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.)

## Contents

* 1 Convex isotoxal polyhedra
* 2 Isotoxal star-polyhedra
* 3 Isotoxal tilings of the Euclidean plane
* 4 Isotoxal tilings of the hyperbolic plane
* 5 Isotoxal tilings of the sphere
* 6 References

## Convex isotoxal polyhedra

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

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  | 75px
Tetrahemihexahedron |
Tetrahemihexacron
p=4
q=3  | 75px
Cubohemioctahedron |
Hexahemioctacron

Octahemioctahedron  |
Octahemioctacron (visually indistinct from Hexahemioctacron) (*)
p=5
q=3  | 75px
Small icosihemidodecahedron |
Small icosihemidodecacron (visually indistinct from Small dodecahemidodecacron) (*)
75px
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  | 75px
Great stellated dodecahedron
{5/2,3}

3 | 2 5/2

| 75px
Great icosahedron
{3,5/2}

5/2 | 2 3

| 75px
Great icosidodecahedron

2 | 3 5/2  |
Great rhombic triacontahedron
75px
Great icosihemidodecahedron |
Great icosihemidodecacron
75px
Great dodecahemidodecahedron |
Great dodecahemidodecacron
p=5/2
q=5  | 75px
Small stellated dodecahedron
{5/2,5}

5 | 2 5/2

| 75px
Great dodecahedron
{5,5/2}

5/2 | 2 5

| 75px
Dodecadodecahedron

2 | 5 5/2  |
Medial rhombic triacontahedron
75px
Small icosihemidodecahedron |
Small dodecahemicosacron
75px
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  |
75px
Great ditrigonal icosidodecahedron
3/2 | 3 5

Great triambic icosahedron
75px
Ditrigonal dodecadodecahedron
3 | 5/3 5
| Error creating thumbnail: Unable to save thumbnail to destination
Medial triambic icosahedron
75px
Small ditrigonal icosidodecahedron
3 | 5/2 3

Small triambic icosahedron

## Isotoxal tilings of the Euclidean plane

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  | Error creating thumbnail: Unable to save thumbnail to destination
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 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

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

1. ↑ Peter R. Cromwell, Polyhedra, Cambridge University Press 1997, ISBN 0-521-55432-2, p. 371
2. ↑ 2.0 2.1 "duality". http://maths.ac-noumea.nc/polyhedr/dual_.htm.

* Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 0-7167-1193-1. https://archive.org/details/isbn_0716711931. (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, doi:10.1098/rsta.1954.0003, ISSN 0080-4614

* v
* t
* e

Tessellation

Periodic

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

Aperiodic

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

Other

* Anisohedral & Isohedral
* Architectonic & catoptric
* Circle Limit III
* Computer graphics
* Honeycomb
* Isotoxal
* List
* Packing
* Problems
* Domino
* Wang
* Heesch's
* Squaring
* 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
* 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
* (4.∞)2
* 54
* 55
* 56
* 5∞
* 5.4.6.4
* (5.6)2
* 5.82
* 5.102
* 5.122
* (5.∞)2
* 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
* 812
* 8.62
* 8.162
* ∞3
* ∞4
* ∞5
* ∞∞
* ∞.62
* ∞.82

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/List of isotoxal polyhedra and tilings. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=List_of_isotoxal_polyhedra_and_tilings&oldid=229133"
*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template