List of isotoxal polyhedra and tilings

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.

Convex isotoxal polyhedra

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

Isotoxal star-polyhedra

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
			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	75px Small triambic icosahedron

Isotoxal tilings of the Euclidean plane

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 {6,3} 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}

Isotoxal tilings of the hyperbolic plane

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 are six (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}	r{p,q}	Dual r{p,q}
Coxeter-Dynkin
[7,3]	{7,3}	{3,7}	r{7,3}
[8,3]	{8,3}	{3,8}	r{8,3}
[5,4]	{5,4}	{4,5}	r{5,4}
[6,4]	{6,4}	{4,6}	r{6,4}
[8,4]	{8,4}	{4,8}	r{8,3}
[5,5]	{5,5}	{5,5}	r{5,5}

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
(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

Isotoxal tilings of the sphere

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.

• hosohedron {2,q}
• dihedron {p,2}

References

• Peter R. Cromwell, Polyhedra, Cambridge University Press 1997, ISBN 0-521-55432-2, p. 371 Transitivity
• (6.4 Isotoxal tilings, 309–321)