As promised in the talk, here are the 12 lines of code. The 13th line is an example and take the smallest 3 dimensional manifold G, the 16 cell which is a small example of a 3-sphere. The fiber bundle P has 384 elements in this case. Every single particle moves on a geodesic with length 8. The 14th line tries to find the longest 3 particle trajectory on G. it is 1568. This was actually my main motivation to look at other particles. We do not have a Hopf Rynov theorem in a finite geometry if we insist that there is an exponential map that is locally defined: there are simply just finitely many geodesics that start from a point then. But this is not enough to reach every point in G unless we have very special situations like an ergodic geodesic flow, in which case also reaching from x to a very close by y, could take a very long time.
[ Update Mai 30, 2025: a write-up [PDF]. It has also been submitted to the ArXiv but as expected is there on hold (probably for a while by experience with such a crackpot topic). The worst is if you hit two crackpot indices at once, like in this topic, when I wrote this 9 years ago, which hit both the Caldwell crackpot index as well as the Baez crackpot index. Being suspicious in such topics is healthy as we are bombarded almost weekly with papers claiming to prove an important open problem in mathematics. We live in a world with 6 billion people and maybe a million mathematicians (folks who have proven a theorem) and probably a 100’000 PhDs from which by pure statistical reasons some become insane or desperate. What helps therefore is to have working code that illustrates a topic. As for the Hadron structure of quaternion primes, which I worked about 9 years ago, one can also test the topic by running a few lines of code. It has a chance to convince, especially if it involves only a handful of lines of code. “Show me the code!” is a healthy attitude and allows a third party to see what is happening or at least to see that there is something interesting going on.]
Generate[A_]:=If[A=={},{},Sort[Delete[Union[Sort[Flatten[Map[Subsets,Map[Sort,A]],1]]],1]]];
Whitney[s_]:=Map[Sort,Generate[FindClique[s,Infinity,All]]];L=Length;Co=Complement;Po=Position;
Facets[G_]:=Select[G,(L[#]==Max[Map[L,G]] ) &]; RL=RotateLeft; RR=RotateRight; S=Sort;
Bundle[G_]:=Module[{F=Facets[G]},Flatten[Table[Permutations[F[[k]]],{k,L[F]}],1]]; Se=Select;
OpenStar[G_,x_]:=Select[G,SubsetQ[#,x]&]; Stable[G_,x_]:=Complement[OpenStar[G,x],{x}];
M[z_]:=Module[{U=Co[Stable[G,z],{S[z]}]},Table[First[Co[U[[j]],z]],{j,L[U]}]];
a[x_]:=Module[{y=x,u},u=M[Delete[x,1]];If[L[u]==1,y[[1]]=u[[1]],y[[1]]=Co[u,{x[[1]]}][[1]]];y];
A[{X_,Y_}]:={Table[y=Y[[k]];a[y],{k,L[Y]}],Table[x=X[[k]];a[x],{k,L[X]}]};
B[{X_,Y_}]:={Table[y=Y[[k]];RL[y,L[Se[Map[S,X],S[y]==#&]]-L[Se[Map[S,Y],S[y]==#&]]],{k,L[Y]}],
Table[x=X[[k]];RL[x,L[Se[Map[S,X],S[x]==#&]]-L[Se[Map[S,Y],S[x]==#&]]],{k,L[X]}]};
T[{X_,Y_}]:=B[A[{X,Y}]]; SS[{X_,Y_}]:=A[B[{X,Y}]]; Orbit[{X_,Y_},n_]:=NestList[T,{X,Y},n];
Orbit[{X_,Y_}]:=Module[{U={X,Y},o},o={U};While[U=T[U];Not[MemberQ[o,U]],o=Append[o,U]];o];
G=Whitney[CompleteGraph[{2,2,2,2}]];P=Bundle[G];{X,Y}={{P[[1]]},{P[[9]]}};o=Orbit[{X,Y}]; o
Q=Tuples[P,2];m=0;Do[u=L[Orbit[{{P[[1]],Q[[k,1]]},{Q[[k,2]]}}]];If[u>m,Print[u];m=u],{k,L[Q]}];
I mentioned “Space-Time-Matter”, a book title Herman Weyl chose in 1918 to describe Einsteins then fresh theory of relativity. I had tried to read it in high school (this is an age you want to impress family and friends and take on things which are way over your head). It is of course ridiculous: the bulk of the book can only be understood with a solid linear algebra (and multi-linear) algebra knowledge, good multi-variable calculus skills as well as knowledge in basic differential geometry. It was still a good book to look at as a teenager, because looking at the real stuff brings you down to earth and tells you how much math you are missing. For myself, it needed many years to get comfortable with calculus, algebra and differential geometry and I still feel only having seen a small fraction of the entire picture. If you only look at books like “The character of physical law” or Hawking’s “a brief history of time” (or nowadays watching youtube videos of folks that are successful in popularizing physics) you get the impression that it is possible to understand physics without actually going into the math. Already Galilei was convinced that the universe is written in the language of mathematics. Also Freeman Dyson (who is seen in minute 6 in the following video) cited an episode, where we went to Chicago to talk to Fermi to show him his work and Fermi famously destroyed an entire line of research in a few sentences like ” ‘I am not very impressed with what you’ve been doing.’ And he said, ‘When one does a theoretical calculation, you know, there are two ways of doing it. Either you should have a clear physical model in mind, or you should have a rigorous mathematical basis. You have neither.’ So that was it – in about two sentences he disposed of the whole subject.” This interview with Dyson was 1998. Dyson died in 2020, 5 years ago.
In the context of quantum mechanics, the space time picture of Einstein has been questioned more and more. I myself am not a physicist but I like mathematical models, especially simple ones like the Ising model or a random matrix model. They are simple, make mathematical sense and one can learn from them. Once one has a dynamical model, it is always possible to look at the space time picture later on to reformulate it. We need a dynamical model, an evolution to check the model. Having grown up in mathematical physics and dynamical systems theory, I like to think of time as an action of a semi group on a space because it is what makes up the most successful theories. (see the introduction to this course taught 20 years ago). Having a dynamics is like playing with a toy that works. Whether it is a true model of the world does not really matter (Dyson’s quote: “I give a damn whether it is fundamental!). The word Fundamental is a term Dyson despised, but he acknowledged in the same interview that there are “birds” who look at fundamental things and “frogs” who like to dig in the mud and understand details. Like Dyson, I prefer to be a frog, playing with concrete things, or in the allegory with toys to play with toys that actually work in the sense that they do make mathematical sense, produce interesting questions and allow for rich experiments. Fundamental theories which are floated around including string theory, axiomatic quantum field theories, or loop quantum gravities are for me a bit toys that look impressive but do not actually work (partly because i can not even understand the rules of them as they operate on sometimes shaky mathematical ground). Dyson has the right attitude, when he told Kuhn in an interview “I love details. I like to look at the real thing and try to understand it. I don’t give a damn whether it is fundamental or not.”. I talked here (2015) a bit about the nature of mathematicians (and especially about the “frog” -“bird” allegory of Dyson. He described it also as “Cartesians” and “Baconians”. It is the same allegory if one identifies Rene Descartes as a bird and Francis Bacon as a frog.
Apropos Herman Weyl, one should mention here also that Weyl seriously looked at ways to define what a ‘sphere’ is from a combinatorical point of view. This turned out to be impossible. One can not read off from the f-vector of a space G alone whether it is a “sphere” or not. The most elegant way to introduce a sphere is by using homotopy. A q-manifold is a q-sphere if removing an open star 
[Definitions can be deceptive if they are ambiguous or if different literature uses different meanings. I myself do not use “homotopic to 1” in the definition of “sphere” because is a very hard (NP hard) condition to check, while “contractible” can be checked by reduction to smaller spaces. There is such a huge difference between “contractible” and “homotopic to one” that one has also linguistically separate it in the word space. The terms”contractible” and “collapsible” is just too close in any word distance metric and are also mixed up historically in the literature. One of the biggest problems which science faces is when dealing with terms that are not well defined. Quite many arguments can be traced back to ambiguous or unclear definitions or definitions that do not make sense as they are wrong or mix a result into the definition. An example of a wrong definition is to define a vector as a quantity with magnitude and direction (which fails for the zero vector which does not have a direction). An example of a definition mixing in a result is to define the dot product as x.y = |x| |y| cos(alpha), both things one can find in reputable calculus books. ]
There is a nice answer to the question “What is a sphere?” of Weyl which does not involve homotopy. I think that it can be done by enlarging the scope of “q=sphere” slightly and replace it with “Dehn-Sommerville q-sphere”, which is a space that is completely determined by the f-vector. It has a very elegant recursive definition. The empty complex 0={} is the Dehn-Sommerville (-1) sphere. Inductively a simplicial complex G is a Dehn-Sommerville q-sphere, if all unit spheres 
Back to the model. One can ask why not introduce randomness into the motion as some cellular automata models do (an example is the Frisch-Hasslacher-Pomeau) cellular automaton used to simulate fluids in two dimensions which is a bit better behaved than the deterministic Hardy-Pomeau-Pazzis model. Even so I love random models, it should always be a last resort as it introduces many new parameters.
