# Isotoxal polytope

A polytope is **isotoxal** or **edge-transitive** if its edges are identical under its symmetry group. In other words, given any two edges, there is a symmetry of the polytope that transforms one into the other. Clearly, an isotoxal polytope must have only one edge length. Isotoxal polytopes as a group are much less studied than isotopic (facet-transitive) and isogonal (vertex-transitive) polytopes.

All regular polytopes are of course isotoxal.

## Isotoxal polygons[edit | edit source]

Non-regular isotoxal polygons have an even number of vertices, which lie on two concentric circles and alternate between the two circles in a zigzag. For all such polygons the radii of the two circles can be continuously varied.

## Isotoxal polyhedra[edit | edit source]

Whereas face-transitivity or vertex-transitivity alone are fairly weak conditions on polyhedra, edge-transitivity is a much more restrictive condition. There is only a finite number of isotoxal polyhedra, and none have degrees of freedom.

Assuming polyhedra that are finite, planar, not compounds, and having no adjacent colinear edges, Collins's complete enumeration in 2022 classified the set of isotoxal polyhedra into 9 regulars, 16 non-regular uniforms, 11 non-regular isohedra, and 11 special cases for a total of 47 polyhedra.

It follows from dyadicity that an isotoxal polyhedron can have at most two distinct vertex types and at most two distinct face types (that is, "distinct" according to the polyhedron's symmetry).

Every polyhedron that is both isotoxal and isogonal can have each of its edges bisected into two colinear edges each to produce another isotoxal polyhedron. These are considered degenerate in the enumeration of isotoxal polyhedra as they do not add much interesting diversity to the set.

### Isotoxal + isogonal + isotopic[edit | edit source]

A polyhedron that is isotoxal, isogonal, and isotopic is transitive on all of its elements and thus is weakly regular. The finite, non-skew polyhedra that are weakly regular are simply the regular polyhedra.

However the rhombic tiling is a weakly regular, but not regular, Euclidean tiling.

There are also four polytope compounds which are transitive on all their elements, but not regular. These are the regular compounds, excluding the stella octangula which is regular.

### Isotoxal + isogonal + not isotopic[edit | edit source]

If a polyhedron is both isotoxal and isogonal, but not isotopic, it is quasiregular according to the definition of Coxeter.^{[1]} These polyhedra are:

- Cuboctahedron
- Icosidodecahedron
- Great icosidodecahedron
- Small ditrigonary icosidodecahedron
- Great ditrigonary icosidodecahedron
- Dodecadodecahedron
- Ditrigonary dodecadodecahedron
- Nine of the ten uniform hemipolyhedra (the great dirhombicosidodecahedron is not isotoxal):

### Isotoxal + isotopic + not isogonal[edit | edit source]

The following well-known polyhedra are isotoxal and isotopic, but not isogonal. All are uniform duals of the isotoxal isogonals that are not hemipolyhedra (as uniform hemipolyhedra have degenerate duals):

- Rhombic dodecahedron
- Rhombic triacontahedron
- Medial rhombic triacontahedron
- Great rhombic triacontahedron
- Medial triambic icosahedron
- Small triambic icosahedron
- Great triambic icosahedron

In 2022, Gordon Collins found four variants of Kepler-Poinsot solids that also fit this description:

- Proper great stellated dodecahedron
- Proper small stellated dodecahedron
- Overlapped great stellated dodecahedron
- Overlapped small stellated dodecahedron

They are also uniform duals, but their duals are exotic, being the two dyadic interpretations each of the small complex icosidodecahedron and great complex icosidodecahedron.

### Isotoxal + not isotopic + not isogonal[edit | edit source]

In 2022, Gordon Collins introduced the following previously unknown isotoxals that are neither isotopic nor isogonal, all of which are hemipolyhedra:

- Cubohemiicositetrahedron
- Octahemiicositetrahedron (the only orientable one)
- Small hexagrammic hemihexecontahedron
- Small 3-decagrammic hemihexecontahedron
- Great 4-decagrammic hemihexecontahedron
- Great hexagrammic hemihexecontahedron
- Small 4-decagrammic hemihexecontahedron
- Great 3-decagrammic hemihexecontahedron
- Decagonic dihemidodecahedron (contains colinear edges)
- Hexagonic dihemiicosahedron
- Decagrammic dihemidodecahedron (contains colinear edges)

## References[edit | edit source]

## Bibliography[edit | edit source]

- Collins, Gordon. "Edge-Transitive Polyhedra."
- Coxeter, Donald (1973).
*Regular polytopes*(3 ed.). Dover. ISBN 0-486-61480-8. OCLC 798003.