The Wythoffian similarity scale

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

The Wythoffian similarity scale

Postby polychoronlover » Tue Jun 22, 2021 8:33 pm

Many vertex-transitive polytopes are Wythoffian (especially convex uniforms) but not all of them. The Wythoffians have a large number of additional properties regarding the symmetry group as it acts on the vertices. We can list these properties in a chain of implications, where each property is more restrictive than the ones above it. The levels at the bottom of the scale are the most general while those at the top are the most specific.

The examples given for each category are polytopes that don't belong in the more restrictive categories.

1. The symmetries of the polytope allow any vertex to be mapped to any other.
Regular-faced property: Scaliformity
General property: Vertex transitivity (examples: rhombic disphenoid, noble faceting of snub cube, stephanoids)

2. The symmetries of the polytope allow any vertex to be mapped to any other and all elements have the same property (under their own symmetries).
Regular-faced property: Uniformity (examples: most uniform snub polyhedra, gap, polygonal antiprisms, most uniform compounds)
General property: Semi-uniformity

3. Same as 2, but each facet is also specifically vertex transitive under the symmetries of the polytope mapping that facet to itself.
Regular-faced example: Compound of 5 octahedra (the vertices of a face can be mapped to each other but not the vertices of an edge)

4. Each element is vertex transitive under the symmetries of the polytope mapping that element to itself.
Regular-faced property: Local uniformity (examples: compound of 5 tetrahedra, compound of 10 tetrahedra)

5. For any chain of elements e1, e2, ... en where the dimensions increase by 1 each time and each element is incident to the following one in the chain, e1 is vertex transitive under the symmetries mapping all elements in the chain to themselves.
Examples: Crossed quadrilateral, bowtie prism, crossed prism of a semiuniform polygon. (I once heard someone call this "recursively uniform". An equivalent definition is that the polytope is vertex transitive and each facet is recursively uniform under the symmetries mapping that facet to itself.)

6. Same as 5, but the symmetries mapping each edge to itself include reflection across the edge.
Examples: Most petsu (primary tame semi-uniform) polyhedra, uniform polyhedra with butterfly verfs, the sabbadipady regiment, most members of Wythoffian regiments, comb products of Wythoffian tilings


These kind of shapes can be called "fully recursively uniform." I have a conjecture about fully recursively uniform shapes, which relies on the following definitions.

Generalized Wythoffian: I use this term to mean that the polytope is constructed from a fundamental region tiling a space (possibly with overlaps to make a nonconvex tiling) by reflection across its facets, and a point is placed inside the region with edges perpendicular to all the facets the point is not on, but the fundamental region isn't necessarily simplicial. Every ridge between two facets in the fundamental region corresponds to a face of the polytope made from the corresponding edges, a peak corresponds to a cell, etc. Generalized Wythoffian compound polytopes can also be defined by allowing multiple copies of a single tiling of fundamental regions to be superimposed, as long as the symmetry of the resulting tiling acts on the fundamental regions.
1/n-Wythoffian: A shape that is either Wythoffian or formed from a Wythoffian polytope with coinciding elements, where all but one copy of each coinciding facet is removed and the facets themselves are similarly "un-coincided".

Conjecture: For finite n-dimensional polytopes whose vertices lie on an (n-1)-sphere, property 6 is equivalent to being generalized 1/n-Wythoffian.

If true, this seems like a big deal, because most uniform polyhedra and polychora in Wythoffian regiments seem to have property 6. It would be cool to have kaleidoscopic constructions of them from non-simplicial fundamental domains. I already found that double coverings of the lepidotrapeziverts (sroh, sird, ri, giddy, etc.) are omnitruncates from butterfly-shaped fundamental domains. One of them (2ri) is even in the same family as the semiuniform ditti.

There seems to be debate about whether kaleidoscopic constructions with non-simplicial domains count as "Wythoffian." I propose that Wythoffian is used only for simplicial domains because otherwise it would (perhaps unknown to most people) include shapes that are not traditionally considered Wythoffian, including ditti, sabbadipady, and (I suspect) many typical uniform polytopes.


7. Polytopes that are 1/n-Wythoffian (with simplicial domains only).
Examples: thah, cho, sidhid, tho, firp, oh, hehad, rhom (compound of 5 cubes)

The non-compound examples are hemi-Wythoffian, while rhom is 1/3-Wythoffian. 3rhom is an omnitruncate from 15 copies of the Schwarz triangle tiling of cuboid symmetry. Each tiling has mirrors in the exact same position as two others, but a point inside the domain is rotated by 0, 120 and 240 degrees. To form 3rhom, the point is in the center of the domain, so the cubes formed by each of the 3 domain tilings completely overlap.

8. Wythoffian polytopes (with simplicial domains)
Examples: Platonic and Archimedean solids (except for snubs), colonels of typical uniform regiments, three nonprismatic compound polyhedra (stella octangula, truncated stella octangula, and rhombihexahedron), flag-transitive compounds and their truncations.
Climbing method and elemental naming scheme are good.
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