Very interesting. I'd recommend following the conventions on 'abstract polytopes' and including an extra index for the bottom of the poset (which can be viewed as a (-1)-dimensional cell). The advantage stems from the fact that you can simply reverse your signatures to get the signature of the dual polytope, and thus self-dual polytopes have palindromic signatures. Anyway, for pentamorphic polyhedra, the possible signatures are: 1:2:1:1:1 -- https://en.wikipedia.org/wiki/Quasiregular_polyhedron 1:1:1:2:1 -- duals thereof 1:1:2:1:1 -- https://en.wikipedia.org/wiki/Noble_polyhedron The only convex pentamorphic polyhedra are the rhombic dodecahedron, rhombic triacontahedron, cuboctahedron, icosidodecahedron, and the disphenoid tetrahedron. There are many more nonconvex examples. Your elegant argument that the only pentamorphic polygons are 1:2:2:1, and notation for describing polygons as a cyclic list alternating between vertices and edges, should make classification straightforward. If you just look at the vertices, then either we strictly alternate between A and B (in which case the only pentamorphic examples are (AaBb)^n) or we do not. In the latter case, the edge types are redundant and determined by the neighbouring vertices. We can wlog assume there are no BBs (because there cannot be all three of AB, AA, and BB). Moreover, every A must be adjacent to a B, so we only get the following possibilities: (AaBb)^n (BbAaAb)^n so your classification is completely exhaustive (for *convex* pentamorphic polytopes). Best wishes, Adam P. Goucher
Sent: Thursday, August 30, 2018 at 9:43 PM From: "Allan Wechsler" <acwacw@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] Oligomorphic polytopes
I've gotten interested in enumerating symmetric convex polytopes in various dimensions, intrigued by the fact that in dimensions as low as 5, we don't yet have a proven-complete census of the uniform polytopes.
Every polytope in N dimensions has a single N-cell, and then a graded collection of cells of dimensions from N-1 down to 0 (the vertices).
The symmetry group of a polytope divides this collection of cells into transitivity classes, and I propose to focus on polytopes with a small number of classes of cells. Since symmetries never change the dimensionality of a cell, the number of classes is always at least one more than the dimension of the polytope as a whole.
I've chosen to apply the term "oligomorphic" ("few-forms") to a polytope that is interesting because its cells fall into a small number of symmetry classes.
The only monomorphic polytope is the trivial 0-dimensional isolated vertex.
The only bimorphic polytope is a line segment; its two vertices clearly form one symmetry class, while the body of the segment provides the other. We can say it has a "profile" of 1:1 because it has 1 kind of 1-cell and 1 kind of 0-cell.
The regular polygons form an infinite call of trimorphic polytopes with profile 1:1:1 (1 kind of 2-cell, 1 kind of 1-cell, 1 kind of 0-cell). There are no other trimorphic polytopes.
When we come to consider tetramorphic polytopes, there are the 5 classical Platonic solids in 3 dimensions, with profile 1:1:1:1, but there are also some interesting 2-dimensional examples. A non-square rectangle and a non-square rhombus are examples of trimorphic polygons with signatures 1:2:1 and 1:1:2, respectively. There are similar examples with any even number of edges. (1:1:2 means 1 kind of 2-cell, 1 kind of 1-cell, and 2 kinds of 0-cells.)
Up until this point I am pretty sure that I have enumerated all the possibilities. But when we get to pentamorphic polytopes, I quickly start to lose confidence. Here is my list, but except in four dimensions I do not know if it is complete:
In four dimensions there are the six "classical" regular polychora, profile 1:1:1:1:1.
In three dimensions the only ones I have been able to think of are: the cuboctohedron and the icosidodecahedron, with profile 1:2:1:1 (because they have two kinds of faces); their duals, the rhombic dodecahedron and the rhombic triacontahedron, 1:1:1:2 (because they have two kinds of vertices); and the "isoceles" tetrahedron, 1:1:2:1 (with two kinds of edge). Note that the duality operation reflects the profile, except for the boilerplate leading 1. The dual of an isoceles tetrahedron is a different isoceles tetrahedron. (In combinatorial algebraic topology, they often add on a single "-1-cell" with which every vertex is presumed to be incident, just to complete this kind of duality symmetry.)
In two dimensions there is an interesting little menagerie of pentamorphic polygons. The simplest is the isoceles triangle, with two kinds of edges and two kinds of vertices, profile 1:2:2. This can be generalized into a family of polygons with 3k sides, with a vertex-edge pattern (AaAbBb)^k. (I hope it is clear what my silly notation means here). There is another family, of which the (non-rhombic) parallelogram is the simplest exemplar. Here, bigger examples have 2k sides, with a vertex-edge pattern (AaBb)^k. The profiles 1:1:3 and 1:3:1 are clearly impossible, since a single class of edge cannot be incident with three classes of vertex, and vice versa.
It would absolutely not surprise me if someone were to come up with pentamorphic examples in 2 and 3 dimensions that were not covered by my sketched catalog above.
Regarding hexamorphic polytopes, I have no confidence whatsoever that I have managed to exhaust all the possibilities, except that we know there are only three hexamorphic 5-dimensional polytopes. The isoceles trapezoids and their dual kites are 2-dimensional examples; right prisms and their dual right bipyramids are 3-dimensional ones; I cannot produce a single 4-dimensional one. Can anyone help complete the enumeration of hexamorphic types?
The snub cube is octomorphic, with profile 1:3:3:1. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun