Shashibo Geodesics

Shashibo Geodesics

The 16 cell, a 3 -sphere and its dual, the tesseract which is a CW complex realization of a 3-sphere.
Shashibo polyhedra

The Shashibo puzzle is a game in the category of geometrically realized simplicial complexes of dimension q=3. One can also play it combinatorially, meaning to disregard geometric realization and just look at finite set of finite sets. One can see it as a playing with 3-dimensional complexes generated by twelve 3-simplices G_3 in which twelve edges (and so their vertices) are identified glued together by factory. It is possible to glue together two 2-simplices in G_2 which are already joined by one of the fixed edges E. The goal is to build some nice shapes, in particular manifolds with boundary like the cube. The puzzle is harder than the combinatorial one because the tetrahedra G_3 are geometrically realized as congruent solids, each of the solids has 4 faces where two pairs are isomorphic. There are 3 magnets in each of the tetrahedra so that once two faces which are allowed to get together are close enough, they stick. There are reportedly 70 different simplicial complexes which can be realized. (There are more if one does not insist on stability). Some of them are 3-manifolds with boundary, most of them are not. No manifolds without boundary can be realized. Proof: The smallest 3-manifold without boundary is the 16 cell, which is the 4-partite graph K_{2,2,2,2} and which has the f-vector (8,24,32,16) which means already 16 three dimensional simplices, more than the Shashibo puzzle has. Note that the 16-cell is a 3-dimensional sphere. It is one of the 6 regular Platonic 3-polytopes first classified by Ludwig Schlaefli, a high school teacher in Bern. It is dual to the tesseract, which a priori is only a 1-dimensional simplicial complex as it is triangle free. But the tesseract is naturally a CW complex with f-vector (16,32,24,8). Both are naturally delta sets. Duality is an other place (like sub simplicial complexes or product complexes) where are thrown out of the paradise of simplicial complexes (a category with only 1 axiom! 5 times less than the axioms of Euclid for planimetry, but which allows to model all smooth manifolds of any dimension. There is no other geometric construct (not even graphs) which beats simplicial complexes in simplicity as there is no interesting axiom system with 0 axioms and simplicial complexes are a 1-axiom system. Delta sets win in generality (as they are a topos) and graphs win in clarity (as kids can understand networks like trees or street maps without any training about set theory [P.S. I was as a “new math” kid exposed to set theory and Venn diagrams early in primary school]). I talked once about the trinity of geometric structures in the context of the Unix paradigm simplicity, clarity generality. (See this talk.)

Lets call a Shashibo polyhedron a 3-dimensional simplicial complex for which the dual is a cyclic graph or a path graph. More generally, a Shashibo polytop is a q-dimensional simplicial complex for which the dual is a cyclic or path graph. A special case is a cyclic polytop, which has an additional constraint that all facets have a common (q-2) simplex. Cyclic polytops have become famous because of the upper bound theorem which was proven in the 70ies by Peter McMullen and Richard Stanley. In some sense, they are the most economical polytopes as they produce the largest possible number of faces among all convex q-polytopes with a given number of vertices. Cyclic polytopes play a role in the geodesic story we work on because the dual of a (q-2) simplex x in a q-manifold is naturally a cyclic sphere polytop and so a Shashibo polytop. Note that cylic q-polytopes are never q-manifolds but that if closed (meaning the dual is a circular graph), then they have a (q-1)-sphere as a boundary. I myself learned about them first not in combinatorics but in mathematical physics: Regge calculus is calculus in a discrete 4-manifold in which the Einstein equations are abstractly realized as a condition on the geometric realization. The (q-2) simplices in that frame work are just triangles. The set of all 4-simplices hinging on that triangle form a Shashibo 4-polytop with cyclic dual graph. In every manifold there are many Shashibo polytopes as subpolytopes. In the case of the tesseract there are 32 Shashibo 3-polytopes for which the dual graph is cyclic with 4 elements. They all are 3-balls with an octahedron as a boundary. (the octahedron is a 2-sphere, actually the smallest one).

The graph complement of C_7 is a Moebius strip. It is an example of a flat Shashibo polygon, a 2-dimensional flat Shashibo polytop and so a geodesic.
A geodesic of the form of a Moebius strip in a projective plane. It is an embedding of a flat two dimensional Shashibo polytop. In this case it is even an embedding.

Lets call a Shashibo polyhedron “flat” if its boundary curvature is zero everywhere. An embedding of a Shashibo polyhedron in a q-manifold is called a geodesic. Now this looks very fancy, it is actually very simple. In dimension q=1, every 1-manifold with or without boundary is a geodesic. In dimension 2, we have an embedding of a cylinder or Moebius strip of width 1 which has zero curvature everywhere. Actually all these manifolds are 2-manifolds without boundary. A simple case is the graph complement of C_7 ( GraphComplement[CycleGraph[7]]) which is the smallest Moebius strip. It consists of 7 triangles. It is maybe the most exciting Shashibo game for a flatlander (a mathematician living in 2 dimensional space) as one can not realize it in 2-dimensional Euclidean space. One can realize it in the 2 dimensional projective plane although. Geodesics are only embedded flat Shashibo poyhedra, not necessarily immersed. That is of course the same as in the continuum. Geodesics are not simple curves most of the time. They form very complicated curves.

A 4-sphere with 32 cells and a geodesic seen in its dual

The setup looks complicated, but it is quite simple. It can not be simpler. Here is a 4 line code which allows to compute arbitrary geodesic in any q-manifold. In the following example, we first build a 4 sphere as the join of a 2 sphere with a 1-sphere, there are 32 four dimensional cells. The frame bundle, which consists of all oriented 4-simplices, has 5! =120= 3840 elements. In this very highly symmetric situation, every geodesic has 10 elements. Here is one of them. We start with the facet {3,7,8,6,1} the next point is {7,8,6,1,4}. These two hyper-tetrahedra =4-simplices are attached to each other along a tetrahedron {7,8,6,1}.

{{3, 7, 8, 6, 1}, {7, 8, 6, 1, 4}, {8, 6, 1, 4, 9}, {6, 1, 4, 9,10}, {1, 4, 9, 10, 5}, {4, 9, 10, 5, 2}, {9, 10, 5, 2, 3}, {10, 5,2, 3, 7}, {5, 2, 3, 7, 8}, {2, 3, 7, 8, 6}}

Here is some Mathematica code:

(* Shashibo geodesics in any simplicial complex. 4 Lines. Oliver Knill, January 12, 2025, for Quantum Calculus Blog *)
Generate[A_]:=If[A=={},{},Sort[Delete[Union[Sort[Flatten[Map[Subsets,A],1]]],1]]];
Facets[s_]:=FindClique[s,Infinity,All];Whitney[s_]:=Generate[Facets[s]];OpenStar[G_,x_]:=Select[G,SubsetQ[#,x]&];
Geodesic[G_,x_]:=Module[{t=Delete[x,1],U},U=OpenStar[G,t];Append[t,First[Complement[Union[Flatten[U]],x]]]];
Orbit[G_,x_]:=Module[{y=Geodesic[G,x],o},o={y}; While[Not[x==y],y=Geodesic[G,y]; o=Append[o,y]]; o];

s = GraphJoin[CompleteGraph[{2,2,2}],CycleGraph[4]];G=Whitney[s];F=Facets[s];B=Flatten[Table[Permutations[F[[k]]],{k,Length[F]}],1];
x=RandomChoice[B]; Orbit[G,x]      (* Geodesic in the join Oct * C4  = 4-sphere, B=Frame bundle has size 3840 *)