunderstand Newtons concepts. Bearing that in mind I can now read a translation skeptic ally without credulity obscuring the translators bias.

So I returned to find the section on motive. This had bern a key liberating idea I drew from my translation activities. In essence I took motive to be the " cause" of acceleration on the basis of my translation of one sentence!

Running with this idea and a general paradigm shift on my part from solid mechanics to fluid echanics, I developed a notion of Newyonian fluid motive as cause of fluid motion and accelerations , that is pressures , pressure gradients etc within fluid element boundaries.. In short Netonian fluid motive becomes an alternative " energy" concept.

In a discussion with a " believer" who regarded the ontological argument as unshakeable logically for demonstrating the existence of a " prin" that is first mover who cannot or which cannot itself move, etc I was introduced to aspects of Aristoles research into motion and motive. On the face of it I was able to adduce that Newyon concocted his motive notions under Aristotelian influence. I have yet o thoroughly research hat hpothesis but it seems very plausible.

However it came at last to googles notice that when I googled Newton on motive, that several,scholarly works were available for inspection. In them I found a detailed discussion on Newtons Force concepts in which it became apparent hat motive was part of a triumvirate of concepts of the notion vis and central acting vis at that. Within this context it did not play a causative role at all, but rather a summative role for intel parts of a body . In fact absolute vis was the central cause and accelerative vis embodied effect, with motive vis being the distributed internal action capable of being summed variously.

This was a triumvirate of conceptual action from and around a centre that was purely mathematical nd not physical.

The mathmatical design was so Newton could apply geometrical modelling. Thus absolute vis emanated from the body centre as the cause of all propensities to move within its influence. This then was the ground by which the parts various and any of a body moved in relayed bodily motion acting on each other and other nearby external parts, either in concert or in opposition. Thus these motive parts were admissible of summation relative to each other and the central cause and the external others.

Continuing with this brief drive through newtons vis concepts we come to the accelerative vis. This was an effect of the other 2 in which from the centre and round about it it may be observed that a bounded body as a sum of its parts is in acceleration in a distinguished direction. It chiefly transmits cause on impact with other bodies but as a measure it distinguishes preferred is placements in equal times for the whole body omplexity.

We can see then that Newtons scope, though  founded on central emanations of various sorts was to measure the behavioural interactions of particulate bodies fom,the outset.

Being thus founded on central emanations it freed his system from any particular scale or quantity, and allowed that any body thought of as a whole was reduce yo the summation of its parts should that transpire to be the case.

Motive played the cruise ole of being the summation principle of the parts.

Absolute vis , underpinning everything, no longer needed to be discussed in his economy. Motive and accelerative vis were the bedrock of his measure system.

It is important to grasp that the accelerative vis is the product of the motive vis within a body. Should all the vis act in one way the accelerative vis is a sum of these which assuming uniform parts quickly allows a multiplication form to stand in general.the multiplication fom is crucial to all geometrical models, nd relies on books 2, 5-7 of Euclids Stoikeia if a Pythagorean expianation is required. The alternative explanation comes through the Aristotelian scholars of the Islamic schools. Newton likely studied both, but that is not certain because evidence suggests he relied on a vast information processing network to verify his postulates, which clearly became accepted as propositions of " truth".

The text here comes after a more general introductory scholium in which he sets out his views and aims and several,advisory warnings.

http://books.google.co.uk/books?id=lSoJ2tJKfIEC&pg=PA5&dq=Motive+newton&hl=en&sa=X&ei=3IzRU-zrO4ep7AbakoFQ&ved=0CB8Q6AEwATgK#v=onepage&q=Motive%20newton&f=false

It is a debate amongs some philosophers whether Newton may have had an initial" field" concept mofivating his thinking. However it may be more enlightning to view Faradays concepts in this light, using Newtons concepts as indicative of a non newtonian(!) concept of force equivalent to Faradays rather than Maxwlls academic conception.

Then we can Compare it to one opinion about the generality of aether concepts as mysterious agents, and the specificity of measure concepts which give a quantitative description of some of these mysterious agents effects.

The central tendency behind Newtons measure concepts is indeed a vector field concept, and motive facilitates that vector summation measure. But Newton does not regard the accelerative vis as a vector field. It is a diffuse force or vis inhabiting a body from its centre and extending round abouts imparting bodily motion to bodies in its influence. You have to understand that these are not physical,explanations but mathematical measures. Absolute vis is the only philosophical principal which has causal power.

Newton thus proposed that the motive of a body be the sum of the " motion" of it's parts , and likewise the motive of a system. These systems are understood to be centrally tending, that is essentially within a great sphere towards whose centre all parts are tending.

Secondly he proposes that a body has a central pointing accelerative force / vis that causes local relative " motions" of parts or Bodies within its influence.

Thus we can look, at any body within a system or as a system and apply summation rules and product rules.

That is precisely what he carries on o do.

Now my point of interest is in fact Newtons reationship to Hooke.

There is little dispute that Newton was influenced by Hooke, and that among others he credited his insights on principles established by Hooke. Thus Hookes extensible spring laws served as the basis for his own accelerative vis descriptions and how he proposed to measure accelerative vis. His proposition was simply to balance all these accelerative vis and motive vis within a system of Bodies to achieve what really is a dynamic equilibrium, but called a static equilibrium! In so doing he could use Hookes spring law to characterise a force in a stable system.

Hookes springs revealed the dynamism in stable systems and allowed Newton to formulate a force erasure in terms of his accelerative force concept. In particular a long spring oscillates to equilibrium in an understandable way if accelerative force is quantified by Hookes spring laws.

Thus, just in balance the extension l characterises the force on the body by the general centre and the force on the body by the spring centre. Extending beyond l creates a greater acceleration back to the spring centre, shortening it means the general centre creates a greater acceleration. Without accelerative vis no terminology existed to explain this observed behaviour.

How the measure related to length and time or periodicity was crucial. And Newtons analytical definitions of the velocity which was summed by celerity of the parts and acceleration which was summed by the motive of the parts required a chain link between all three concepts. This was done empirically by measuring the displacement for each quantity of accelerative vis, , the associated velocities and celerity absorbing action of sping resistance past equilibrium, the changes in velocity and a regular periodic measure.

Tabulation of displacement against time is crucial to defining a workable concept of velocity and acceleration. That these were differentials goes without saying, but that an accelerative force should be characterised by Hookes law is apparently a concept of Newton.

The oscillating mass on a Hooke spring illustrates several key concept. Velocity does not stop when acceleration is 0. When velocity is 0 an accelerative force may become free to act. Force erasure links mass and accelerative force as an equivalent or analogous concept of force but also dynamic, resistive forces are in play in all, dynamic systems.

So briefly the dynamic inight revels that for fluids acted upon by a solid in non accelerative motion, the resistive medium would act to drain away celerity to itself. In terms of conservation of momentum we can expect the momentum to the same throughout the interaction. But because the celerity leads to an ever increasing mass of fluid we immediately expect the sold to slow down. In fact we expect the interaction to spread celerity to the whole mass of any resisting fluid making the solid come to near rest simply on conservation of momentum grounds.

Now because we are diffusing accelerative force by celerity transferring to surrounding fluid, that fluid is not going to be bound, but will spread spherically. The exact nature and veracity of that spherical fluid dynamic involves a pressure gradient between fluid elements transmitting different amounts of celerity or transferring celerity at different rates. The principles of motive and celerity are summative so a spherical superposition would be a description of the pressure gradient profile most in keeping with Newtons principles.

Newtons own investigations I have yet to explore fully but one important element missing from his vis triumvirate is the rotational moive and the rotational celerity, along with the rotational accelerative and the rotationl velocity concepts. Consequently, despite the spherical nature and veracity of his " field" concepts rotation does not have an intrinsic place in it. If it did then Magnetic fields around wires would not be misdescribed,,and vortex shedding would not be an unexpected finding .

It is probably clearer to state Newtons triumvirate of vis this way: absolute force is universal and causative and we can only deduce it from instances of it acting in centres of bodies or space/ places. This last attribute explains t
He missing "bodies " as centres of gravitational attraction required for elliptical orbits.

The instances of absolute force thus cause a local effect emanating from the centre of body or space to which bodies are tending ". This local effect is not clearly distinguished yet as centripetal or centric.ugal behaviours because these are foundational postulates which we must agree on or reject. This vis he calls accelerative and it clearly defines a causal field of action associated to the centre of a place or space. However accelerative vis is empirically observable only by the action or behaviours of bodies relative to this centre. Thus it itself is not causative, but rather it is observable by an affect on bodies within a sphere that may be called its field of influence. The only cause then is absolute vis which is universal, and this centre acts as a .conduit to an instance of universal cause in action. The observed accelerative vis emanating from this centre is quantified by the bodies impelled into relative motion, either toward or away from this local centre.

The observed motion of the bodies can be summed and these sums are called motive. Thus motive is a sum of the mass and the acceleration of the mass or masses in some combination. Experiment showed that mas and acceleration were best producted so motive is a summation of products of mass and acceleration.

However absolute vis must also be the cause of velocity, which is similarly a central emanation that moves a body as a whole . But the previous discussion interjects accelerative vis as the cause of change in velocity. It must therefore be that velocity remains in a body maintaining its constant motion. The summation of velocities in a system would then be the celerity of the system.

Again velocity is an instance of cause emanating from a centre, but this time duration of action is significant, and freedom of motion and motive is mediated by the super positions of accelerative force fields and thus motives.

As a kind of second rank behaviour or rather a necessary effect velocity, that observable behaviour of a body in motion whether accelerating, moving uniformly or relatively still , is the measure by which we link our perceptions of motion to space proper.

Celerity as a conception perhaps has its recogniseable expression as " momentum", but it would seem Newton had a much broader conception of celerity in terms of a concept called the quantity of motion measure. We might say he conceived of a body having a rest quantity of motion, a kind of kinetic energy but not as a conserved quantity. This quantity disappeared without trace in interaction.

Again, the lack of any rotational measures in this fundamental system is a serious flaw.

It occurs to me that the concept of action at a distance is not avoided by this structure of the vis, unless the bodies and space in which absolute vis finds its centres are indeed something, call it what you will from space time to aether. It is the absence of this self reflexive and self motivating substance that leads to the inescapable action at a distance.

Also, though Newton dies not make all things clear in this introductory Scholium, he by no means neglects to establish rotational motion as fundamental, and so to establish a further putative system of vis. However it is his " students" who have rejected or not understood his modifications to his principles of Astrology.

As you know, in fluid mechanics pressure takes the leading role. Because Newton actually qualified vis in this way motive vis is in fact a more flexible concept than the current bald F = ma notion!

Motive force is the sum of bodies in accelerative motion relative to a centre. Thus the complete notion is poetical in Spaciometry and entirly consistent ith bubble pressure or fluidic pressure measured at the boundary of a fom by summing the accelerated masses. In fact what is done is a bulk modulus is used to contain this motive force and like a spring this deflection once settled characterises the accelerative force.

The adjustment to surface area that is made to the f = ma concept is in fact a mathmatical method that oes not capture this very well. The introduction of density reveals the underlying philosophical difficulty.

The concept of fluid motive that I derived is exactly the reverse of Newtons concept because I though motive was causative of acceleration. In fact Newton gives no cause beyond bsolute vis. The accelerative vis thus stands on its own as a local instance of the universal cause in action . When density is introduced to distinguish various matters it also distinguishes how the accelerative vis is behaviourally modified by material density distinctions. More work needs to be done on this aspect of materiality because the quantity of motion measure seems to be a good measure of material intensity/ density as a pressure system throughout a volume.

The change in this pressure throughout the volume conveys the ability of the material to spread spherically.

High speed photography confirms this kind of behaviour and perhaps reveals the wave like deformations that are involved in establishing a bubble of a given quantity of motion!

Computational fluid Dynmics has made some remarkable progress, but it's foundations still need to be revised as Claes Johnson points out.
However here is an interesting research project into the plumes and vortices that make up a turbulent boundary layer
http://youtu.be/L9LD5eZicAg

http://www.youtube.com/watch?v=L9LD5eZicAg

Advances in CGI and surface coloration and plotting make clear the filament awry vortices and plumes and how they decompose a spherical or spheroidal volume . These volumes are not neat spheres but deforming ones so they are morphing as they move in the flow, thus characterising turbulence

The evolution of vortices is modelled elsewhere on this site, and it is worth noticing that vortices tend to break up into regions with connections to each other that fade. This is a fluid analogue of a particulate structure, and should inform all particle theories as to the nature of the interactive behaviours of matter.

I am moving toward a fundamental fluid plasma as a primitive aether which underlies the material space. Therefore it is important to realise that this plasma is not a 4 th state of matter, but the fundamental ground of matter

The concepts of temperature and heat need to be revised so as to support endo thermic and exothermic Enthalpy in plasma behaviour.

Chemists know that endothermic reactions occur as well as exothermic ones. Physicists prefer to ignore endothermic reactions, focussing on a concept of entropy tht is biased towards exothermic reactions. It is more reasonable to accept a steady state balance between exothermic and endothermic interactions in plasma , between cold and heat, with stable matter bEing a balanced state of these cold and hot plasma interactions.

This discussion of the classic Helmholtz Kelvin flow resulting from flow over " riblets"  is flawed only by the assumption Helmholtz vorticity kinematics is correct. However the statistical modelling approach means that the results really are closer to " first principles" development rather than fully accepting Helmholtz kelvin kinematic principles. That the same results are achieved does not validate Helmholtz vorticity kinematics, rather it reveals that the boundary conditions are robust in determining the internal behaviours , thus a wide range of boundary conditions compute the same outcome because the superposition in the process tends to a fractal outcome which in these cases are logarithmic in nature.

http://www.youtube.com/watch?v=yf4oCfdPUu8

http://youtu.be/yf4oCfdPUu8

I think I have discussed before the characterisation of shear force by a vertical velocity differential . This is supposedly based on Newtons idea of lubricity in resistive fluids, later replaced by the notion of viscosity. This initial deformation at the boundary underlies much discussion about flows around objects and between objects.

Terminal velocity analysis also links acceleration to velocity in a limiting scenario which defines an exponential velocity as a force equivalent.

My research into Newtonian Vis models or principles has revealed Newtons use of the Galilean principle derived from the observations of the Jovian system and the orbit of Venus, which essentially Galileo  adduced to a fractal system of orbits around bodies in orbits  around larger bodies.

Newton elaborated this principle to a system of absolutes : space, time, force. In particular he posited a centripetal force, as observed. This centripetal force was metaphysically structured into a triumvirate so as to develop a quantitative system of measures that linked directly to the balance of moments or the Hooke spring balance.

The absolute vis or power of a system was structured spatially by an accelerative vis or power of the system so that the absolute power was distributed according to some " law " over the region of the absolute system. This law was accelerative and affected velocity in that region over time. Finally the bodies within that region were characterised by a motive vis which was centripetal and summative , that is a vector sum was assumed for the motive of a body which itself was a product of the bulk and the accelerative velocity of the bulk, towards the centre of the absolute system. Thus weight  was the concept of motive vis Newton sought to on next to, and the velocity which products with the bulk is a velocity difference.

To make full sense of this motive Neton connects it by a system of measures to the fundamentals of bulk and displacement and time, introducing several measure the most addictive of which is the quantity of motion measure which is necessary to support the motive vis.

To this analysis Newton added the Centrifugal Anlysis of orbital motions described by Halley, and do established a centripetal centrifugal description of the accelerative force in a region counteracting the centrifugal accelerative  force in a bulk or corpus.  Thus Newyons description of orbital motion analyses the forces only in the centripetal and centrifugal case, with the resultant velocity being everywhere a consequence of competing forces or vis. Newton does not consider a ritationl vis!

The rotational vis or power is obvious to any empirical observer, but because Newton did not specifically discuss it, many have concluded it does not " exist". It is supposedly an artefact of our observations of radial accelerations, and tangential instantaneous infinitesimal impulses. Things become simpler when one realises that the Newtonisn system is a model of a centripetal / centrifugal orbital velocity system. His model was based on standard geometrical models which take a circular or elliptical locus as conditional on radial constraints. This kind of analyis does not deny the circular or elliptical motion it relates it to radial factors which the observer includes in the description of the motion. The motion itself stands alone!

The imposition of radial measures does not negate the motion that is bing measured, and thus rotational vis is not negated by centripetal and centrifugal vis!

In 1820 Örsted revealed the magnetic behaviour around a current surrounded wire.. Arago demonstrated this to the French  Academy in that year. Only Ampère realised that this was empirical evidence of a rotational force! The rest of the academy, LaPlace,LaGrange,Biot.Savant, all were incredulous. They did not believe in a rotational force.

Ampère devoted a brief but intensive period to establishing the "laws" of this rotational force using Newtonian principles. He declared this as the field of Electrodynamics. There is a ritationl force and it is buried within the obscure mathematics of Electrodynamics. It does not require all that excrudescence to be believed and understood.

In that light the rotational force as a model becomes the fundamental force of the cosmos at all scales. Lineal forces are thus derivable from it according to scales or indices based on the radii of curvature of the rotational force.and in this regard the Reynolds number is a scale factor that indexes these radii of curvature.

The Newtonian concept of Vis is thus very flexible. The kinematics that supports Newtonian mechanics goes into detail of how flexible. The concept that force is just mass by acceleration is misleading. Newtonian force is proportional to acceleration, to velocity and to displacement!.

The above statement does not directly refer to the time constants required to apply the measures because time is considered absolute in the Newtonian system. In a relativistic system time becomes relativistic!

However the vectors are usually in the same direction : acceleration, velocity,displacement. How can the power be directed in any other way? The orientation of the force in a centripetal force region is distributed by the trigonometric "laws" of the right triangle. Mechanically this was a geometric no brainier. Experiment and empirical observation showed that Pressure in the contact region was radially consistent. Thus the vector was a compound of all these radial vectors.  Because of this any vector could be resolved into radial vectors from the same point of contact.

The significance of this is that Pressure is more fundamental than lineal force in the Newtonian system.

While the system resolves the forces to relationships between straight line segments, the nature of these line segments were not explored until Hermann Grassmans Ausdehnungslehre. In particular the trig line segments are analogies not of straight line motion but rotational motion.

There are many assumptions explicit and implicit that underpin the Newtonin system, but fundamentally it uses a polar and Cartesin reference frame in combination to describe physical behaviours. Thus Newton Wren, Halley and Bernouilli may not have specified a polar reference frame, but they implicitly assumed it, and they implicitly assume a rotational force. This is only clear when the reader does not deny the centrifugal firce model of Halley. Newtons orbital system requires both centrifugal and centripetal force measures to conjoined with the motive force measure to describe the orbital motion.

In a fluid, the existence of rotational force explains how a shear force transmits orthogonally to disturb the fluid in a vertical velocity distribution which is a characteristic of the fundamental pressure in this dynamic system. Thus one can resolve the shear force by trigonometric vectors which correspond to the velocity distribution in the fluid under shear.

Now rotational force implies one other aspect of pressure promulgation. In the surface of an expanding radial or contracting radial force, a rotational force will be evident, and in a " circular" behaviour. This is what is observed in expanding bubbles, the elements in the expanding surface rotate within the surface as well as expanding or contracting. Transverse " waves" on a spherical surface thus have a chiralty of rotation, and such "waves" develop into filamentary plumes from the central spheroidal mass.

The fundamental dynAmic structure in fluid Mechanics is the plume and the vortex ring associated with it. It's a kind of " mushroom" structure. It evolves into bubbles. Spinning bubbles either expanding or contracting.

Thïs plasma demonstration of the magnetic space lead to a post on MagneticUniverse.com regarding the fluid Mechnics of plasmas as defined by high energy physics t the LHC.

http://youtu.be/ECokfl2y0Fs
http://www.youtube.com/watch?b=ECokfl2y0Fs

The cathode ray is a plasma and it interacts with a dynmic equilibrium plama flux in the space of a magnet to produce a diffraction pattern. .

This is Norman at his most obscure. I am grateful that he renounces this rhetorical style early in his career and starts to develop a much more inclusive approach.
Nevertheless this abstract concept allows a mechanical description of fluid Mechanics to be connected to Lie groups which are fundamentally groups of trochoidal rotations
http://youtu.be/pqAM4-pic8M
http://www.youtube.com/watch?v=pqAM4-pic8M
 To make use of the abstract format above Norman glosses over the need for a suitable Fourier basis. The Fourier bases are the topic in the thread Twistor in the complex number forum. In that thread I explore a exponential format for rotation. The format formally introduces circular arc line segmnte, analogous to straight line segments

In this video Norman exposits Topological notions that eventually as you go through the course makes sense of Tangent planes to curved surface topologies.

http://youtu.be/kdpbfOzkJzI
http://youtu.be/

Gradually this will lead to an algorithm to model fluid dynamics using Normans Universal Hyperbolic approach nd the chromatic algebras of matrices.
Hopefully applying Grassmans concept of extending magnitude will lead to This  the conclusion of the basic mushroom plume of fluid mechanics.
http://youtu.be/4gHhAA-9o_8

While in the bath, and thinking about the day and bubbles, and my reply to funkyNfresh I drew a single trochoidal locus with a single curl back onto itself and then looping and flirting away, in my mind.

Let that represent a pressure or trochoidal force line segment I said.

Almost immediately I thought, that curl should be the surface tension, and then I thought of a mirror symmetry  trochoidal line segment , and the pair reminded me immediately of the plume and mushroom smoke ring, with the surface tension along the surface of the mushroom..

This surface tension comes from the trochoidal force line as  at curls .

Them I thought : what is surface tension any way? Just the region where the trochoidal force becomes curled , no longer straight out from the centre.

Then I thought about the radial expansion of a pressure/force field,  it was always a totally inadequate representation, because it left out the expansion between the radials . This trochoidal path accounts for this inter radial expansion.

Then I said: it accounts for surface tension. And then I said : Newtons tangent decomposition of a circular force represents surface tension. Until now I had to understand that as a velocity resulting  from an initial outside push or pull. But now I can conceive of it as a force resultant , the surface tension resulting in circular or spherical acceleration that makes the surface an incredibly active region!

Without further ado I realised that a pressure field must not only have radial line segments but also line segments between the tips of the radials . The centripetal and centrifugal forces may be static in dynamic equilibrium but there is nothing to stop the surface tension acceleration. So within that boundary expansion  the surface generates a rotation within the surface .

The consequence of this is that deformations called " wave" deformations are not just transverse or just longitudinal but a combination of the two which results in trochoidal deformations in space. These deformations we see as filamentation and vortex ejections if energetic enough, otherwise as an incredible surface tension

Ummm...

EUREKA!

I hope you didn't run through the streets naked to go make that post   :stickingouttongue:

Interesting parallels though, and somehow reminds me of certain bits of String Theory with it's curled up dimensions and hidden vibrations.

 ;D

My wife restrained me!  :embarrass:

I can not express it yet, but I perceive the underlying formal structure (not space ) to be based on trochoids . I would call them Grassmann Trochoids but I have to find the direct link yet.

At this stage in my research Hermann is still using simple elements. Trochoids are the next step up from the simple circular arc segments, and they can go all the way to the n-th stage.

String theory as a model has fabulous adjustability potential, a Fourier like transformation that can reproduce any "shape" in space. But it becomes fantastically complex very quickly. That is why a fractal description is so crucially important to its application .

Why would I need a product or a multiplication process in describing a physical system behaviour?

The nature of dynamic systems is that they vary. They are thus naturally expressed in differential likenings..

Newton's Fluxions are based on the binomial product design. Thus his solution to differential likenings will be a binomial product of some description.

Product design is also conflatable onto function theory or vice versa..

Thus if you can design the product for the system you wish to explore you do not need to use the inferential equation approach.

In general products are the way any system likenings capture rotation in a system as well as extension/ intension

While I have posted this video in V9 thread it naturally belongs here.

http://youtu.be/eCJxhxDMIb0
http://m.youtube.com/watch?v=eCJxhxDMIb0

The hyperbolic design is reversible in the sense that flow through a small hole or venturri channel should work optimally if designed accordingly.

Stress stressors topology and magnetic bonds.
One of the plane facts is that matter is variable in its intensity and that it has continuous and discrete forms .

Further we recognise energy -phase changes that is to say simply : solid liquid gas plasma. States of matter with different vibrational signatures.

It was determined by Newton et al that heat was best described or pictured mentally as a vibration in a corpuscular region corpuscles were thus conceived as flexible plasmas from the nascent biological discoveries of the time.their flexibility was merely a way of accommodating the vibrational modes conceived as heat .
We have since developed the heat model into the Energy ,Odell.nsuch a model really has its foundation in Gilbert's magnetic philosophy which recognised 2 forms of magnetism: crystalline lattice or mineral magnetism and biological magnetism, that is organic lattice type. The one was called the iron or ferro magnetism, the other Electra magnetism or amber magnetism.

These specs were later intensively studied and distinguished into models that were fundamentally mysteriously related. However the various guilds of magnet makers and electric charge makers  jealously hung on tontheirbtrade secrets and further distinguished the common link, that is the fundamental material experience and phases of matter at different energy or vibrational states.

Matter and space were philosophically divided in the most natural way , but space was called empty for religious reasons. It was desired that spirit be distinct from matter, and invisible able to manifest to conscious souls, themselves spiritual. 

Such a model kept the material world that technology was exploiting as a distinct environment that mankind could "play" with like a child, but spirit was the playground of the gods and those ascended humans who struggled to grasp it.

Of course the material philosophers were brought up in that Matrix of ideas ad models, but younger more ambitious material scientists saw no valid reason to limit material science in this way and technology indeed advanced to miraculous inventions gadgets and contraptions which supported that viewpoint. 

Unfortunately the mathematical clan or guild retained enough obscuranting mystics to cover the material  viewpoint with a pseudo mystical or religious analogue of religious theology . Bishop Berkely wrote a powerful attack on mathematical religious views pointing out that material scientist of the mechanical philosophy were in fact no more clear of superstitious nonsense or beliefs and practices than religious theologians and clergy

Accepting that point we may progress toward a common view proposed in detail by Hegel et sl and eastern mystics since way back, that spce is in fact something that is connected to a singular concept that contains all: if you like spiritual and material are two aspects of an entity with many aspects.

The Aether or Ether philosophies attempted to place that view as foundational to the material sciences, but all clans and guilds fell apart in disarray: a modern Babel of jargon and gobble de gook .

We do not have to be so inclined.  A simple and useful combinatorial model underpins the technological achievements of mankind. It is rather humble in its scope and often  symbolically and ritually obscure. Attempts to make it clearer usually do the opposite, academically. But those who actually make things using their whole organism develop inexpressible expertises that often seem to those not involved in the physical actions of making , well they seem like magic, even magycke of the old kind.

Regularly organic and rock crystals cry out to us ! We call them TV's and Radios!! 

So we can take material science or specifically the science of material and develop an expertise that Tribology or tribologists express. This level of expertise is an art form found among the few material scientists with a broad enough range of experiences and understanding. It is eclectic and very much in its infancy .
For example, topology would help to provide a structural overlay akin to corpuscular philosophy of Newtons time. Topology fundamentally is a combinatorial art that gives forms or formats for space in its most general Aetheric model . These formats may then be used to count and measure regions and distinguish behavioural changes in material over time, circumstance and internal and external conditions. 
Topology allows us to place atomic or molecular models into regions of space so we can characterise certain regions by their phase and nearby status and their visual characteristics . The stresscandvstressors in such a space become characteristics of the phase of the element or molecular material as well as contact behaviours between such characterised materials.

Clearly a chemistry and organic/ biochemistry based on these characteristic measurements and observations has proved very fruitful in organising our technological advances. But we have found some who want to isolate these behavioural and technical expertises away from common knowledge to support some particular world view or religious ideology. 

While the knowledge is presented as iconic, eclectic, difficult to understand, even iconoclastic, it is in fact not. The simplicity of the energy/ vibration - phase model of matter is how it has been hidden in plain sight.

It is a simple robust and powerful combinatorial system that was advanced by Navier Stokes and yes even Newton into the fluid dynamic model. Helmholtz and Kelvin added the Vottex model rules (inaccurately: see Claes Johnson work) but all of this was obscured by the Mathrmaticians. Helmholtz following perhaps Euler and Lagrange felt that the differential equation format was the safest way to express natural laws of motion in space-like material especially if they were fluid.

In a sense , given the prevailing attitude that aether was basically solid billiard balls at the femto metre scale( an assumption since demonstrated to be misleading especially by quantum Mechanics) his reliance on the continuous nature of differential equations by design does preserve the fluidity of the foundational corpuscular theory of Newton et al. 

The use of hard billiard ballasts representing corpuscles is a mistake we can trace back to Leibniz et sl . But even then he was not insisting that matter was thus. The mathematical topology was simpler if this was assumed , and produced useful models. It also produced strange aberrational models which rightly were ignored! Now they are mystically clumped together in the quantum mechanical world viee.

The fluid mechanical expertise is very difficult to establish except in the simplest cases, but in fact computationally it has provided us with some excellent models within the boundary conditions appropriate to the study. The advent of more powerful computers and computational schemes has made it possible to model complex data increasingly accurately, by applying a whole mish mash of equations topologically to a spatial region.

The one fundamental breakthrough for physical topology is ascribed to Benoit Mandelbrot. His Fractal Geometry  inspired many computer geeks to model the particular types of inductive or iterative( recursive) equations that hitherto were avoided by Mathematicians as monsters. Ad yet we know Focault and many others were studying them quietly prior to the wars with little academic support . 

We can laugh, but it was mathematicians that were the Luddites when computers were first introduced!! 

Today I watch a lovely flowing scene of actual waves lapping onto a shore with all the sound effect on a flat, crystalline organic matrix presenting the display of raw data as processed by billions of logic gates in some silicon(rock) chip processor with a certain vibration/ energy which has to have the heat removed by invisible gases passing over it as winds! 

The fluid dynamic models expressed as differential Equations may be nonsense. But the expertise that organises itself in this way around the energy phase model really has a powerful set of models with which to develop further models and technological advances .

Why have I not mentioned Maxwell, and Einstein? Because they represent a deliberate mystification of the simple model. Faraday was not happy with how Maxwell fundamentally distorted his ideas. In fact Örsted Faraday, Ampère Volta and others held that a circular force o rather a rotational force was fundamental to our understanding of all models built on observation. Newton, unlike his successors took a circular or rotational force for granted. 

This simple topological element distinguishes models that account for quantum behaviours from those that do not. 

And yes the mathematics becomes confusing and complicated, but only when the circle and the trochoids based on circles are excluded.

Include trochoids in space in your models and you can model magnetic behaviours topologically in a consistent way.

The gravity of this point should not be missed, as Gravity to this day is still touted as a mysterious force,  all forces are fundamentally mysterious characteristics of topologically described space. Their iscavsimple reason why conic curves describe both gravity and magnetic attraction and repulsion. The topology of space that best describes them is rotational or circular. Add to that fractal geometrical topologies and you can model fluid dynamics in appropriate small boundary conditions.
Add phase changes at different energy and thus vibrational states( frequency amplitude and trochoidal topologies) and you arrive at discrete regional distributions with phase changes marking the boundaries of regions fractally.
What we know about bonds is dependent on these fractal regional surface phase change behaviours at differing energy/ vibrational statuses

We require a new label for application of fractal geometry to real physical behaviours and forms : fractal topology .

Claes Johnson helped to pioneer the finite element model of computational solutions to partial differential equations . Another way of saying that clearly is the Fractal Element Model.

We can see how complex the tedious algorithms are , but the results are not only intuitive but spectacular for fluid dynamical modelling.

I really feel that we can dispense with the high mathematical presentation and get down to brass tacks. Fractal applications should be able to implement the regional designs to produce the same results .

http://m.youtube.com/watch?v=hQjo4DkSFyA
https://m.youtube.com/channel/UCCPenA6XwM3REzmqoAXMw5A

Just wanted to tell you that I plan to read this thread in the future, but at the moment I can't say anything yet about it.

Welcome aboard!
I hope others too will contribute to the thread.
I post occasionally as insights come to mind and to keep the thread alive.

Currently I am gradually exploring the progress in this field to glean insights especially for understanding some fractal sculpture outputs, but any contribution or questions are welcome

http://publishing.cdlib.org/ucpressebooks/view?docId=ft4489n8zn&chunk.id=d0e16359&toc.id=&brand=ucpres
This guide to the Natural Philosophy of the Pronciple for Astologers breaks down the key structures of Newtons presentation . It is important to read the Latin presentation before assuming the English translation is accurate.
Now my conception is that our proprioception of space and spatial dynamics is necessarily representational of a fractal mesh of sensory experiences synaesthesically ombined . Thus a superpositional or interference pattern of sensory signals underpins our conception of extension and intensity of magnitude( that is extrnsive and intensive magnitudes, so called) . The consequence of this is that the simple must be conceptually derived from the complex by analysis. We thn construct the complex by synthesis of our analytical products.

So Newton attempts o didactically impart this notion in synthetical steps after he had done the Anlysis privately and to his satisfaction.
It is the classical duty of all Rhetoricists to impart instructive information at the level of their audience!

Thus we find axiom, definition, proposition, Lemma , demonstration all woven together as set out by some of the greatest classicl master taught. We name only a few such as Plato and his version of Pythagorean scholasticism, Aristotle and his critical revision of Pythagorean ideals, Eudoxus,Euclid and Apollonius .
To these we may add some of Newtons contemporaneous teachers and Philosophers especially Wallis, Barrow and Hooke et al.
For me then a concept of pressure precedes any concept of force, just as a concept of the experience of magnitude precedes any concept of motion .

Finally the concept of spatial distribution of experiences of magnitudes now distinguishable by dynamism into at rest or in motion relative to the observer and perceivable as both intensive experience of magnitude nd or extensive experience of magnitude precede the concpt of Tyme or in modern guise Time.

Tyme itself is a conception of record keeping , whereas by the epoch of Galileos pendulum and Huygens formulaic expression of the pendulum Period time had become an experience of the regularity of motions.

We may go back to Timmaeus and other nicest engineers of worders who expressed the records of spatial positions by ratios of lengths of arcs on the circular perimeters of wheels and gears thus animating ' dead and lifeless " records and creating miniature models of their "Universe"
It is eye opening to realise that our brightest minds only a century ago thought the Milky way was the universe and now we think it is only a small local part of it!!

In defining pressure therefore I acknowledge that a space or region in space is always assumed, and itching that region a surface divides space into  two relative to which side of the surface the remaining space is situated.
Thus a surface is hardly defineable if it does not have the extensive magnitude that dissects spatial perceptions into at least 2 relative positions we may call " sides. The record of point like objects such asvstarsvor planets therefore are conceived to be on or in a surface and on one side of that surface. Classicàlly that surface was universally designated as a spherical one with the gods being on the other side of that surface essentially unknowable but conceptually present.
As we have pushed back the radius of that sphere, the diameter though huge has not expunged the notion of sides to surfaces in our universal knowledge .

Today we have an alternative to this central topology and that is a fractal topology. We can now think of fractally distributed regions of space like objects the totality of which is not conceivable and hich is not bounded by a spherical surface and thus has no sides but one corvl spatial distribution within which surfaces and ides are locl nd relative characteristics of a surface.

Pressure is now conceptual usable as acting on th sides of a surface and as regionally differentiable( differing in some way determined by the side of the surface it acts on, and the region the surface dissects, whether open or closed.
 A closed region now underpins notions of orporeslity or corpuscularity and the pressure inherent in such regions are often considered to be energies, body forces, tendencies and tensions .

We may derive from sides to surfaces notions of orientation hich then feed into notions of direction. Thus we now are able to speak of surface relative irections and distinguish between sides of. Surface and notions of a point in a surface!
The motion of a point is therefore derived from the general notion of a surface in space evn though a general concept of a point seems to be obvious in space! This is because in identifying a point in space the surfaces associated with that point always guide the perception of the precise point.

The action of pressure may now because of a surface be described giving pont direction and effect of that pressure

It is when we come to effect of pressure that we enter into a synthesis that is more complex and yet still fundamental. In particular, the dynamics b hich side ha Ben is cussed nd indeed perceived are revealed to be founded in the general dynmic concept of rotation, which itself in generality is now discussed under the heading of Roulettes, but previously was discussed as Tochoids.

It is this complex trochoidal behaviour we call chaos or turbulence and yet we May approximate to it by Fractal dynamical means computationally.

http://m.youtube.com/watch?v=WmKgT0a2KiY
Claes Johnson has substantially improved this method for computing fluid flows including Virtex outcomes in the potential flow solution
http://m.youtube.com/watch?v=CMz8KtvWKW4
A fractal generator cannot depict these situations with specificity, Yet it can still provide interesting outcomes that visualise these flows on the negative! ( photographic negative)
The iterated equation therefore has to describe the flow under the fractal generator set up. That is indirectly the or Mandrlbrot format With the bailout restrictions set up a circular/ spheroidal boundary condition that relates to certain flows of fluids in a region. .

http://m.youtube.comwatch? V=/QPTjuS5G3dY
Claes explains clearly if disappointedly the way fluid dynamics for aeroofoils( and thus boundary condition flow in hydrofoils and other situations ) was hindered by the lack of understanding . But that itself is an untenable position. Planes flew, boats sailed because experimentalists conformed to nature not theory or computation!
Today our computers can process the complexity of the descriptions we give them, but they can not predict an outcome. We do that as experts in factual reality not Theiretical mythology!

Is myth less useful than fact? Clealy not. Our humanity utilises both to make life more interesting. However to obscure myth by calling it law and fact by denying it is observable by at least a few others is a recipe for shock, disaster and perplexity.

For Claes to expect to be revered for solving a mathematical conundrum they thought they had already solved is the sad part. His work ithbHoffmann is truly groundbreaking but one of those small but vital details, like your little toe, that just keeps getting overlooked!
Johnson and Hoffmann have transformed the finite element method into a powerful tool for all sorts of materials, especially fluids .
Despite the jargon, it is essentially a fractal generator optimised for this purpose .
The equation controls the material motion or behaviour under iteration , the bailout defines the boundary conditions and can be made as complex or conditional as your application allows , and the surface plotter colour cycle reveals not just the extension of the calculation, but also the intensivity of the resultant outcome.

All three have to match a physical example in the limit to model that behavioural outcome , and that still does not guarantee that the computational model is applicable to any other situation!
How nice to be able to say , by fiat, this is a universal law!

Clearly one can not replace fact by such mythology no matter how useful that myth is in its original environment.

http://m.youtube.com/watch? V=B2XOrNyirhc
Claes explains the basics . The one dimensional coordinate system actually includes curvilinear coordinates.
In fluid dynamics Eulers reference was the particle moving with the fluid.LaGrange adopted the reference of the particle measured by an observer I dependent of the fluid and the particle. However the particle then becomes a means of describing the fluid flow at a reference point!

Now we can isolate the particle ( Euler data) and the fluif( LaGrange data) to recombine them by superposition to give a more informative flow description.

The Finite Element method does not allow its elements to flow with the fluid, so it records a laGrangian or Laboratory measurement of the flow. So a test particle has to be supposedly introduced to indicate the effect of the laboratory forces on the particle.

If the test particle itself recorded position and velocity by GPS we would develop a far more useful description of fluid flow.

Currently weather satellites compute wind flow patterns in the atmosphere based on best fit for observed motion of an identified structure or phenomna in the data.  These are always checked against ground measurements by Nasas dedicated teams of verifiers to give us confidence in the model data .
Every do often they update the model to make it even better at describing the observed measurements.

Our weather map models are now increasingly fractal in nature, design and execution.

The myth is that we can then predict the weather!! What we can do is simulate the fractal pattern evolving for a few more time step iterations , but then we have to reconnect to factual day to keep relevant!

http://m.youtube.com/watch?v=LNLXdlXKvL8

Now we know Cles's view, we see that time steps as update sequencers are fundamental to fractal generators making them ideal for performing all the elements necessary for a computational solution to any integral or differential equation. In addition the color cycle maps can display that information visually, and the surface plotters sculpt out the consequences of such motions on an initial uniform fractal mesh.

The issue I am addressing is the comprehension of the desired or final product of a fractal rendering. Is the art just art or is it a model of surfaces that could be generated by applying our current laws of motion to familiar topological starting points?
For example do plants grow by just reproduction, or is some external varying bailout shaping the outcome as well? Do the shape and colours of leaves for example reflect this combination and a colour distribution based on chemical density of all those elements that have moved a certain distance from the roots?

These are simple suggestions to highlight how the three main factors of a fractal generator may be shaped to model real life outcomes. The question is : would that significantly enhance our understanding of nature and natural laws  or would it mislead us into thinking mathematics is gods language for Nature! ?
 God forbid that we should ever subscribe to that bit of Mathmythics !!

http://m.youtube.com/watch?v= NPJImYibx48
Some teachers are more accessi LE than others.
In this case I wanted to get a feel of all the constraints one has to place on a fractal generator to get it to calculate a recognisable sculpture.

The surfaces plotted represent points that satisfy all the constraints for each region but how the points are defined in a region is based on a calculation sequence truncated or equated to a condition . That sequence of calculations is not defined in a fractal generator, but may be defined in a Computational Fractal Element Dynamic calculation as a Euler stream line .

The mass of calculation results are isolateable in many ways, but how representative of a physical situation it is depends on more detailed constraints.

It occurrs to me this morning that the Grassmann Twistors now come into the discussion.
These computational fractal elemnt fluid Dynmical calculations have one simple idea. Specify the form( surface topology) so that boundary conditions can be defined .
Whatever else happens happens or is defined within this context or topological mosaic.

The Grassmnn Twistors as explained in the thread called Twistors, are Fourier transforms , they are specifically Quaternion Fouriet transforms.
The simplest expression of the idea is to : do a quaternion Fourier Anlaysis of the form or topological mosaic.
Having the opology in Grassman twistor form now confines us o setting constraints for : phase, Frequncy and Amplitude within the quaternion argumnt of each term in the Euler: Cotes expression of the Grassman Transform.

Clearly the arguments are not simple variable strings , but rather complex conditional statements . Each condition is an evaluation process so the computation is in fact a complex evaluation process rather Han an arithmetical number crunching exercise!

What numerals come out of the process relate to extension or inanity at each iteration step., and the "solution" of a differential expression or an intgrl expression for that iteration step.

The display magic comes when rendering takes those outputs and defines a surface plot for the display constraints, and the colour cycle decoders display the data in regional mosaic form highlighting extensivity and or intensivity.

Finally doing or running this many times , or rather evaluating after each iteration enables  key fram video sequence to be captured that may( or my not) give insight into internal Dynamics as expressed in the Grassmann Twistor depiction.

While we can not say that this process is how it happens, we can by taking smaller and smallet scale sizes( specifying a kind of fractal zoom!) provide more detailed tracking of th processing outcome and compare that with an ultra fast video capture of the dynamics frame by frame.

Of course, our digital signal processing allows us to use the processed data and the captured data ( after digital  signal  processing too!) as interchangeable.

While at first this sounds ideal, it poses a philosophicl problem: are we redefining reality, limiting our apprehension to the mythical output of a computational processor? Is a human. Biological computational processor, and oes this if the case, justify accepting this kind of complex out put as Truth ( Wahrheit) as Grassmann put it?

Even in 1844 Grassmann expressed his hope that these types of complex combinatorial systems( now commonly called Matrices) could be populated or internally decorated ( entdeckt) with the truth .

Should we accept this, now it has been realised In our time without philosophical debate?
Somehow as a fractal Philosopher I do not think we should just accept anything!

It has been a while since I posted here, but that is not been because I have lost interest.

In fact much has bern learned in the interim.

The main thing was to unlearn a reliance on mathematics as a revealer of physical  reality, and to put it in its place as a method of measuring by quantifying distinctions the amount and dynamic amounts over time of these differing quantities .
The establishment of princles and processes of quantification that address the spatial dynamics that we visually observe, and to apply these sketchy interpretations to the experience of proprioceptive signals as well as audio and kinaesthetic ones .

Because of the difference in general in these sensory signals no one representation is best for all, but one process applies to all and that is counting .

By counting we can use metrons  appropriate to each signal system, but the counting process itself is idealised by sequenced lists of names, and those names in sequence allied with our appropriate metrons in each ignl system allow us to transfer to the visual system a sketch of what we experience in all and every ignl system .

Fluidity and fluid mechanics requires the adoption of dynamic Arithmoi, that is specialised or normalised mosaics to support the measurement of dynamic variation over time.

And now we see through these types of accommodations that technology has enabled these mosaics to capture the fleeting ness of dynamic change, and gifted us with multiple opportunity to count the changes we discover in extension, form and dynamism.

In fluid mechanics, curvilineal surfaces are more fundamentally useful that straight lines or plane surfaces. And regional concentrations and dispersions of these curvilineal surfaces best depict the dynamics of fluids.

http://m.youtube.com/watch?v=-wokVaaGFgA
Here Norman shows how the strain ellipse and ultimately the strain ellipsoid can be encoded by a generalised dot product form
The significance of this is that currently fluid mechanics is using these forms to describe fluid behaviours in laminar flows without a great deal of intuitive background. Setting it in the format of lineal rational trigonometry and algebra gives access to Normans powerful insights.

In addition, Claes Johnson et al. Using there pioneering work in finite element modelling in fluid dynamics can now derive simpler formulae for faster calculations of the dynamic evolution of a turbulent flow, especially round aerofoils and hydrofoils general magnetic phenomena at all scales .

http://youtu.be/llqBmy_Ax50

http://m.youtube.com/watch?v=llqBmy_Ax50

Turbulence has an almost inverse square law, again suggestive of rotational dynamics affected by trichoidal behaviours. Not easy to see unless you are looking for it as a fractal spaciometry.

The Stoikeia, an introductory discourse on the Pythagorean philosophy at an undergraduate level guides the student into thought patterns that make these dynamic relationships apprehensible. The relationships themselves are scale invariant, and so fractal in that sense, but definitely fractal in the sense of approximation to a smooth continuity. Discreetness of form is not discontinuity perse but it is akin to discontinuity of perception. The isolation of the invariant forms , distinguished only by scale and orientation is akin to discretisation of an otherwise imperceptibly continuous dynamic.

And it is form that we perceive, however fleetingly , however dynamic. To denote this as a fractal perception, a rough and ready approximation of a continuous experience of extensivity and intensivity , curvilineal in generality is one of the intellectual tour de force of modern natural philosophy .

While Cantor and his set theoretical infinities are a misleading attempt to describe continuous dynamics , nevertheless the attempt to capture these notions of perceptible reality are pertinent even if esoteric.

We can not count on mathematics to depict this if Mathmythics is allowed to obscure the inquisitive intellect of Natural Philosophers.

How we may apply ourselves to investigating and depicting these dynamics is given in the Stoikeia, the Conics and the Spherae of Pythagorean scholars down through the ages, of whom the most Noteable is Newton, and of course Benoit Mandelbrot.
http://youtu.be/TnkgG0wSVng
http://m.youtube.com/watch?v=TnkgG0wSVng

http://youtu.be/jiMyzYf_COA

http://m.youtube.com/watch?v=jiMyzYf_COA

http://youtu.be/SKoaFgvsogs

http://m.youtube.com/watch?v=SKoaFgvsogs
An inverse law that indicates rather than defines rough behaviours. But what ubbdelies these behaviours is trochoidal dynamic.
Why?
Frequency distributions are not smooth as frequency changes relative to each other. There are dynamic ranges where smooth changes do occur suddenly to be interrupted by wild incoherent motions.

Many fractal sculptures show this "uninteresting" result which hitherto I had not understood the significance of .

http://youtu.be/NjHUi1BvXRg

http://m.youtube.com/watch?v=NjHUi1BvXRg

I have said it many times : the simple fractal generator is a tool fr studing dynamic and physical, chemical systems, in particular as I have extensively meditated upon in the magnetic universe forum. The phenomena of Magnetic behaviour and electrodynamic behaviour, sound , heat and light.  Google jehovajah a sound basis of rotational dynamics

THank you for the links and the topic....my brain has drifted towards the fluid dynamics of fractals too....and am working on how to integrate that into my art

You are Welcome .

As the new Fractalforums goes through beta testing I am gradually preparing for a new progression of this topic there. This will be left as a resource hopefully inspiring art and computational development .
I am currently reviewing Benoits own interview on his life and works.

It makes these sculptures so much more accessible.

http://youtu.be/FNap0r2XSpg

http://m.youtube.com/watch?v=FNap0r2XSpg
Here Sheldrake does what many philosophers do: he formalises an aspect of Fractal geometry thus killing it! Any time a philosopher mentions Platonic, they are foisting a view of Platonism that is essentially deathly. Benoit was Alice at the time and actively continuing his development of fractal Geometry. And the nature ofFractal geometry is dynamic, not only in spatial form, but in scale, in affinity and in time.
The phrase "almost self similar" is essential to apprehending a fractal dynamic and so a fractal topology or manifold. Both these ideas of dynamic and topology deeply imply trochoidal behaviour.

That. If I may be do bold, is the direction for your artistry.