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

[jehovajah]

An interesting theory of the fractal description of notFS. 


http://www.euclideanrelativity.com/idea/index.asp

[jehovajah]

So to follow on from a vortex process operating in an iterative way in set FS leads me to re emphasise newtons laws of motion in FS. A body continues in a state of vorticular motion iteration by iteration unless impressed upon by a force. A body impressed upon by a force changes its motion in proportion to the force and along the vorticular path of the force. And finally the impressed or drawn force is opposed by an equal and opposite force acting on the bodies involved  .

Newton accepted the states of motion and rest, but in set FS the iteration of the vortex is the source of all motion by fractal entrainment. A body is only at rest to an observer with the same vorticular motion, and i will discuss this more when i consider equilibrium and inertia.

A body at rest is in a state of force equilibrium which is to say that all forces acting on the body cancel out. Only when this condition fails does a net force impress upon or draw upon a body in a "right" direction in the newtonian sense. This right direction is not a straight line but a perpendicular direction to the place of contact of the force. In a vortex field this approximates to a straight line as a first order approximation, but it is more accurately a logarithmic motion.

That iteration is the fractal entrainment for motion will be a basic axiom of the set FS and will be a development of axiom 2.

[jehovajah]

i want to note 2 things also for further development.

The cultural iteration '+1' the counting or numbering, that is number naming iteration is based on a very 'natural' process of aggregation as well as a human process of collecting, sequencing, sorting and aggregating. Subtraction or the '-1' iteration is the logical opposite and is based on disaggregation. Natura/Yh~h however provides another formative iterative experience that of splitting into parts. This is not just a biological process but also a physical process of disintegration, found when say a plate drops and breaks into pieces or a rock is shattered into smithereens. It is important to note the wide generality of these 2 processes on which our cultural iterations of addition and fractions  are based. The notion of proportion i have discussed under equivalence in a previous post. it would appear that disaggregation is a similar process to disintegration but at a different scale. The scale however is a psychologically determined unit, which is to say it is a perception determined unity. This of course is just another way of describing the iterative processes involved in the perception process discussed in an earlier post regarding axiom 1 .

Our notions of relativity are inherently based on these iterative perception processes.

The second important point derives from studying the formulae and programmes to produce the mandelbrot set and i will see if it extends to all fractals: a simple aggregation rule is necessary and a simple splitting process is required. To these two requirements a simple iterative force is added to drive the two procedures repeatedly- now i am thinking of waves on a sandy sea shore! By extension all the processes of erosion. In the sea i can visualise a process where single celled creatures are clumped together and dissolved or pulled apart periodically, that is iteratively. When i look for these three things i find them abundantly around me, thus i expect to find real life examples of fractal sculpting. Fractal sculpting is an analogy i have been developing in order to understand what the programmes are doing and how.

So now i am thinking of limestone caves with stalactites and stalagmites as fractals and showing fractal sculpting and structure.The periodic flow of groundwater swelled by seasonal rain, disintegrates the limestone by dissolving the calcium carbonate- that is splitting at the molecular level. Then evaporation aggregates the calcium carbonate at the molecular level causing precipitation of the calcium  carbonate. This means that stalactites and stalagmites are the escape ejecta of the fractal which will be in the cave roof structure. So like rendering can add ornamentation to a fractal by visualising the escape trajectories, or escape speeds stalactites and stalagmites do the same thing.

Another fruitful area for search will be dunes, and wind erosion sites.

It should be possible to use a 2 or 3 variable function to describe the mapping of condensation or crystallization of  molecules in a suspension medium onto a substrate and then through julia iteration to build the shape of the crystal or condensate. Julia iteration is particularly apt for this type of model as a linear function is always a good first approximation.

[jehovajah]

Equilibrium http://en.wikipedia.org/wiki/Mechanical_equilibrium and inertia http://en.wikipedia.org/wiki/Inertia are 2 sides of the same coin in set FS.
The first thing to acknowledge is that Newton's Laws of motion are the first complete axiomatic field theory. The axioms deal with the elements of motion and force.However the notions of motion and rest are not explored directly in the axioms. Newton was keenly aware of relative frames of reference but still accepted that the true frame of reference would be where his god lived. This he thought of as absolute reference frame in which all measurement and force would have its "true" value.

By stripping down the system to these few axioms, Newton followed euclid in his classical organisational approach to the study of the forcefield and its effect on an object. But newton and his contemporaries had to reduce an object to  point mass to enable this classical euclidean arrangement to utilise as much of euclidean geometry as possible.Mass therefore lost its significance in the pure axiomatic theory to be interpreted as momentum and inertia. Motion also was not defined or explored but assumed and assumed to be modifiable by an invisible hand pushing or pulling through bodily contact.

Newtons third axiom deals with mechanical equilibrium at the points of contact and this was to be the basis of his notion of rest as others derived it. Newton himself saw rest or uniform motion as special solutions to equations making them equivalences in some way that would not be explicit until Einstein's axioms of relativity and relativistic frames of reference.
In setFS motion is the result of iteration and is a vorticular displacement at every scale by fractal entrainment. The fundamental attributes of  Fractal space are axial orientation, axial extension and axial rotation, which is to say spinning around an axis. From these three i derive a special axis for orientational rotation which i define as a norm to the plane of rotation of an axis. A rotating axis does not have to rotate in a plane but my definition of orientation as a fractal uses this special case to exemplify it. Simi8lar dwfinitions have been culturally used to define angle measure etc.,and without further elaboration i use a measurement fractal to measure axial extension, an angular measurement fractal to describe orientation rotation in a quantised form and also axial rotation or spinning. I further establish a foundational orientation and a foundational rotation called the cycle which is measured in radians as 2π by definition. see polynomial rotations
Of these three motion attributes of space orientation is the psychological one in that i as an observer choose outside of any system of axioms i develop a foundational orientation which i call and experience as "fixed" even if it is not or it is only apparent relative to me.

Apparent extension can be modeled in setFS by the relation
                         
                   apparent extension=k*ln(local extension)

which is to say that the static observer sees a different measurement to the observer moving in the direction of the extension at the pace of movement of the extension. In other words, the observer in the moving reference frame sees a different measurement to the static observer within whose frame of reference the other observer is moving.

The iteration producing the vorticular field of motion in setFS is not identified here but my intention is to define it in sub planck limit  space and or super universal space. However the motion fractally entrained in this iteration is thus by definition inertial.

The motion is also held in equilibrium by the fractal entrainment acting on a region in a coherent way. Thus the disturbance of this equilibrium leads naturally to forced behaviour and inertial or momentum transfer. This occurs not only in extension but also in rotation.

[jehovajah]

Just a quick note for further development. Taking the planck limit as the size of a "pixel", i will describe geometrical space as being a pixellated structure or organisation or arrangement. I propose that this is all that there is and so that there is no empty space in which this structure sits: just to be clear and logically finite. An alternative proposal would be to have this structure nested within an infinite fractal structure in a self similar way with boundarisation at iteration level boundaries. I would prefer the infinite iteration level model, but this logically implies that space is this infinite structure within which my geometrical space sits,but it also implies that it is not empty.

Whichever proposal i adopt, motion in geometrical space is through the pixels switching on and off or varying the energy levels they emit or absorb. This is radiative emission or absorption by neighbouring pixels. A collection of pixels cohered in some way will take on the organisation, properties and attributes of quantum space at some iteration level and then at some other iteration level the properties of classical relativistic space. The pixels at the planck level thus do not "move" in the classical sense but impart  movement to objects at a higher iteration level through boundarisation changes.

The "pixels" represent the measurable level at which the iteration process is "impressable" which is to say that my definition of any universal iteration processes cannot be distinguished below this length. This iteration process then i am thinking of as  occuring at sub planck lengths and the the planck length pixellated structure of the universe sequences this iteration process fundamentally, through a radiative process.

[Timeroot]

...but that interpretation removes Gaussian (or is it Netwonian?) invariance. By viewing the Planck length as a pixel size, you make certain directions behave differently than others. Really, the Planck length works more like the smallest distance you can measure... but then, you can turn 10 degrees and go back to where you came, and "probably" you'll be back where you started.

[kram1032]

the planck lenghts seems to form a grid, but a totally isotropic one.
There is no such thing, yet known, in 3D though. - Well, yet known to me^^

However, I played a bit with spheres and based on the fact that an infinite dimensional sphere neither has an infiniteD Hypervolume, nor a corresponding surface (the general sphere volume and surface formulae have peaks at 5 or 6 dimensions or so and then steadily decrease...), together with an assumption for an 11-or-more-D Universe/Multiverse and the not yet found solution for higher-than-3D- close sphere packings, I had the idea, that maybe at higher dimensions, there is such a packing

[jehovajah]

Hello timeroot and kram1032 and anyone else who would like to comment or contribute. Your thoughts are very welcome. Please do not be afraid of the title maths as i pdsted recently, maths is a certain type of thinking based on a subset of language which focuses on iteration and the product of iteration namely fractals. The fact that i write this definition is due to the work of artistic people here and through the generations and throughout the worlds languages.

Language itself is a model of my experiential continuum and a model of my conscious and unconscious experiences. My thought today is about identity, that which i call "i". Simply put and to be elaborated at a later post : identity and interface are equivalent notions. Thus using the computer programming paradigms it is possible to trace the development of identity over time in the model of the development of programming interfaces. We recognise this 'identity' immediately: as a mac user i know a mac interface from a windows interface! Of course the interface itself and below/before the interface are collections of seething sequences of iterative processes.

What pops up on my screen is very much a symbiotic relationship between these iterative processes and the grid structure of the display. More relevantly the display algorithms and mechanisms that drive the display grid.

My point about planck length is thus illustrated by this paradigm: below the display grid pixel size and in a sense before and beyond it much is going on that i am not aware of directly, and the interface only gives me a coherent selection of the maelstrom that lies at the heart of a processor and memory configuration. I do not mean to rely on this paradigm,but to utilise it to convey a notion that is guiding my exploration. That is why i welcome any and all comments clarifications constructive criticsms contributions and collaborations!

The set FS is a model of notFS which i am building and exploring in an axiomatic approach, so i am looking for axioms to further the 11 or so that i have outlined so far. These represent my best guess at axioms and are open to modification.

For example i recently realised that i have not fully explored the basis for the SI units for time. Time for me is any periodic motion. By a fractal splitting method i develop a scale for fractions of that motion and by fractal aggregation i extend the scale.Quite naturally i transpose this scale into a geometric linear form,but often without acknowledging that this is a construction which may or may not have geometrical validity. The fact that it is culturally endorsed does not make it suitable for all uses. For example i am given to understand that the western concept of "time" is linear, while the indian and far eastern concept is cyclical or circular. With this conception i am able to trace the western notion of time back to persian Zoroastrianism through so called gnostic groups.

My point is everything is not always as it seems and thus it is valid to explore and question and propose and construct. In any case its fun too.

[jehovajah]

Bootnote

Wen-Hann Wang may be the director of research for both circuits and systems at Intel Labs, but he made it clear where his sympathy lies when he said: "A circuit is at the bottom of the totem pole. It does all the work and no one appreciates it. The system gets all the glory."

[jehovajah]

I make no claim to originality, but rather seek to collaborate and explore. In this light the following reference highlights similar thinking about the foundations of Einsteinian space time (which term by the way i have traced so far to Wiliam Hamilton as originator). Because of its overt "mathematical" origin, i justify its inclusion in the setFS exploration.
  http://theresonanceproject.org/research.html

[jehovajah]

My idea of planck length pixels forming an implied grid is just one possible outcome for sub planck length processes. It may in fact be a more kinetic structure for which a thermodynamic model would be more appropriate. However it is to be noted the fractal self-similarity in these suggestions, which is to say that a scheme for a larger scale is being applied to a different scale. How can i do other? and this again begs the question of the construction of all this fractality!
Never mind, i live in acceptance of my place in the scheme of things!

[jehovajah]

  That's what i am talking about,Willis!

This is my preferred fractal universe structure seen in a plane. In the universal void this would extend in any orientation one chooses and look like a great vortex. From the apparent centre of the vortex an anti twisting higher rotational vortex would jet out, and in the centre of that a higher rotational vortex might jet in or remain stationary. Each level of vortex will establish a coherent set of cylinders that will allow for stable universes of long endurance.

What was that bit about a range of frequencies to bring out the details?

[kram1032]

I really wonder what paddelbrot could do with the 3D variants. He seriously mastered the Mset and several alternative formulae.
That one's just breathtaking!

[jehovajah]

Yes! Very creative images in his gallery!

The thought occurred that iteration requires an iterator. So to make that explicit for setFS i realise that the only natural/Yh¶h iterators i know are circular or elliptical or vortex motion. All of these are conical section curves so i especially choose the conical helix as the motion iterator for setFS as i think the other two periodic motions can be derived from it. i will propose this as an axiom for set FS When i have tidied it up a little.

The image of the fractally structured univese is so suggestive as it indicates regions flowing into other regions which are bounded iteratively by vortices. This to me is a powerful realisation of the big bang and inflation happenning within a local universe separated by turbulent boundary conditions from neighbouring universes.

[jehovajah]

A property of all conics is that they are surface-hugging. By this i mean that they lie in a surface that is conical, so i would expect a kind of coherency in their "action". I would expect them to form discrete layers for example and any spiral to gravitate toward a nearby surface to maintain ts behaviour, I would also expect them to be space filling at all scales while maintaining extreme coherency. Thus a system of circles will densely pack but still be discrete. This will also i suspect be true for a spiral leading me to suspect  that it is possible to have a spiral counterspiral arrangement which would enable the energy concentrated at the apex to be dissipated back down through a spiral in a densely packed spiral system. So perhaps the power in a maelstrom may exhibit this phenomena  .