List of isotoxal polyhedra and tilings

The dual of a convex polyhedron is also a convex polyhedron.^[2]

There are nine convex isotoxal polyhedra based on the Platonic solids: the five (regular) Platonic solids, the two (quasiregular) common cores of dual Platonic solids, and their two duals.

The vertex figures of the quasiregular forms are (squares or) rectangles; the vertex figures of the duals of the quasiregular forms are (equilateral triangles and equilateral triangles, or) equilateral triangles and squares, or equilateral triangles and regular pentagons.

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

The dual of a non-convex polyhedron is also a non-convex polyhedron.^[2] (By contraposition.)

There are ten non-convex isotoxal polyhedra based on the quasiregular octahedron, cuboctahedron, and icosidodecahedron: the five (quasiregular) hemipolyhedra based on the quasiregular octahedron, cuboctahedron, and icosidodecahedron, and their five (infinite) duals:

Form	Quasiregular	Quasiregular dual
p=3 q=3	Tetrahemihexahedron	Tetrahemihexacron
p=4 q=3	Cubohemioctahedron	Hexahemioctacron
	Octahemioctahedron	Octahemioctacron (visually indistinct from Hexahemioctacron) (*)
p=5 q=3	Small icosihemidodecahedron	Small icosihemidodecacron (visually indistinct from Small dodecahemidodecacron) (*)
	Small dodecahemidodecahedron	Small dodecahemidodecacron

(*) Faces, edges, and intersection points are the same; only, some other of these intersection points, not at infinity, are considered as vertices.

There are sixteen non-convex isotoxal polyhedra based on the Kepler–Poinsot polyhedra: the four (regular) Kepler–Poinsot polyhedra, the six (quasiregular) common cores of dual Kepler–Poinsot polyhedra (including four hemipolyhedra), and their six duals (including four (infinite) hemipolyhedron-duals):

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 dodecahemicosahedron	Great dodecahemicosacron

Finally, there are six other non-convex isotoxal polyhedra: the three quasiregular ditrigonal (3 | p q) star polyhedra, and their three 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 at least 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}