END OF AN ERA, FRACTALFORUMS.COM IS CONTINUED ON FRACTALFORUMS.ORG

Warm air  being less dense rises, or is pushed up by denser air pressurising it out of the way, and the only way is up.

Well in an earlier post i described this as an inadequate description of the dynamics, particularly as it did not explain angular momentum of the warm air bubble. I did not explain angular momentum then but simply transported it from on existing angular momentum in the heat/ light source to a resultant angular momentum in the warm air bubble, that is a relativistic motion transfer of angular momentum fom the light to the gas.

Now of course the traditional explanation of the angular momentum of the warm air bubble is the Coriolis force transmitted from the earths rotation, and this is used to explain and describe water behaviour as it flows down a plug hole.

Conservation of angular momentum is a curious thing. Newton could never quite get to grips with it, and although Bernoulli derived differential equations for fluid flow he could not understand the dynamics of of water or air flow because the computation was so complex. Even the Navier Stokes Equations make simplifying assumptions to describe fluid and gas behaviour only in very limited and tightly bounded situations.

We have very powerful computing platforms and they can give simulations to flows that are very realistic and descriptive of behaviours say in a nuclear explosion event, but they cannot yet predict the weather, or an actual outcome for a given real life explosion. Still, they are good enough for short scale short term description and analysis and prediction.

So it is safe to say that angular momentum is an unfamiliar aspect of dynamic systems which has only recently begun to be understood and applied to describing behaviours of systems in dynamic equilibrium, and many assumptions are still made in its application.

The chief assumption has been Newtonian in classical Mechanics, but this has been supersceded by a quantum description of angular momentum and spin with a more elaborate set of vectors and descriptors.

So Angular momentum simply describes one part of the fundamental equilibrium nature of the motion field.

I cannot say more than it is axiomatic that everything is in motion and that motion must be in any direction and in any rotation.
My interaction with the motion field through the Logos Response generates a set of relations inherent within my neurology which allow the development of a spaciometry and a self within a spaciometry to exist.

The spaciometry and the self are an indivisible construct with many attributes including the perception and the perceiver and the projection and the projector.

That being said the set of relations imbue an order  continuum with static order as one pole and explosive disorder as the other and static and dynamic equilibrium ranging between the two poles. As a secondary response the self imbues a complexity continuum which relates more to ease of measurement, comparison, description and analysis than to any actual order or disorder.

We also attribute many many other things to our self /spaciometry construct which influences our experience of it.

Thus dynamic equilibrium is a concept of order in which we note that a moving object continues to move in that path whatever it may be unless it is perturbed by some pressure. And static equilibrium is that order in forms i see that tends to remain motionless relative to me and yet press upon other forms through contiguity. I may discount a static system as such until it moves in some way revealing that it has in fact been in equilibrium! Thus i may as well assume that all systems are in dynamic or static equilibrium if they possess some form of order that is not identical to the poles of my measurement continuum.

This makes an assessment of order just that, a subjective assessment, and i do not have to assume that the abstract poles of my continuum would ever be realised, even at absolute zero degrees Kelvin!

Thus the notion of complexity becomes useful for those dynamic systems which appear to be disordered but seem to have some purpose and those Static and dynamic systems that appear to be tightly ordered but exhibit unpredictable behaviours.

Predictability is an expectation notion we develop from experience which covers interpolation and extrapolation of behaviours. Behaviours are the actual motions we experience .

There is one equilibrium system that is assumed but not properly described and hat is the rotational equilibrium system. I assume it under the dynamic equilibrium, but it deserves a special place as it is fundamental to any understanding of dynamic equilibrium. In fact i do not think dynamic equilibrium can be properly understood without it.

If you have ever seen smoke rings or bubble rings you will understand what a fascinating example of dynamic equilibrium rotational equilibrium is.

So now i have the framework for the motion field set up it is easier to explain that angular momentum is conserved as part of a dynamic equilibrium system within a motion field. Thus such systems do not exist in isolation as it is commonly posited, but in a context which involves  whole network of equilbria. This is usually called the inertial frame and in itself allows the orbits of planets to continue as they do, and posits an equilibrium pressure to maintain all motions. Therefore motions do not alter unless the equilibrium pressures are altered. And if the equilibrium pressures are altered the  motion is apparently generated by a "force" , but in fact is generated by a pressure variation to which the equilibrium system responds to restore equilibrium.

However if the pressure is such as to destroy the equilibrium system the motion is not free, but rather subject to the larger equilibrium system it is in and to any adjustments that cascade through the system to establish a new equilibrium state.

Thus motions of regions are governed by a network of equilibria, and these are in a fractal pattern throughout space.

I believe Newton derived his laws through trigonometric manipulation and reference, and thus did not posit an eternal motion in a straight line as some later interpreters did. Rather he referred directly to the tangent to any curved or linear motion and the right angled triangle which defined the tangent. By this construction he could compound the curved motion from infinitesimal tangents and an orthogonal string acting as means of pulling an object round a curve.

While this was adequate he was never personally satisfied with the explanation, as it required action at a distance to explain planetary motions. Cotes he hoped had an explanation not requiring this based on a logarithmic relationship not a reciprocal one.

The logarithmic relationship linked directly to Kepler's law and thus linked the two formulations in an interesting way involving imaginary magnitudes and direct trig ratios. However Cotes died before he could explain much of it to Newton.

I feel Cotes would have made Newton more at ease with his analysis by linking it easily to a great Authority to Newton, namely Kepler, but i do not think they would have advanced much in the explanation of gravity beyond that .

Descartes had the prevailing idea of Vortices, but no real mechanism or rigorous description. Cotes and Kant both showed that it was not as sound or measurable a theory as Newton's, but Newton had no "medium" to apply pressure to keep regions in place, and so posited internal attraction between bodies acting at a distance. Very unsatisfactory, and in fact embarassing to him, but still far superior to Descartes theory of vortices.

Angular momentum therefore has this undefined quality that all these notions had until custom and practice settled them to their fields of applicability.

For me the motion field view of space provides all the necessary elements, without having to have an aether or action at a distance. I posit only a fractal distribution of equilibrium systems throughout space, their status of being dynamic or static determined relative to the observer, and a special fundamental dynamic equilibrium state known as rotational equilibrium.

In spaciometry rotation is formally described as apprehending the boundary. Thus traveling along a straight line boundary is a rotation, as much as watching a bounded object pass you as the observer in a straight line relative to you. But complete rotation involves apprehending a "closed" boundary.

This is a fascinating property of dynamic spaciometry, that rotation actually has different attributes the tighter the closed boundary!

We observe and experience the differences and can measure them in spaciometric terms, but i can hardly believe my eyes at what i observe. Newton too, was fascinated by rotation and made many observations of it, including his bucket on a rope experiment. What few realise is that Newton actually sun the bucket of liquid about it axis formed by it hanging on the rope, observed the vortex behaviour of the liquid and then pulled the spinning bucket around in a great circular motion, noting what happened to the vortex in the bucket!

He puzzled over the behaviour of liquids in this vortex system for the rest of his life, but he did predict tidal phenomena on planets in such a system!

Newton could not get the bucket to keep on moving round him in a circular motion without constantly tugging it round, that is providing a pressure along a rope that pulled the bucket toward him but also slightly past him, and it did this by itself spinning, So Newton observed that if a spinning centre could act on a body at a distance it would drag that body round the spinning centre. If the body is itself spinning then its behaviour is relative to its own axis of spin combined with the external axis of spin. Thus the water in the bucket still spins around the original axis of spin, even if that axis is spinning around some other body.

This clarified things in his mind a lot but had one flaw: there is no string holding planets in circular motion! Therefore, somewhat unsatisfactorily he had to posit an action at a distance.

However he had observed the vortex behaviour of liquid in a bucket, but could not relate it to his needs. We can actually now use Boyles law to explain the behaviour of spinning liquids in buckets as well as the work now done in fluid mechanics, but we have to posit a dynamic  density field with liquid/ gas like behaviours. Providing the surrounding space provides a boundary condition that equals th pressure within a spinning gas or liquid will maintain its spin. The factors effecting the spin will be the viscosity, the elasticity and the permeability and the density of the materials within and without the boundary.

Now as Newton observed a spinning system requires a spinning centre, and the question is how does this occur? The anwer is that rotation occurs naturally and spontaneously because it tis the de facto motion of a motion field .

We in fact have it the wrong way round: it is straightness and straight lines which are special and unusual.

So why is not everything spinning? This is of course a matter of relative perception: Everything is "spinning", but spinning or rotation is the apprehension of boundaries, some of which are infinitely large and others of which are infinitesimally small. The ones in between which we observe are all "spinning" at varying rates some very fast some extremely slowly. and all relative o ones own rate of "spin". As outlined above "spinning" or rotating is not necessarily circular. One may "spin" around the block for example and complete a rectangular trip around a set of buildings.

Bearing that in mind why do we need to keep tugging objects in circular paths?

This actually relates to free choice, free will, and freedom of action within an otherwise complex stochastically deterministic system.

If a deterministic system is sufficiently complex it allows a range of behavioural choices within its overall functioning, these choices require an agent with free will. (that is self originating energy) to access, and the consequences of that interaction are free of the prevailing deterministic pattern.

Thus a fractal equilibrium system determines the behaviour of ech and every part at all scales but is sufficiently complex to allow a free choice causing a free action.

So if i want a bucket to go round me in a circle i have to chose and determine every part and component of that action. Therefore i need some string to impose my will on the bucket, otherwise the bucket will follow the deterministic route set for it by the prevailing conditions.

The prevailing conditions are hat everything moves in a rotational system in dynamic equilibrium unless altered by some other external pressure.

Pressure is the dynamic motion fields footprint in space, and the smaller the volume the higher the pressure if the mass is maintained. Similarly the smaller the volume the higher the angular rotation if the motion and mas is maintained.

The two are linked.

When i sat down this morning i was actually going to consider Newton's third law, as the basis for the notions of static and dynamic Equilibrium. I shall get to that anon.

However in glancing over law 2 i am struck s
as to the pure geometrical nature of it.
The geometry Newton uses, and therefore sets as axiomatic is that of the right or rectilinear form. This is a common Euclidean form, to which he add the law of the Cosine for a triangle hidden within the parallelogram rules.

The cosine law for a triangle is a generalisation of Pythagoras theorem.

However in introducing the notion of compounding forces (vis) Newton defines "vector addition".
Thus showing that vectors are an alternative form of good old geometry!

Of interest is the use of the word compound. At the time compound interest had been shown to be an impressive method of producing growth, but it had also been shown to solve certain mathematical problems simply. Therefore, astronomers in particular were interested in its use in describing the orbits of the planets. Newton's idea was by applications of infinitesimal additions he might compound them to the curve of the orbits. Using thi idea in part he developed the infinitesimal calculus. Of course the conception had to be translated from fiscal theory to geometry, and for this purpose Newton and other of his time used the tangent to a curve. Newton's use of the prallelogram rule, was therefore in and of itself not novel, but its application in infinitesimal proportions was, and his paradigm was compound interest.

Hmmm... interesting!

I have completed the transfer of this thread to a blog found here .

Unfortunately it does not currently support Latex.

My next step is to revise and transfer the axioms of the setFS.

I will continue to post here but my more free flowing thought will be done in the blog space i have set up.

The work i have done on polynomial rotations and The Bombelli operator will continue here on fractalforums.com and i invite again any contributors.

I thank you for your patience, kindness and interest and i  hope that my musings may be of some use to you in your fractal explorations, at least as much as they have in mine.

This is not a blog, but a thread of thoughts on the fractal foundation of Mathematics. Any child, adult, rebel or erudite scholar may therefore add to it and contribute. But i particularly want those who are artists and artisans to find utility here, and freedom of thought in their work.

However this is a blog right here, and i hope you can tell the difference. LoL!

When mathematicians were human beings they used to talk to each other.
They were people like you and me, carpenters, artisans, builders architects, singers, dancers, artists, genuinely curious people who wanted to know. So they talked. They swapped observations.

They still talk, but not in human codes!

I do not think they all mean to be rude (some do). I think they have a combination of autism and peer pressure.

So i am not surprised that a scientist who loves physics would come in from the fringe and talk in understandable terms.

Enjoy your studies!

<a href="http://www.youtube.com/v/wMm-V8JSRF0&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/wMm-V8JSRF0&rel=1&fs=1&hd=1</a>

I am currently studying Aggregation structures and the Lorenz transformation came up again.

From my view point the use of+-*/ has been so heavily propgandised that it is hard not to have a knee jerk reaction to them. This notation was only gradually introduced and standardised, and during the process certain uses were proscribed. However Hamilton shows the fluidity of thought that existed in his time and allowed these signs to be used suggestively and indicatively, but without fixed presupposition: thus + has the more general force of a connective rather than a summation, and -  a relational sign rather than a subtractive. *  has always had many representations reflecting the many rhythmical situations it is recognised in and / has the general notion of "casting out" over the notion of division, or modulo over sharing.

I hope that you can see the great confusion we have been subjected to in our mathematical education after Bombelli.

So my point today is simply that in deriving a solution to   we should not be surprised that the answer is not a number, numeral but a scalar called the geometric mean of 1*-1, because after all we derived the equation in geometry, so our solution should have geometric significance.

The fact that Euclid an others went on to abstract from geometry the notion of "numbers" is a non fact! Euclid dealt with Arithmoi, that is articulated forms, and their attributes. Some of these attributes were clearly abstracted from the arithmoi, and the greeks acknowledge such in their definitions and propositions.
it is we later translators and interpreters who have foisted the twisted notion of number onto our generations. Arithmop not only form a apprehensible basis for magnitude, but tie together all the arts and retain "vector" properties throughout mathematics. enabling not concealing the geometrical dynamism of "reality".

There is an old long held notion from the ancient past: "nothing can come from nothing". This is so patently obvious that we can assume that men as wise(but biased) as Brahmagupta understood this, and greek thinkers also. Thus to characterise  Brahmaguptas advance in deriving the "misfortunate" ciphers as part of a correct description of Brahma(, the universe) as a mind boggling concept is incorrect. I am afraid we have been the subjects of unbridled supremacist propaganda, foisted on us and later generations by those who vaulted all things greek above all other wisdoms, bigoted views which toned down the progressive link between early greek thinking and influence on Indian Culture, resulting in advances both for traditional Indian and Greek conceptions, in order to validate a then current and western view of superiority.

We cannot accuse Arabic culture of making this mistake. The Dark Ages in the west were truly dark.

a+ib, then may as well be written a,ib for all the use of the + added to it. Today we write (a,b) and add a property in the "multiplication" or more accurately group action to lay down the relationships we mean to utilise. These relationships are group symmetries which highlight particularly quarter turns around a centre and reflection in a centre, and we recognise the modularity of what we are doing. All of this is of course Euclidean Geometry but mediated precisely symbolically, and distinguishing reflection in and through a centre.
When i measure any object, i perform these group actions and modulo extractions without even batting an eyelid, and consequently i was not aware of what i was doing formerly. Thus explicit detailed notation has and should be used to elucidate my actions and activities, and is the background that supports aggregation structures, and is not to be held hostage by "number" conceptions. Dynamic magnitude is sufficient a basis on which to formulate all our mathematics.

I have completed a rough draft of the theory of space paying particular attention to the density function.
http://my.opera.com/jehovajah/blog/2011/08/12/when-younger-i-was-introduced-to-algebra
The reason why i mention it here is because of its deep fractal significance. Some very simple rules iterated again and again underlie the dense structure of space, and also the rarefied structure of space. Thus a simple fractal generator could easily model the quantum behaviour of space as well as the classical behaviour of space.

I have spent some time exploring the relevance and meaning of the √-1, and number in general. I feel confident in saying to any artist or artisan if you understand geometry in a pragmatic way especially trigonometry, there is no need to worry about i. Just get out there and do crazy things with ratios and have fun with geometry!