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.

Collins considers isotoxals degenerate if any two adjacent edges are colinear. Without this constraint, taking a regular polytope and subdividing every edge into two congruent edges would produce an isotoxal polytope.

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
Isotoxal polyhedra have some rather strong constraints. All faces must be isotoxal, and any face that is not regular must have its containing plane passing through the center of the polyhedron. Thus any isotoxal polyhedron that does not have all regular faces must be a hemipolyhedron.

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).

By Collins' review in 2023, the isotoxals break down into 9 regulars, 16 non-regular uniforms, 10 non-regular isohedra, and 11 special cases, for a total of 46 isotoxals. The set has not yet been proven complete. There are more known isotoxals if compounds and tilings are allowed.

Isotoxal + isogonal + isotopic
A polyhedron that is isotoxal, isogonal, and isotopic is transitive on all of its elements and thus is weakly regular. The finite 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, it must be uniform, and an even number of faces meet at each vertex with the two face types alternating in the vertex figure. The polyhedra that meet these criteria 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):
 * 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. Three are uniform duals, and the other three are stellations of the icosahedron:


 * Rhombic dodecahedron
 * 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:


 * 11 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