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

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

Whereas face-transitivity or vertex-transitivity alone are fairly weak conditions on polyhedra, edge-transitivity appears to be a quite restrictive condition. Only a finite number of isotoxal polyhedra are known, and none of the known figures have any degrees of freedom.

Assuming polyhedra that are finite, planar, not compounds, and having no adjacent colinear edges, Collins' review in 2023 classified the known 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. The set has not yet been proven complete.

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

Main article: Weakly regular polytope § Weakly regular polyhedra

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

If a polyhedron is both isotoxal and isogonal, but not isotopic, an even number of faces meet at each vertex with the two face types alternating in the vertex figure. The known polyhedra that meet these criteria are all uniform:

• 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):
  • Tetrahemihexahedron
  • Octahemioctahedron
  • Cubohemioctahedron
  • Small dodecahemidodecahedron
  • Small icosihemidodecahedron
  • Great dodecahemicosahedron
  • Small dodecahemicosahedron
  • Great icosihemidodecahedron
  • Great dodecahemidodecahedron

Isotoxal + isotopic + not isogonal

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 2022-23, 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

Isotoxal + not isotopic + not isogonal

In 2022-23, Gordon Collins introduced the following previously unknown isotoxals that are neither isotopic nor isogonal, all of which are hemipolyhedra:

• Cubohemiicositetrahedron
• Octahemiicositetrahedron
• Small hexagrammic hemihexecontahedron
• Small 2-decagrammic hemihexecontahedron
• Great 4-decagrammic hemihexecontahedron
• Great hexagrammic hemihexecontahedron
• Small 4-decagrammic hemihexecontahedron
• Great 2-decagrammic hemihexecontahedron
• Decagonal dihemidodecahedron
• Hexagonal dihemiicosahedron
• Decagrammic dihemidodecahedron

References

• Collins, Gordon. "Edge-Transitive Polyhedra."