In geometry, isotoxal polyhedra and tilings are edge-transitive. An isotoxal polyhedron or tiling must be either isogonal (vertex-transitive) or isohedral (face-transitive) or both. Regular polyhedra are isohedral (face-transitive), isogonal (vertex-transitive) and isotoxal. Quasiregular polyhedra are isogonal and isotoxal, but not isohedral; their duals are isohedral and isotoxal, but not isogonal.
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There are nine convex isotoxal polyhedra formed from the Platonic solids. The vertex figures of the quasiregular forms are rectangles, and the vertex figure of the duals of the quasiregular are rhombi.
| Form | Regular | Dual regular | Quasiregular | Quasiregular dual |
|---|---|---|---|---|
| Wythoff symbol | q | 2 p | p | 2 q | 2 | p q | |
| Vertex configuration | pq | qp | p.q.p.q | |
| p=3 q=3 |
Tetrahedron {3,3} 3 | 2 3 |
Tetrahedron {3,3} 3 | 2 3 |
Tetratetrahedron (Octahedron) 2 | 3 3 |
Cube (Rhombic hexahedron) |
| p=4 q=3 |
Cube {4,3} 3 | 2 4 |
Octahedron {3,4} 4 | 2 3 |
Cuboctahedron 2 | 3 4 |
Rhombic dodecahedron |
| p=5 q=3 |
Dodecahedron {5,3} 3 | 2 5 |
Icosahedron {3,5} 5 | 2 3 |
Icosidodecahedron 2 | 3 5 |
Rhombic triacontahedron |
5 nonconvex hemipolyhedra are based on the octahedron, cuboctahedron and icosidodecahedron:
| Form | Quasiregular | Quasiregular dual |
|---|---|---|
| p= q= |
Tetrahemihexahedron |
Tetrahemihexacron |
| p= q= |
Cubohemioctahedron |
Hexahemioctacron |
Octahemioctahedron |
Octahemioctacron |
|
| p= q= |
Small icosihemidodecahedron |
Small icosihemidodecacron |
Small dodecahemidodecahedron |
Small dodecahemidodecacron |
There are 12 formed by the Kepler–Poinsot polyhedra, including four hemipolyhedra:
| Form | Regular | Dual regular | Quasiregular | Quasiregular dual |
|---|---|---|---|---|
| Wythoff symbol | q | 2 p | p | 2 q | 2 | p q | |
| Vertex configuration | pq | qp | p.q.p.q | |
| p=5/2 q=3 |
Great stellated dodecahedron {5/2,3} 3 | 2 5/2 |
Great icosahedron {3,5/2} 5/2 | 2 3 |
Great icosidodecahedron 2 | 3 5/2 |
Great rhombic triacontahedron |
Great icosihemidodecahedron |
Great icosihemidodecacron |
|||
Great dodecahemidodecahedron |
Great dodecahemidodecacron |
|||
| p=5/2 q=5 |
Small stellated dodecahedron {5/2,5} 5 | 2 5/2 |
Great dodecahedron {5,5/2} 5/2 | 2 5 |
Dodecadodecahedron 2 | 5 5/2 |
Medial rhombic triacontahedron |
Small icosihemidodecahedron |
Small dodecahemicosacron |
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Great dodecahemidodecahedron |
Great dodecahemicosacron |
There are a final three quasiregular (3 | p q) star polyhedra and their duals:
| Quasiregular | Quasiregular dual |
|---|---|
| 3 | p q | |
Great ditrigonal icosidodecahedron 3/2 | 3 5 |
Great triambic icosahedron |
Ditrigonal dodecadodecahedron 3 | 5/3 5 |
Medial triambic icosahedron |
Small ditrigonal icosidodecahedron 3 | 5/2 3 |
Small triambic icosahedron |
There are 5 polygonal tilings of the Euclidean plane that are isotoxal. (The self-dual square tiling recreates itself in all four forms.)
| Regular | Dual regular | Quasiregular | Quasiregular dual |
|---|---|---|---|
Hexagonal tiling {6,4} 6 | 2 3 |
Triangular tiling {3,6} 3 | 2 3 |
Trihexagonal tiling 2 | 3 6 |
Rhombille tiling |
Square tiling {4,4} 4 | 2 4 |
Square tiling {4,4} 2 | 4 4 |
Square tiling {4,4} 4 | 2 4 |
Square tiling {4,4} |
There are infinitely many isotoxal polygonal tilings of the hyperbolic plane, including the Wythoff constructions from the regular hyperbolic tilings {p,q}, and non-right (p q r) groups.
Here's 4 (p q 2) families, each with two regular forms, and one quasiregular form. All have rhombic duals of the quasiregular form, but only one is shown:
| [p,q] | {p,q} | {q,p} | t1{p,q} | Dual of t1{p,q} |
|---|---|---|---|---|
| Coxeter-Dynkin | ||||
| [5,4] | {5,4} |
{4,5} |
t1{5,4} |
|
| [5,5] | {5,5} |
{5,5} |
t1{5,5} |
|
| [7,3] | {7,3} |
{3,7} |
t1{7,3} |
|
| [8,3] | {8,3} |
{3,8} |
t1{8,3} |
Here's 3 example (p q r) families, each with 3 quasiregular forms. The duals are not shown, but have isotaxal hexagonal and octagonal faces.
| Coxeter-Dynkin | 3|node_1|p|node|q|node|r.png]] | 3|node|p|node_1|q|node|r.png]] | 3|node|p|node|q|node_1|r.png]] |
|---|---|---|---|
| (4 3 3) | 3 | 4 3 |
3 | 4 3 |
4 | 3 3 |
| (4 4 3) | 4 | 4 3 |
3 | 4 4 |
4 | 4 3 |
| (4 4 4) | 4 | 4 4 |
4 | 4 4 |
4 | 4 4 |
All isotoxal polyhedra listed above can be made as isotoxal tilings of the sphere.
In addition as spherical tilings, there are two other families which are degenerate as polyhedra. Even ordered hosohedron can semiregular, alternating two lunes, and thus isotoxal.