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

[jehovajah]

 I feel the logos response compels a spaciometry/geometry, and a geometry compels an algebra; and it is the algebra that compels a language including the so called mathematical language.

The algebra compels vocabulary, syntax. phonemes/graphemes, parsing, grammar and composition; as well as verbs and adjective/adverb reference. There are also prepositional rules and conjunctive rules guiding the use of all these elements, and of course a community of practitioners overseeing and enforcing certain cultural standards.

The fascinating thing about this all to me now is that this represents tne result of processing ratios in my CNS, neural network distributed, processing system; which iteratively samples notFS and reveals a spaciometric fractal design based on spaciometric rotation and extension.

Which seems to imply a vorticular arrangement and entrainment of all things.

[jehovajah]

So looking again at the ying yang sculpture i can now see that they are concentric rocking circles! Thus the centre disc looks like an ellipsoid because of perspective , but it also is a good approximation to the ying Yang in 3d .

http://img1.jurko.net/2554.gif

Hey  look what i have just found! http://www.geodynamics.org/cig/community/workinggroups/mc/workshops/litho2008/presentations/Tackley_MCW_Davis08.pdf/view.  chilli

Check it out. Not my spiral reference frame but inspiring to me. 

[jehovajah]

Just thought that the sin and cos functions would be a helical function when extended to 3d. Then i could use Fourier 3d helical function interpolation to model vorticular wave behaviour.

<a href="http://www.youtube.com/v/Qif8K7fc6uY&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/Qif8K7fc6uY&rel=1&fs=1&hd=1</a>




                                                                          <a href="http://www.youtube.com/v/LeIF1mG500o&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/LeIF1mG500o&rel=1&fs=1&hd=1</a>   

if you have epilepsy do not watch this one!

<a href="http://www.youtube.com/v/6l0Ul6AgBqM&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/6l0Ul6AgBqM&rel=1&fs=1&hd=1</a>


http://www.avantgravity.com/3d_gravity.html

spiral packing

  Hey just surfing man! Check out the ying yang crop circles!
http://kylepounds.org/science/crop%20circles.html

Is this guy me or what? spiralman

[jehovajah]

some more good stuff  nanomagnets


http://vimeo.com/groups/114/videos/8979430

http://www.chinesefortunecalendar.com/YinYang.htm

[kram1032]

the sin cos function in 3D is probably closely related to the complex log screw

Ha, nice
Just a few days ago, I compared a tennisball to Yin-Yang and stated, it's like a 3D version of it. I actually meant it as a joke but now you send this pdf. Nice.

[jehovajah]

Hiya Kram1032, yes very nice. 

I am looking at weaving now so some more good stuff.
baskets from spiral wraps

Also there is a relationship between the Archimedian type spiral length and the euler formula which i have not worked out yet.

[jehovajah]

Just realised that if i can define a perpendicular to a spiral loci i can uniquely reference every "point " in a region by a parameter ζ the length along the spiral and a parameter x the length along a perpendicular to the spiral at that length thus reducing 3d to 2d, just like that, as Tommy Cooper used to say. so using a regular helix and extending the radius to any length x gives me roughly

ζ=ø+ø/2π*pitch and the coordinate system S(ζ,x). with x= r the radius of the helix defining the axis of the helix. Although this is not a conical helix reference frame it is a start!

[jehovajah]

The great thing about this "slinky" reference frame is it does not have to be straight, and it can follow the contours  of an object while it is itself being referenced by another reference frame, slinky, polar, cartesian, elliptical etc. For a relativistic reference framework i need at least two reference frameworks that are linked. They can be the same but with a transformational difference, giving at least a stereoscopic view of the tensors,flexors and growors.

Now i have to say that function is a bit of an old fashioned word and so i guess i am going to use Rule as a simple designate, and Linking as another. Then i can have linkings which are the rules if there is no simple sequential or programmable action or iteration on the elements of one parameter relating it to another parameter and its elements.

So i can still have fun linking things even if i cannot give a concise rule of what is being linked: these will be like parameter data sets.

[jehovajah]

Here is a question for cosmologists. If a vorticular spiral locus , that is a conical helix locus is used as a reference frame does that support or trash an expanding sub atomic "particle" theory of gravity?

I don't know the answer so i guess i will look at it when i get through working out the relativistic motion model i am intuiting.

[jehovajah]

russian spiral

<a href="http://www.youtube.com/v/M_sHmpagq5Y&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/M_sHmpagq5Y&rel=1&fs=1&hd=1</a>

http://suzannecooper.com/classroom/spiral.html

Just some practical examples of Links and Rules, and it is clear that these are classed as Instructions and  a particular type of Instruction would be a Recipe the finished product is a Construction. So doesn't that sound better than "function"?    What do you think?


my favourite spiral wrap so far.

Instructions of a more general nature, then applied to a specific construction.

Rule guided automatic machine  This is an automatic machine so it has an algorithm that links the elements of the parameter and produces the construction. Algorithm then is just another way of saying Rule.

So i guess i can use more accessible language to talk accurately about mathematical ideas.

[jehovajah]

There is a certain set of numbers/numerals/polynomial  power series expansions/ratios- which are  fundamentally linked to rotational motion in spaciometry. These ratios are called logarithms. These logarithms are on a curious base which is called e the Euler constant.

The relationship was first devised by John Napier using a projection of a surd line onto a parameter line. Napier, interestingly describes these ratios in terms of motion along the parameter line. He describes the whole interaction in terms of variable speeds along the parameter line. He was indeed an admirable fellow! 

[jehovajah]

So now the laws of thermodynamics are for increasing entropy systems but what laws are for decreasing entropy systems ?

I will call them electrodynamic laws, and these laws will govern decreasing entropy and self orgamisation and self asembly arrangements.

So a little background musing to illuminate my thought path.

So i take the hypothetical electron and ask does the hypothetical electron travel in a straight line? So i accept that it may be good enough to think of it doing this in a vacuum, but i do not accept it as the accurate description. soo now taking it as moving in some straight line what happens when it encounters a dense substance? does it travel in a straight line? Well no, if we use the rutherford model of the atom and the molecular structure of compounds it is accepted that the electron interacts with the substance by either passing through the "space" between the atomic nucleus and the electron shells or being deflected by the electron shells or joining a particular electron shell around a particular atom. so the fate of the electron is to be interactive with the existing molecular structure and so does not travel in a straight line.

The next question is does the electron that goes in even ever come out? Well again the accepted idea is in a lattice structure the electrons form a kind of cloud around the nuclei and this is how electricity flows through certain conductors. Well a cloud is bot a good analogy as the flow appears to be incompressible, so a fluid analogy seems more appropriate, and the principle seems to be that if an electron goes in one end an electron comes out the other end of the conductor. All pretty much like plumbing,except that the electron that come out is unlikely to be the electron that went in, because of interaction.

We see this more clearly when i look at a liqud structure supporting a current flow through an applied potential difference as the flow is now described as ionic and as a fluid flow of ions. So now is the electron that goes in the electron that comes out as we are ususally shown? Well no because the chemist will point out the chemical interation among the ions that promote the formation of certain compounds in solution. even some of the compounds may precipitate forming a more solid but amaorphous structure that settles to the bottom of the fluid away from the action potential.

In a case like this it is even less likely that the electron that goes in ever comes out,

But what we do have is a catlysis of a reaction by an electric potential.

Now i jump to an enzyme whose job is to provide substrates with a place that lowers the action potential for their reaction . thus i see that a particular molecular structure is in fact acting like an electric potential difference faciitating specificc kinds of reactions.
i then jump to these enzyme "batteries" having a specific electro potential that varies around the shape of the molecula r enzyme whic encourages certan other electro potentials to either bond or deflect from certain sites on the molecular structure. That whwn the right potentials bond with the right sites the battery circuit is complete and an actio potential as a current of ions flows between the two parts eithe r bonding or breaking bonds. The product then falls a way from the site of biding because it develops a repulsive or non attractive electric potential. If the bonding site is open a neutral electical potential will be easily pushed out of the way. If the site is sunken a repulsive potential may be needed or a change in the molecular arrangement that expels it. For example a potential may cause the molecule to coil like a spring which then returns to its normal state expelling a neutral potential.

So in this case the electron that comes in i involved in and mediates chemical reactions on a molecular template called an enzyme. Such enzymes if they are highly iterative will confer a self arrangement onto the substrates and thus induce self arrangement due to electrodynamical resonance modes solely.

Thus i have an inkling of hoe electrodynamic resonance may lead to self assembly and self organisation which is an decrease in entropy.

So now if i run through this again but use quantum dynamic descriptions of electron transport i think it is obvious that the electron that goes in is unikely to be the electron that coemes out, but an electron comes out in an electron transport system whatever the phase of matter , and this weaving of the electrons is what creates order.

Now if the electron is actually a vorticular wave?

 to be corrected

[jehovajah]

Check out this cone.
This is how Yh¶h does vorticular wraps:
   note the loxodromes start from a ring not a point.


This is  a picture of an atomic nucleus made up of vortices:   

                                                         

This is also a snapshot of a vorticular wave:

.

Pretty neat for a pine cone huh!

[jehovajah]

Now one particular vortex fantasy i have just realised is the  translational speed fantasy. Vortices do not create terrific  translational speed as the object nears the vortex centre. A little drawing showed me that a region following a spira path undergoes spaciometric rotation, and that this rotation accelerates the closer the region gets to the spiral centre. Quite simply a spiral or vortex path is an accelerator of rotational motion, and there is no upper bound to the spaciometric rotation rate of change, as following a spiral path accelerates by definition.

with this in mind i move to an explanation of solenoid magnetism. The effect of a spiral is to generate spaciometric rotation in regions that follow that path. Now magnetism is experimentally proportional to electron spin alignment. So if electron spin is aligned magnetic force is increased. The helical wrap of a coil entrains all the electrons to spin in the same direction at the same rate, simply as a consequence of spaciometric rotation. The electrons that flow around the conductor interacting with the atomic structure   gain a constant rate of spin spaciometrically, in the same vorticular direction. Thus a vorticular resonance is set up which encourages a vorticular coherence, that is a vortex spirals out in the helical direction. This effect is the basis of masing and lasing. Thus the usual magnetic force line diagram is a vortex cross section.

Now when a suitable material is placed in the vortex field its free electrons are entrained in the spaciometric rotation  enhancing the rotation spin density and alignment thus extending the vortex. The free electrons interact with the atomic structure so that atomic spin is also enhanced so diamagnetic effects are also enhanced.

Tighter coils should increase spaciometric rotation, but a conical helix wrap should provide the greatest acceleration.

Now in the light of this an alternating current will develop a vortex  which will collapse and then be established in the opposite direction . This alternating vortex field has its own effects on neighbouring regions.

The Rodin coil wrap has 2 spaciometric rotation effects that oppose each other. Thus the electron spins are aligned and cohered but the spin is accelerated and decelerated regularly thus generating a variable vortex field that is forming and collapsing, thus a potential null vortex field. But in addition to the spaciometric rotation being canceled by the wrap the pulsed signal resting on the 3rd cycle skews the vortex round the torus and accentuates the central vortex which is a distinct punch in a direction determined by the spaciometry of the wrap. It seems mono polar but in fact it is coherent field formation and collapse in resonance mode.

[jehovajah]

I think that one point of view is that multiplication is repeated addition, but that is a powerful but inadequate description of multiplication, because of the primes and because of cell division. You see the primes seem to indicate there is a structure to bundling regions together, and every so often one has to start a new thread, so to speak which i will call a prime thread. The ratio description of notFS admits of equivalence classes which are thread like. By not cloaking the ratio aspect of notFS and my logos response i hope to explore this structure.

The bundling together process, is an aggregation, but the division process is a disintegration, and both multiply but in differing ways structurally. Ratio analysis may indicate how, but essentially disintegration requires exponent bases to describe, while aggregation requires mod or clock arithmetic bases to describe.

So we need ratios written in fraction form, and then bundles of fractions are soon measured fractally in exponential denominator schemes 1/2^n ,1/3^n,1/5^n,1/7^n.. all prime number numerators.


Similarly we have used mod(1), mod (2),mod(3),mod(6), mod(7), mod(eight), mod(10), mod(12), mod(16),mod(20) mod(22),mod (24)mod (28),mod(29),mod(30) and mod(31),mod(220) mod(52),mod(365), mod(1760),etc to name but a few.

Currently i aggregate time under a mod (60)+ mod (60)+ mod(24)+ mod (7 )+mod(29,30,31)+ mod( 12) system .

So the structure of my aggregation is described by the collection of clock arithmetics used to specify a bundle.

The decimal system itself is a mod(10^n) bundle aggregation with a collection rule that orders them and sequences them into a polynomial series .

This way i can construct an arbitrary  mod(n) arithmetic bundle collection  and define a rule that orders and sequences them and then use them as an aggregation scheme .

Polynomials then are an aggregation scheme based on logarithmically varying bundles like mod(10^n) of arbitrary fixed base called x.

So aggregation schemes based on bundle size x are mod(x^n) clock arithmetics with a collection rule that orders the series logarithmically , and sequences them irrespective of the surface they are written on, or the region the bundles are collected in.

These have traditionally been called polynomials and used in all sorts of ways to describe aggregates generally.

An interesting polynomial system for me would be a mod(e^n) bundle clock arithmetic collection with a logarithmic power order rule, and a normal sequence pattern.