Fractal Foundations of mathematics: Axioms notions and the set FS as a model

[jehovajah]

What happens when these tightly packed and coherent conic section curves are intersected? I would expect the disruption to generate "conicsectional chaos" by which i mean seemingly chaotic curves which are in fact some polynomial combination of the conic secrion curves aith an additional impetus from the intersecting curve. It may be that a taylor or maclaurin expansion of the conic sections might reflect this or a fourier interpolation based on conic section curves may be possible to describe these chaotic curves.

So for example the boundary between one spiral and another will be a turbulent region of conic curves where the surfaces interact and give more than one possible surface to cling to. The variation in the density field at such a surface should be sufficient to entrain conicsectional curved behaviour proportional to ,in a dynamic mechanical system, the energy transfer between the the two regions in contact.

The combination of these conical section curves would look lie a vortex in geometrical space.

[jehovajah]

Just to note that given a conicsectional iterator certain things are straightforward corollaries. But first in my system there is no movement in space. Instead there is movement of space. Thus the conicsectional iterator promotes conic section motions. Thus these surface hugging motions are hugging movements of space that are conicsectional in nature. When i jump and return to the earth it is along a conicsectional curve. Einstein introduced the notion of curved spacetime, but i use a notion of a conicsectional iterator. All motion will in this view no matter how chaotic be some combination of conical sectional motion. Variation in density is involved and proportional to the amount of energy in the space under equilibrium.

[jehovajah]

If i applly this conception aximatically, that is to say as a first principle when analysing a physical event i get atural results in thought experiments.
So throwing a stone into water means the entry velocity and pathe will be parabolic, the movement of the water will instantaneously be conicsectional and should from this initial condition lead to conic sectional results in the first instance. I would expect the water surface to be pushed apart and the underlying water to be radiated downwards in conic sectional motions.

 The sufface would be impelled outward in an elliptical shape depending on how the parabola of initial motion lies on the cone. The circle-ellipse will be the intersection of the cone of the parabola with the surface plane of the water and will move in such a way as to realise this cones surface.

Consequently when energy density equilibrium is achieved by instantaneous viscosity friction forces the motion of the object will become either instantaneously zero or uniformmotion in the direction of greatest instability. The cone of the opened surface will invite conicsectional motion as it returns to equilibrium above the sinking object. Thus a pattern of spiralling vortices will radiate inwards and outwards at all levels .A radiation of concentric circles will form around the site of entry, high impact cylinders of material will form aeound the main instantaneous vortex and die away parabolically radiating hyperbolically. At the initial instance of impact as the water surface is first broken a hyperbolic compression wave radiates out depending on the angle of incidence, becoming ellipsoidal or circular the steeper the angle of incidence.

The conicsectional motions also generate horizontal vortices within the "splash wall" which dissipates the viscosity energy hypebolically into the surrounding denser material. The high energy instantaneous vortices on impact create a jet which shoot some of the less viscous surface up out of the impact site back along the parabola of entry

[jehovajah]

just now had  great thought that given a straightening out of mass/density and friction/viscosity and definitely removing the "nothing" absrraction for space, conicsectional motions and curves could combine elecromagnetism and gravity through variation in density and viscsity. Thus light for example would be high energy hyperbolic radiations in a rarefied density medium with very low viscosity. The density will be assumed at plank length sided volumes and the viscosity will be proportional to the planck length distances between between these densities.

[jehovajah]

Of course i have to take into account the redefinition of "time" that conicsectional movements imply when thinking about velocity etc on a conic surface. Conic sectional movements have a repetetive movement called a period. If i draw a "grenwhich" line from the apex of the cone tthen i can charaterise this period as a distance moved on the surface of the cone that intersects the grenwich starting from the grenwhich. This imediately distinguishes circles ellipses and spirals. Of the three spirals are distinguished by a variation in this distance due to the loop not closing and so crossing the grenwhich at a different distamce s along the grenwhich as measured from the apex.

Now to construct our current notion of time i need motion. In set FS i always have motion as an entrained result of the iterator the universal vortex. The final thing i do to construct time is to take a standard period at a standard value of s along the grenwhich and use the observed motion there to define a unit of time. I then use fractal processes to define my subdivisions and multiplications of scale.

What happens if i use a spiral as a standard period? The time i construct from that will have curious properties but what? and remeber i can not talk about standard speeds as ths is meant to define a standard, but i can compare and contrast relativistically between a standard defined say on a circle with circular motion.

These proportional comparisons would come under the earlier discussion about equivalences, and are a feature of the definition of standards. So mass for example as defined is in reality a standard density and force is in fact a standard pressure, but that is not so easy to see until i define pressure using newtons axiom for motion under force which has the word "impressed " included in it.

[jehovajah]

Now helical curve/motion has a special relationship to conicsectional curve/motion in that a helix intersects a conic surface in one or 2 places, and if we extend the conic surface through the apex then always in 2 places.  However there is a special case when a helix is tangental to a surface i the  cylindrical direction of the helix. in this case the helix can touch a conic surface in one limit point or along a conic sectional curve of limit points.

the other classical object that interacts with a conic section is a sphere and a spiral on the surface of a sphere. this again intersects a conic urface in a maximum of 2 places with tangential limit points of a circle or at least 1 limit point. I use the term limit point to emphasise the iterative nature of these observations, but all is in fact iterative .

Now the intersection of these surfaces with the conic surface in general is giving a conic curve or a pair of conic curves which are linked by the intersecting surface. These then are candidates for linked or entangled motions and probabilties and also tunnelling effects of electron  holes.

In my observation this also explains the observation of cylindrically stable regions inside a vortex. As a geometrical space object the surface of a cone rotating will entrain all internal cone surfaces. which i will observe as stable cylindrical regional motion which is a result of stable helical motion on a cylinder at that regions conic surface reference s along the greenwhich line. This will also be affected by the viscosity of the space being rotated, so that theoretically infinite helical motions will devolve into a chaotic cylindrical surface of a given thickness and same viscosity .         

The observation of tangential helical motion/curve to a conic surface also allows me to see a helical motion radiating out from a conic section impact event along with all the  conic sectional events. However due to the stability of the helical motion on a conic suface this is likely to be an extremely short radiation close to the origin of impact. The helical cylinders however are likely to propogate along the vortex wave propogation effect that radiates out from the impact region,decaying under a conic sectional curvr law, probably hyperbolic . 

[jehovajah]

I was going to put this inthe above post but i will isolate it to this one.

As mentioned i have thought that abstract concepts of force and  and mass are not accurate concepts fr the set FS, because i cannot define mass without volume, or force without surface area of application. For me Density and pressure are more applicable conceptons. a;ong with this i generalise friction to viscosity and generally adopt the fluid mechanical descriptions.

The thing about using this paradigm is that motion of dense objects in an equilibrium pressure field is towards regions of lower pressure, hus pulling a dense object in a pressure field which is in equilibrium is creating a small region of lower pressure into which the dense object is pushed. Depending on the viscosity interaction between the dense object and the pressure fieldis how much and how fast the movement occurs. The natural inertia of the dense object is then a combination of the underlying iterator entrainment and the equilibrium this sets up within the variable densities region and the viscosity of the pressure field medium the dense object is in.

Now if in general movement is toward lower density how is condensation possible? My only suggested solution is in the behaviour  of conic sectional motion. Providing a vortex motion exists at a centre in the lower density and that vortex motion is of a higher speed than the environmental vortex entrained motions the space will condense around that lower density core until an equilibrium is reached between the vortex energy and the surrounding low pressure field energy.

If this is the case then the core of a condensed object should have a lower density than the surrounding "medium" which in turn should have a higher density than its surrounding medium enabling an equilibrium/ stability wave to be established. Thus peak density should occur between the centre and the limit surface of the condensed object and should be susceptible to the conic section curve motions in the system.

Why in lowest density space do objects condense into a spherical shape given the the underlying conicsectional curve motions? My suggestion is that the shape is not  sphere but a torus but with a high speed vortex at the core making for a closure at the vortex tops/bottoms in a dual cone i apex arrangement with motion being directional in that condensing motion occurs on one cone system and evaoration/sublimation on the opposite cone system   balanced by an anti spiral interaction going the otherway dealing with the low density material in the system which arises from interactions of different densities "releasing" higher density material back into the void, but as a superfine spray or mist of enormous velocity.

These incredible densities with such enormous velociies are the basis in set FS of electromagnetically discrete behaviours, and are way way way below sub planck length in dimension which is one reason why we generalie them as field effect phenomena. I might add hat a candidate for aether they maybe,but aether is not a needed concept im set FS as i do not have a "nothing" to fill. All of these effects are just variations in density motion viscosity and conicsectional and helical rules of motion under a universal vortex iterator,pressure being a consequence of motion variation within and on the surface of a region. The basic axioms of geometrical space being extension along conic sectional curves, rotation around axes that point along or tangential to conic sectional curves with no fundamental orientation all being relativistic, and finally iterator entrained motion along conic sectional curves due to a universal vorticular iterator which generate helical and toroidal curves as tangent to or intersections with conic surface curves/motions.

[jehovajah]

According to Bayley, the target of this scene is projective geometry, a subject that involved concepts that Dodgson would have found ridiculous, particularly the "principle of continuity." Jean-Victor Poncelet, the French mathematician who set out the principle, described it as follows: "Let a figure be conceived to undergo a certain continuous variation, and let some general property concerning it be granted as true, so long as the variation is confined within certain limits; then the same property will belong to all the successive states of the figure."

When Poncelet talked of "figures", he meant geometric figures, of course, but Dodgson playfully subjects Poncelet's description to strict logical analysis and takes it to its most extreme conclusion. He turns a baby into a pig through the principle of continuity. Importantly, the baby retains most of its original features, as any object going through a continuous transformation must. His limbs are still held out like a starfish, and he has a queer shape, turned-up nose and small eyes. Alice only realizes he has changed when his sneezes turn to grunts.

[jehovajah]

As usual Taking the time to question fundamentals produces interesting results. This is the periodic motion our standard of time is constructed from. Of interest is the helmholtz coils used to trap the caesium particles: another spiral!

http://www.nrc-cnrc.gc.ca/eng/projects/inms/fountain-clock.html.

So the iterator in the system is a stable quartz crystal resonator of 5mhz, so how are these mhz measured without a definition of time? Not obvious but the solutions discussed here are iterative solutions! Start with the old definition and iterate it to the new more accurate definition.

A feedbackloop is an essential part of the design and formally represents part of the iterative process within the design. the motion of the particles are conic sectional motions and the force used is clearly referenced as a pressure. The density within the system is controlled by a cooling system using lazers to effect the necessary condensing forces within the magnetic  helmholtz field which is spiral in nature. 

[jehovajah]

So watching Richard Hammond's invisible world with interest and noticed the raindrops and milkdrops sequence. Surface tension and viscosity are related. The spherical shape maintaining its shape at freefall speed depending on droplet size and air density and viscosity is interesting,as was watching it disintegrate when equilibrium conditions were exceeded. The instantaneous shockwave curve seemed to initially be a bezier type curve of a high order before being overtaken by a radiating conic curve going from a bezier suface to a conic surface. By this i simply mean that the underlying curves for thr observed shapes moved froma bezier description toa conic sectional one.
I noted that the milk dropped vertically followed a curve that was conically either straight on the conic surface or direct from the apex of a conic surface through  evry connectected entrained conic apex within a given conic surface. therefore iwas eager to see if cylindrical and helical motion ensued on contact with the viscous surface. Further analysis is required but it looked consistent with expectations, with the elastic bounce effect being entirely consistent with anti vortex behaviour at the instantaneous centre point of impact.
 The pressure shock wave was also evident in the water surface and quickly seemed to damp whether that was hypebolically or harmoniously requires further analysis.
I understand references to mass as a volume reference to a dense material and force as a pressure per unit area but wonder if pressure per unit volume might not be more in keeping? Although force is a derived unit as is pressure as is acceleration, and really is a quantification of motion and arises out of observations of motion, i feel that the conditions under which the observations took place are not fully reflected dimensionally in the definition of force if the definition of mass is changed as i advocate to a density per unit volume of some substance based either on Avagadros constant or a Planck unit.

Things in the set FS are ordered by the conicsectional ,helical and spherical curve behaviors entrained by the universal vortex . Anyone of these curves can destroy the order of a system if they move in a non equilibrium direction. In that sense order and with it life and computational consciousness is very fragile,and it is very reassuring to see a strong region of coherence in vortex behaviour.

[jehovajah]

i proposed somewhere on a physics site that the proper paradigmatic medium for light waves is light, just as ocean waves are in te ocean etc. So now the shock wave it seems to me represents a boundary condition for any advancing medium with idiosyncratic wave behaviour being either side of the boundary. The boundary in set FS is always a fractal structure and thus represents intensified energy conditions in this structure. I would expect light to produce a shockwave in its advancing edge and thus as is currently stated this advancing edge would travel at or above the speed of light with the following wave behaviour averaging out at the speed of light. The advancing shockwave is what would initialise all the properties of light such as reflection, refraction etc.

Of course this would be generalised for all electromagnetic radiation.

Now a question: are the magnetic field lines around a solenoid a helical torus structure? This is to say would a test electron hypothetically take a helical path around the solenoid passing through the open or vacuum core of it repeatedly as it traverses the toroidal surface ? And would the toroidal surface the electron travels on helically through the core, depend on the energy of the electron?

 

[jehovajah]

I want to redefine force as pressure per unit volume P with the actual definition being

           P=

Where p is the "gas" pressure and the is the measuring volume of the gas and A is the surface area of the measuring volume.

this then leads to
 
P =  (1)

For a constant pressure this has a solution

P= == =

Where is the "gas" density and is the observed mean acceleration of expansion in the gas which integrated over the surface area of the measuring volume will be a constant that is non zero but small. This represents an inertial /equilibrium solution with non equilibrium or non constant solutions resulting in resolved component acceleration.

by Boyle's law we can link  to the mean gas temperature, T:=k.T

I think that the simplest sequential method of integrating over a surface is by means of a surface spiral. Any space filling curve could be used.
The is an operator in this instance as it is undefined for calculation. As soon as a product can be passed into it it can receive a defined calculation procedure, in this case sum spirally around the surface area. As the product admits no change on an infinitesimal basis if is constant the result should be . However the mean acceleration is not a constant and so a small variation in the sum should be expected as the infinitesimal areas go to limit. However i would need to have a function for acceleration of expanding gas based on surface area expansion.

[jehovajah]

I want to redefine mass as density per unit volume D which is written in this way:

D= = ==

Where is Avogadros constant is the volume of the substance containing Avogadros number of elemental or compound particles as a molecular weight in grammes and is the measuring volume for the standard and molecular weight includes atomic weight for elements.
This is an iterative redefinition of mass so that the initial definition of mass for calculating the atomic weight is the current one.

[jehovajah]

The cultural iteration +1 is an example of specialism, in that it defines  counting as addition. A similar structure defines multiplication as counting in fixed value groups; +2,+5,.... being the 2 times and 5 times table numerals. General addition then is characterised by adding variable value groups and s called aggregation.

Similarly the iteration -1 defines counting in reverse as subtraction and -2, -5, .... as division which is subtracting fixed value groups and noting the remainder and how many subtractions took place. `in multiplication we note the final value and how many additons took place. General subtraction involves subtracting variable value groups and is called disaggregation.  

Now mathematicians have modeled a form of aggregation which is variable but in a specialist way : the variable groups are formed by  "multiplying" each variable group by the "group size", so these are termed power law additions and condense the notion of addition into a new process called multiplication by a base.

Distinguishing multiplication in this way tends to obscure the essential iterative base  of it so that it seems to be a binary operation when in fact it is a unary operation on a binary base operation of counting.

 

Binary operators require 2 elements to define there operation. A unary operator requires only "one" to act as a modiier. So the operator 1*, 7*,... makes no sense without the base binary operation +  Which instructs me or another to combine 2 elements by adding group size to a previous named element and to continue to do so until told to stop, calling out the names as i go.

Aggregation is specified by the 2 element names and a counting iteration starting at one numeral and continuing +1 the second element numeral of times. In the base operation the 2 elements define an iteration instruction or procedure one being the starting numeral the other being the iteration control numeral. This says stop after this numeral has been called out in the counting iteration while you have been continuing the initial counting iteration from the name of the first numeral. Thus 2 iteration counts are established the second controlling the first. This can be reduced to one counting iteration mod( the size of the aggregation of the 2 elements) . So 2+7 is not only 9 but also +1 mod(9) which is count or iterate forever mod 9 .

Multiplication operates on this binary base not on another element of a set, so in this strict sense multiplication is a unary operation on a binary addition iteration, groupsize . The elements of the set being operated on are usually 0 and .

Multiplication by a base is discrete as far as counting iterations go, as each numeral in the value name space has to be arrived at by its own independent counting iteration. There is no counting iteration that gives the sequence of numerals  in a power law addition sequence. Because of this power law sequences are useful as linearly independent basis vector forms or polynomial forms. This type of sequence of aggregation  products is a fractal iteration arrangement allowing independent counting to take place in each part of the sequence such that the power numerals record the cycles of the itereations in the preceding power numerals, ie 2 based power laws  can record  normal counting as cycles of 2 iterations.

The reverse of this leads us to look at disaggregation to find this special form of power law subtraction and in fact we find the basis for understanding this as a new process called division or better inversion. For inversion to make sense i have to have a constructed set of counting names that deal with the enumeration of splitting or breaking a boundarised object into pieces either by a natural destructive action or as a biological division process. This started to be attempted under the name of fractions but continues under the name rational numbers. The construction of this type of numeral space is a power law tour de force.

[jehovajah]

It seems clear to me,but then i have thought about it for a while, that Newtons laws are of motion but that motion is de facto  Euclidian which is why it is not as accurate as Einsteins which appears to be Riemannian.  The more general Riemann geometry is not hard as it is the geometry of plants and trees,and bulbous fruits etc. However we now have fractal geometry which is more general still and may lead to a more accurate description of motion.