List of isotoxal polyhedra and tilings

In geometry, isotoxal polyhedra and tilings are defined by the property that they have symmetries taking any edge to any other edge.[1] Polyhedra with this property can also be called "edge-transitive", but they should be distinguished from edge-transitive graphs, where the symmetries are combinatorial rather than geometric.

Regular polyhedra are isohedral (face-transitive), isogonal (vertex-transitive), and isotoxal (edge-transitive).

Quasiregular polyhedra are isogonal and isotoxal, but not isohedral; their duals are isohedral and isotoxal, but not isogonal.

The dual of an isotoxal polyhedron is also an isotoxal polyhedron. (See the Dual polyhedron article.)

## Convex isotoxal polyhedra

[edit]

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

## Isotoxal star-polyhedra

[edit]

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 dodecahemidodecahedron |
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

## Isotoxal tilings of the Euclidean plane

[edit]

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}

## 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 |  |  |  |
[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 isotoxal 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

[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 be semiregular, alternating two lunes, and thus isotoxal:

* hosohedron {2,q}
* dihedron {p,2}

## References

[edit]

1. ^ Peter R. Cromwell, Polyhedra, Cambridge University Press 1997, ISBN 0-521-55432-2, p. 371
2. ^ a b "duality". maths.ac-noumea.nc. Retrieved 2020-10-01.

* Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 0-7167-1193-1. (6.4 Isotoxal tilings, 309–321)
* Coxeter, Harold Scott MacDonald; Longuet-Higgins, M. S.; Miller, J. C. P. (1954), "Uniform polyhedra", Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences, 246 (916): 401–450, Bibcode:1954RSPTA.246..401C, doi:10.1098/rsta.1954.0003, ISSN 0080-4614, JSTOR 91532, MR 0062446, S2CID 202575183

* v
* t
* e

Tessellation

Periodic

* Pythagorean
* Rhombille
* Schwarz triangle
* Rectangle
* Domino
* Uniform tiling and honeycomb
* Coloring
* Convex
* Kisrhombille
* Wallpaper group
* Wythoff

Aperiodic

* Ammann–Beenker
* Aperiodic set of prototiles
* List
* Einstein problem
* Socolar–Taylor
* Gilbert
* Penrose
* Pinwheel
* Quaquaversal
* Rep-tile and Self-tiling
* Sphinx
* Socolar
* Truchet

Other

* Anisohedral and Isohedral
* Architectonic and catoptric
* Circle Limit III
* Computer graphics
* Honeycomb
* Isotoxal
* List
* Packing
* Pentagonal
* Problems
* Domino
* Wang
* Heesch's
* Squaring
* Dividing a square into similar rectangles
* Prototile
* Conway criterion
* Girih
* Regular Division of the Plane
* Regular grid
* Substitution
* Voronoi
* Voderberg

By vertex type

| Spherical|

* 2n
* 33.n
* V33.n
* 42.n
* V42.n

|

Regular|

* 2∞
* 36
* 44
* 63

Semi-
regular|

* 32.4.3.4
* V32.4.3.4
* 33.42
* 33.∞
* 34.6
* V34.6
* 3.4.6.4
* (3.6)2
* 3.122
* 42.∞
* 4.6.12
* 4.82

Hyper-
bolic|

* 32.4.3.5
* 32.4.3.6
* 32.4.3.7
* 32.4.3.8
* 32.4.3.∞
* 32.5.3.5
* 32.5.3.6
* 32.6.3.6
* 32.6.3.8
* 32.7.3.7
* 32.8.3.8
* 33.4.3.4
* 32.∞.3.∞
* 34.7
* 34.8
* 34.∞
* 35.4
* 37
* 38
* 3∞
* (3.4)3
* (3.4)4
* 3.4.62.4
* 3.4.7.4
* 3.4.8.4
* 3.4.∞.4
* 3.6.4.6
* (3.7)2
* (3.8)2
* 3.142
* 3.162
* 3.∞2
* 42.5.4
* 42.6.4
* 42.7.4
* 42.8.4
* 42.∞.4
* 45
* 46
* 47
* 48
* 4∞
* (4.5)2
* (4.6)2
* 4.6.12
* 4.6.14
* V4.6.14
* 4.6.16
* V4.6.16
* 4.6.∞
* (4.7)2
* (4.8)2
* 4.8.10
* V4.8.10
* 4.8.12
* 4.8.14
* 4.8.16
* 4.8.∞
* 4.102
* 4.10.12
* 4.122
* 4.12.16
* 4.142
* 4.162
* 4.∞2
* 54
* 55
* 56
* 5∞
* 5.4.6.4
* (5.6)2
* 5.82
* 5.102
* 5.122
* 64
* 65
* 66
* 68
* 6.4.8.4
* (6.8)2
* 6.82
* 6.102
* 6.122
* 6.162
* 73
* 74
* 77
* 7.62
* 7.82
* 7.142
* 83
* 84
* 86
* 88
* 8.62
* 8.122
* 8.162
* ∞3
* ∞4
* ∞5
* ∞∞
* ∞.62
* ∞.82

*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template