List of isotoxal polyhedra and tilings

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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[edit]

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
Uniform polyhedron-33-t0.png
Tetrahedron
{3,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
3 | 2 3
Uniform polyhedron-33-t2.png
Tetrahedron
{3,3}
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
3 | 2 3
Uniform polyhedron-33-t1.png
Tetratetrahedron
(Octahedron)
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
2 | 3 3
Hexahedron.svg
Cube
(Rhombic hexahedron)
p=4
q=3
Uniform polyhedron-43-t0.png
Cube
{4,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
3 | 2 4
Uniform polyhedron-43-t2.png
Octahedron
{3,4}
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
4 | 2 3
Uniform polyhedron-43-t1.png
Cuboctahedron
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
2 | 3 4
Rhombicdodecahedron.jpg
Rhombic dodecahedron
p=5
q=3
Uniform polyhedron-53-t0.png
Dodecahedron
{5,3}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
3 | 2 5
Uniform polyhedron-53-t2.png
Icosahedron
{3,5}
CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png
5 | 2 3
Uniform polyhedron-53-t1.png
Icosidodecahedron
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png
2 | 3 5
Rhombictriacontahedron.svg
Rhombic triacontahedron

Isotoxal star-polyhedra[edit]

5 nonconvex hemipolyhedra are based on the octahedron, cuboctahedron and icosidodecahedron:

Form Quasiregular Quasiregular dual
p=
q=
Tetrahemihexahedron.pngTetrahemihexahedron vertfig.png
Tetrahemihexahedron
Tetrahemihexacron.png
Tetrahemihexacron
p=
q=
Cubohemioctahedron.pngCubohemioctahedron vertfig.png
Cubohemioctahedron
Hexahemioctacron.png
Hexahemioctacron
Octahemioctahedron.pngOctahemioctahedron vertfig.png
Octahemioctahedron
Hexahemioctacron.png
Octahemioctacron
p=
q=
Small icosihemidodecahedron.pngSmall icosihemidodecahedron vertfig.png
Small icosihemidodecahedron
Small dodecahemidodecacron.png
Small icosihemidodecacron
Small dodecahemidodecahedron.pngSmall dodecahemidodecahedron vertfig.png
Small dodecahemidodecahedron
Small dodecahemidodecacron.png
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.pngGreat stellated dodecahedron vertfig.png
Great stellated dodecahedron
{5/2,3}

CDel node 1.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.pngCDel 3.pngCDel node.png
3 | 2 5/2

Great icosahedron.pngGreat icosahedron vertfig.png
Great icosahedron
{3,5/2}

CDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.pngCDel 3.pngCDel node 1.png
5/2 | 2 3

Great icosidodecahedron.pngGreat icosidodecahedron vertfig.png
Great icosidodecahedron
 
CDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node 1.pngCDel 3.pngCDel node.png
2 | 3 5/2
DU54 great rhombic triacontahedron.png
Great rhombic triacontahedron
Great icosihemidodecahedron.pngGreat icosihemidodecahedron vertfig.png
Great icosihemidodecahedron
Great dodecahemidodecacron.png
Great icosihemidodecacron
Great dodecahemidodecahedron.pngGreat dodecahemidodecahedron vertfig.png
Great dodecahemidodecahedron
Great dodecahemidodecacron.png
Great dodecahemidodecacron
p=5/2
q=5
Small stellated dodecahedron.pngSmall stellated dodecahedron vertfig.png
Small stellated dodecahedron
{5/2,5}

CDel node 1.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.pngCDel 5.pngCDel node.png
5 | 2 5/2

Great dodecahedron.pngGreat dodecahedron vertfig.png
Great dodecahedron
{5,5/2}

CDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node.pngCDel 5.pngCDel node 1.png
5/2 | 2 5

Dodecadodecahedron.pngDodecadodecahedron vertfig.png
Dodecadodecahedron
 
CDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node 1.pngCDel 5.pngCDel node.png
2 | 5 5/2
DU36 medial rhombic triacontahedron.png
Medial rhombic triacontahedron
Small dodecahemicosahedron.pngSmall dodecahemicosahedron vertfig.png
Small icosihemidodecahedron
Small dodecahemicosacron.png
Small dodecahemicosacron
Great dodecahemicosahedron.pngGreat dodecahemicosahedron vertfig.png
Great dodecahemidodecahedron
Small dodecahemicosacron.png
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.pngGreat ditrigonal icosidodecahedron vertfig.png
Great ditrigonal icosidodecahedron
3/2 | 3 5
CDel 3.pngCDel node.pngCDel d3.pngCDel rat.pngCDel d2.pngCDel node.pngCDel 5.pngCDel node 1.pngCDel 3.png
DU47 great triambic icosahedron.png
Great triambic icosahedron
Ditrigonal dodecadodecahedron.pngDitrigonal dodecadodecahedron vertfig.png
Ditrigonal dodecadodecahedron
3 | 5/3 5
CDel 3.pngCDel node.pngCDel 5.pngCDel rat.pngCDel d3.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel 3.png
DU41 medial triambic icosahedron.png
Medial triambic icosahedron
Small ditrigonal icosidodecahedron.pngSmall ditrigonal icosidodecahedron vertfig.png
Small ditrigonal icosidodecahedron
3 | 5/2 3
CDel 3.pngCDel node.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.png
DU30 small triambic icosahedron.png
Small triambic icosahedron

Isotoxal tilings of the Euclidean plane[edit]

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
Uniform tiling 63-t0.png
Hexagonal tiling
{6,3}
CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
6 | 2 3
Uniform tiling 63-t2.png
Triangular tiling
{3,6}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.png
3 | 2 3
Uniform tiling 63-t1.png
Trihexagonal tiling
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
2 | 3 6
Star rhombic lattice.png
Rhombille tiling
Uniform tiling 44-t0.svg
Square tiling
{4,4}
CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.png
4 | 2 4
Uniform tiling 44-t2.png
Square tiling
{4,4}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
2 | 4 4
Uniform tiling 44-t1.png
Square tiling
{4,4}
CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png
4 | 2 4
Uniform tiling 44-t0.svg
Square tiling
{4,4}

Isotoxal tilings of the hyperbolic plane[edit]

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 CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png CDel node.pngCDel p.pngCDel node f1.pngCDel q.pngCDel node.png
[7,3] Uniform tiling 73-t0.png
{7,3}
Uniform tiling 73-t2.png
{3,7}
Uniform tiling 73-t1.png
r{7,3}
Order73 qreg rhombic til.png
[8,3] Uniform tiling 83-t0.png
{8,3}
Uniform tiling 83-t2.png
{3,8}
Uniform tiling 83-t1.png
r{8,3}
Uniform dual tiling 433-t01-yellow.png
[5,4] Uniform tiling 54-t0.png
{5,4}
Uniform tiling 54-t2.png
{4,5}
Uniform tiling 54-t1.png
r{5,4}
Order-5-4 quasiregular rhombic tiling.png
[6,4] Uniform tiling 64-t0.png
{6,4}
Uniform tiling 64-t2.png
{4,6}
Uniform tiling 64-t1.png
r{6,4}
H2chess 246a.png
[8,4] Uniform tiling 84-t0.png
{8,4}
Uniform tiling 84-t2.png
{4,8}
Uniform tiling 84-t1.png
r{8,3}
H2chess 248a.png
[5,5] Uniform tiling 552-t0.png
{5,5}
Uniform tiling 552-t2.png
{5,5}
Uniform tiling 552-t1.png
r{5,5}
Uniform tiling 54-t2.png

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 CDel 3.pngCDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node.pngCDel r.png CDel 3.pngCDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.pngCDel r.png CDel 3.pngCDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.pngCDel r.png
(4 3 3) Uniform tiling 433-t0.png
3 | 4 3
Uniform tiling 433-t1.png
3 | 4 3
Uniform tiling 433-t2.png
4 | 3 3
(4 4 3) Uniform tiling 443-t0.png
4 | 4 3
Uniform tiling 443-t1.png
3 | 4 4
Uniform tiling 443-t2.png
4 | 4 3
(4 4 4) Uniform tiling 444-t0.png
4 | 4 4
Uniform tiling 444-t1.png
4 | 4 4
Uniform tiling 444-t2.png
4 | 4 4

Isotoxal tilings of the sphere[edit]

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.

References[edit]