Plato Vortices KAM

In January-February 2007 Ivars Fabriciuss and I had e-mailconversations about some of his interesting ideas and how they mightbe related to Platonic Regular Polytopes, Compton Radius Vortices,Kolmogorov Scaling, and KAM. Here are some excerpts from my messagesthat I sent to him during those conversations:
"...The "small circular helical pipe" sounds to me likethe world-line of a fermion particle or antiparticle,the particles having + helicityandthe antiparticles being mirror image with - helicity.The picture sounds sort of like what I write about athttp://www.valdostamuseum.org/hamsmith/Sidharth.htmlIt seems to me that you are looking at things in termsof rotations in 3 or 4 dimensions.That can give you part of the symmetries that we see inexperiments, such as the Lorentz group of spacetime,and if you gauge them plus translations you can get gravity.If you go to 5 dimensions, you can get a version of electromagnetismin addition to gravity.In my work, I had to go to 8 dimensions to get a rotation groupwith enough content to give me all the forces and particles thatwe see by experiments.It is really hard to visualize in 8 dimensions,so in order to describe stuff clearly it is necessary to usethe mathematics of Lie groups, particularly SO(8) etc,and that math background is not something that people usuallyget without studying math at least at an advanced undergraduate college,or graduate school, level.You can learn it yourself by reading books, but it is not easy to do.As you see, even in 3 or 4 dimensions it is hard to describestuff in words without using so many words that readers tendto get lost, and even then word descriptions are not as definiteand clear as math descriptions (in terms of Lie groups etc).So, I think that your intuition is good and that you are on theright track in intuitive understanding ...You start with "aether self resonance twisted helical pipe structure".The twist can be either right-handed or left-handed,sothere are only two basic structures to start with:left-handed (particle)andright-handed (antiparticle).Now, what kind of spin should such a fundamental thing have?If it only knows what direction it is going,then it is spin-1 (vector),butsince it has helical twist it also knows how it is oriented.Things that know how they are oriented are spin-1/2 spinor fermions.For pictures of how that works, seehttp://www.valdostamuseum.org/hamsmith/play456.html#SpinorWhat we have now is physically likeneutrino left-handed (particle)andanti-neutrino right-handed (antiparticle).The spinor fermions that we don't have yet are like electrons and quarks.To get electrons, positrons, muons, and tauons, you must give themelectric charge (+ -).To get quarks, you must, in addition to electric charge, give themcolor charge (red, blue, green).You say "Honestly, I can not understand charge",butyou need charge to get the rest of the spinor fermions,sothink about what you mean when you say"Charge is ... a ... sum of all vortexes on shapes surface ...which cancel each other out,come in pairs...There is also internal time ... which is time it takesfor a pair of fractional vortexes of certain scale to formafter interaction with environment. ...".Compare that to how Feynman sees charge, which is thatcharge is related tothe probability of a charged particle tosend out a force-carrier gauge bosonthatconnects with another charged particle and sets up a force between them.In your picture,the more force-carrying vortices that live on the surface of the fermionandthe faster-in-time the fermion can replace a force-carrying vortex thathas been sent out to connect with another particle,the stronger the charge of the fermonand the stronger the force carried by the force-carriers,and that is consistent with how Feynman sees such things.For example,if the neutrino with no charge looks like a uniform sphere with noforce-carrying vortices,thenmaybethe electron looks like it has a force-carrying vortex on one hemisphere,where one hemisphere is for - charge and the other is for + charge,anda quark looks like an electron with additional structure like the6 faces of a cube, which can be considered as 3 pairs of faces,each pair being one of the ones you see and its opposite hiddenface on the back as shown here:     x-------x    /       /|   /   R   / |  x-------x  |  |       |G |  |   B   |  x  |       | /  |       |/  x-------xNote that at each vertex of the cube there are 3 square faces,each belonging to one of the 3 pairs of faces,and I have labelled them R (for red), B (for blue), and G (for green)to correspond to quark color charges.In other words,quark electric charge would be due toan electric-force-carrying vortex living on either the hidden hemisphere(the hidden 3 faces of the cube) or the hemisphere (3 faces) shown asnot hidden in the figure above,andquark color charge would be due to a color-force carrying vortexliving in one of the three (either R, B, or G) faces of the surfaceof the sphere corresponding to the faces of the cube.This picture would also probably explain why the quarks carryonly fractional amounts (1/3 or 2/3) of electric charge.Now that you have the electrons and quarks,you can put them together in a condensate to form spacetime,and then see how force-carrying gauge bosons work, etc.Your ideas and way of thinking are very original and I think very useful.I hope that more people will consider them seriously and study them. ...you might ask why stop at the quark color force cube     x-------x    /       /|   /   R   / |  x-------x  |  |       |G |  |   B   |  x  |       | /  |       |/  x-------xand why not go on to other more complicated forces.My guess would be that, living in 3 space dimensions,there are only 3 types of regular polyopes:tetrahedron - 4 faces, 2 pairs of faces, 2 hemispheres, electric + -.cube (includes its dual the octahedron)            - 6 faces, 3 pairs of faces, color R B G.docecahedron (includes its dual the icosahedron)            - 12 faces, 6 pairs of faces - this gives you gravity.To see how it gives you gravity, I need to talk about Lie groups,and you can ignore that if you don't want to get into that math,butI do want you to know that your idea can be naturally extendedto get gravity as well as the other forces,andthat you do NOT get any extra unobserved forces,soyour picture is very realistic physically.Also,note that looking at regular polytopes in 4-dimensional spacetime wouldgive the same result, because the only new type of regular polytopethat appears going from 3-dim to 4-dim is the 24-cell, and it issort of a combination of the 3-dim ones, so it does not bring inanything new. It only shows in more detail how they all fit together.Here is (in case you want to see it) the Lie group stuff thatleads to gravity:The 4 vertices of the tetrahedron correspond to the generatorsof the Lie group U(2) = U(1)xSU(2) involving the electric charge,the U(1) photon and the SU(2) weak bosons.(Half of those 4 vertices, 2 of them, would give the root vector systemof the Lie group U(2), and I think that the half part may be due to thefact that the tetrahedron is self-dual.)The 6 vertices of the octahedron (dual to the cube) correspond tothe 6 root vector system of the Lie group SU(3) of the color force.The 12 vertices of the icosahedron (dual to the dodecahedron) correspondby a Buckminster Fuller tensegrity transformation (see attached image)to the 12 vertices of a cuboctahedron which in turn correspondto the root vector system of the conformal group Spin(2,4) thatgives gravity by the MacDowell-Mansouri mechanism.Note that Spin(2,4) acts linearly on a 2+4 = 6-dimensional space,in which the 6 dimensions correspond to the 6 pairs of dodecahedron faces,or, equivalently, to 6 pairs of icosahedron vertices.... the point that IS easily understandable is that your picture also works for gravity and is therefore very realistic physically....There are 5 Platonic solids, or regular polytopes, in 3 dimensions:tetrahedron (4 faces and 4 vertices)cube (6 faces and 8 vertices) andits dual the octahdron (8 faces and 6 vertices)dodecahedron (12 faces and 20 vertices) andits dual the icosahedron (20 faces and 12 vertices)The cuboctahedron is not regular since its 14 faces are not all the same.It has 6 square faces and 8 triangular faces.It has 12 vertices, which is how it can be related tothe icosahedron (which also has 12 vertices) by Fuller's transformation.
If you go to four dimensions,you get analogs of the five Platonic solids, plus one new one,the 24-cell which sort of shows how they are all related to each other.------------------------I don't understand Kolmogorov scaling very well,particularly whether and/or how it might be involved in goingfrom the scale of the electron Compton radius about 10^-11 cmto the Planck scale about 10^-33 cm,buthere are some comments.A web page that at one time was atgrus.berkeley.edu/~jrg/ay202/node166.htmlsays in part:"... the total kinetic energy of turbulent eddies of an isolated fluiddecreases with time due to viscous dissipation.Hence a turbulent fluid can be maintained in a steady state only ifenergy is continuously fed into the system so thatthe energy injection rate equals the rate of dissipation.... Kolmogorov ... proposed a theory to calculate the energy spectrum ofsuch a system and began a new era in the theory of turbulence. ...Hence, L. F. G. Richardson's corruption of Dean Swift's sonnet:``Big whirls make little whirls which feed on their velocity;Little whirls make lesser whirls, and so on to viscosity.'' ...the velocity associated with eddies of a particular size is proportionalto the cube root of the size -- a result known as Kolmogorov's scaling law ...We now have an outline of a scheme for turbulence:energy must be fed at some rate  per unit mass per unit timeat the largest eddies of size  and velocity ...This energy then cascades to smaller and smaller eddies untilit reaches [the smallest] eddies ... which dissipate ...[the energy]...in order to maintain the equilibrium...the Reynolds number associated with the largest eddies determineshow small the smallest eddies will be compared to them ...".You might think of the Compton radius as the largest eddy size,and the Planck scale as the smallest eddy size,butif you do that you have to explain the source of the driving energyand also how, as you say,"...  energy of Aether eddies would be dissipated ...".If there is no realistic way to get such an energy flowto drive such a dissipative system,thenyou might consider the electron Compton radius vortex asa conservative system.A book describing such conservative systems is Deterministic Chaos,by Heinz Georg Schuster (VCH 1988). ...Here are some quotes from the book Deterministic Chaos,by Heinz Georg Schuster (VCH 1988):"... the transitions to chaos in dissipative sytems only occurwhen the system is driven externally, i.e. is open....conservative systems ... the solar system ... motion in particleaccelerators ... display (in contrast to dissipative systems) ...no attracting fixed points, no attracting limit cycles, andno strange attractors ... Nevertheless, in conservative systemsone also finds chaos ... i.e. there are "strange" or "chaotic"regions in phase space, but they are not attractive and can bedensely interweaved with regular regions....we investigate ... an integrable Hamiltonian and consider theeffectof a small nonintegrable perturbation ... One of the simplestexamples ... a torus for two harmonic oscillators ...Closed orbits occur only if ...[the ratio of frequencies is]... rational... For irrational frequency ratios,the orbit never repeats itself but approaches every point onthe two-dimensional ... torus ... infinitesimally close in thecourse of time. In other words, the motion is ergodic on the torus....What happens if an integrable system with ...[frequency ratio]...close to an irrational value is perturbed ... ? The ... questionis answered by ... the ... KAM theorem ... Kolmogorov ... Arnold ...Moser ...[if the conditions of the theorem hold, then]...those tori, whose frequency ratio ... is sufficiently irrational ...are stable under ...[a small]... perturbation ...[A]... large enough ... perturbation ... destroys ALL tori.The last KAM torus which will be destroyed is the one for whichthe frequency ratio is the "worst irrational number" (sqrt(5) - 1)/2... golden mean ... which has as a continued fraction... 1 / ( 1 + 1 / ( ...[related to the]... Fibonacci numbers...The destruction of the KAM torus shows some similarity tothe Rouelle-Takens route to chaos in dissipative systems ...the decay of the last KAM trajectory shows scaling behaviorand universal features....[in] the Rouelle-Takens route ... after three Hopf bifurcations,regular motion becomes highly unstable in favor ofmotion on a strange attractor...when ... frequency ratio ... is rational ... the original torusdecomposes into smaller and smaller tori. Some of these newlycreated tori are again stable according to the KAM theorem ...and other tori decompose into smaller ones according tothe Poincare-Birkhoff theorem. This gives rise to ...self-similar structure ...in conservative systems regular and irregular motion aredensely interweaved....So far ... we have only dealt with systems having two degreesof freedom for which the two-dimensional tori stratify thethree-dimensional energy surface ... The irregular orbits whichtraverse regions where rational tori have been destroyed aretherefore trapped between irrational tori. They can only explorea region of the energy surface which .. is ... disconnected fromother irregular regions ...For more degrees of freedom ...the tori do not stratify ... the energy surface ... The gaps thenform one single connected region. This offers the possibiliy of... "Arnold diffusion" of irregular trajectories...in quantum mechanics ... defined by ... Planck's constant ... h =/= 0... only phase points with a rational ratio ... are allowed ...This means that the points with irrational ratios, which werethe only ones ... in the classical cat maps ... which lead to chaotictrajectories, are forbidden in quantum mechanics....no quantum system seems to exist which exhibits deterministic chaos... in ... quantum systems whose classical limit displays chaos ...the finite value of Planck's constant leads, together with theboundary conditions, to an almost-periodic behavior of the quantumsystem even if the corresponding classical system displays chaos....Gutzwiller ... calculat[ed] ... for an elecron which is scatteredfrom a non-compact surfdace with negative curvature ... that thephase shoft as a function of momentum is essentially given bythe phase angles of the Riemann zeta function on the imaginary axis,at a distance 0.5 from the famous critical line. This phase shiftdisplays features of chaos because it is able to mimick any givensmooth function. It, therefore, seems that the chaotic nature ofquantum systems whcih are described by wave mechanics is of arather subtle and "softer" kind than the chaos in classical mechanics....another major area is the question of chaotic behavior of quantumsystems with dissipation, such as lasers or Josephson-junctions ..."....". 

 


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