Regular polytope compound

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Regular polytope compounds, or simply regular compounds, are a generalization of the idea of regular polytopes which includes polytope compounds. However the generalization differs rather significantly from the simple extension of flag-transitivity to compounds (See ยง Flag-transitivity).

Definitions

Both Coxeter and McMullen provide definitions for regular compounds. These definitions are very similar but differ in some subtle ways.

Inscribed polytopes

Related to both definitions is the concept of an inscribed polytope. We say that an n -polytope, ๐, is inscribed in another n -polytope, ๐ , written ๐๐ , if the vertex set of ๐ is a subset of that of ๐ . Regular compounds are inscribed in regular non-compound polytopes.

Coxeter

In the book Regular polytopes, Coxeter gives the following definition for a regular compound.

• A vertex-regular compound is a compound of regular polytopes whose convex hull is a regular polytope.
• A facet-regular compound, ๐, is a compound such that it can be arranged with a regular polytope, ๐, such that every facet of ๐ is parallel to a facet of ๐.

A regular compound is a compound which is either vertex-regular or facet-regular.

McMullen

In 2018, following the work by Coxeter, McMullen presents a slightly adjusted definition of regular compounds. McMullen's definition is very similar to, but ultimately stricter than, Coxeter's.[1]

• A vertex-regular polytope compound is a compound of regular polytopes, ๐ = ๐ i , with symmetry group G , such that:
• G  acts transitively on the components of ๐.
• There exists a regular polytope ๐ such that ๐๐ and G  acts transitively on the vertices of ๐.
• A polytope compound is facet-regular if its dual is vertex-regular.

A regular compound is a compound which is either vertex-regular or facet-regular.

Related concepts

Flag-transitivity

The usual notion of a regular polytope is one whose symmetry group acts transitively on its flags. It is natural to extend this notion to polytope compounds, however it is quite different from the notion of a regular compound.

The finite flag-transitive polygon compounds are exactly regular polygon compounds. However for higher ranks the notions begin to differ more significantly:

• Of the polyhedra the only (planar) polytope compound which is flag-transitive is the stella octangula. This is a regular compound in the sense of Coxeter and McMullen, however it is one of five regular polyhedron compounds. The other four are of course not flag-transitive.
• Of the polychora the compound of two 5-cells is flag-transitive, but because it's convex hull is not a regular polychoron, it is not a regular compound by either of the definitions provided.

Weak regularity

Weak regularity is the condition that the symmetry group of a polytope or compound acts transitively on each rank of elements. For example a weakly-regular polyhedron is vertex, edge and face transitive. This is a weakening of the flag-transitivity as all flag-transitive polytopes are also weakly regular.

The weakly regular compounds end up being a better approximation of the regular compounds than the flag-transitive compounds were. The weakly-regular polygon compounds are exactly the flag-transitive polygon compounds are exactly the regular polygon compounds. In addition it turns out that the weakly-regular polyhedron compounds are exactly the regular polytope compounds.

However this pattern does not extend to rank 4. The compound of two 5-cells is naturally weakly regular, however as before it is still not a regular compound because of its hull.

Regular compounds by rank

Regular polyhedron compounds

There are 5 finite planar regular polyhedron compounds. These are exactly the weakly-regular polyhedron compounds. The first, the stella octangula is truely flag-transitive. The second two the chiricosahedron and the icosicosahedron are both vertex-regular and facet-regular. The last two are a dual pair, one being vertex-regular and the other facet-regular.

Regular polyhedra compounds
Name Picture Compound of Convex hull
Stella octangula[note 1] 2 tetrahedra Cube
Chiricosahedron[note 2] 5 tetrahedra Dodecahedron
Icosicosahedron 10 tetrahedra Dodecahedron
Rhombihedron 5 cubes Dodecahedron
Small icosicosahedron 5 octahedra Rectified dodecahedron

Notes

1. โ Stella octangula is flag transitive.
2. โ Chiricosahedron is chiral and thus its pair is also a regular compound.

Bibliography

• Coxeter, Donald (1973). Regular polytopes (3 ed.). Dover. ISBN 0-486-61480-8. OCLC 798003.
• McMullen, Peter (2018). "New regular compounds of 4-polytopes". New trends in intuitive geometry: 307โ320.