Isohedral figure

Short description: ≥2-dimensional tessellation or ≥3-dimensional polytope with identical faces

A set of isohedral dice

In geometry, a tessellation of dimension 2 (a plane tiling) or higher, or a polytope of dimension 3 (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent but must be transitive, i.e. must lie within the same symmetry orbit. In other words, for any two faces A and B, there must be a symmetry of the entire figure by translations, rotations, and/or reflections that maps A onto B. For this reason, convex isohedral polyhedra are the shapes that will make fair dice.[1]

Isohedral polyhedra are called isohedra. They can be described by their face configuration. An isohedron has an even number of faces.

The dual of an isohedral polyhedron is vertex-transitive, i.e. isogonal. The Catalan solids, the bipyramids, and the trapezohedra are all isohedral. They are the duals of the (isogonal) Archimedean solids, prisms, and antiprisms, respectively. The Platonic solids, which are either self-dual or dual with another Platonic solid, are vertex-, edge-, and face-transitive (i.e. isogonal, isotoxal, and isohedral).

A form that is isohedral, has regular vertices, and is also edge-transitive (i.e. isotoxal) is said to be a quasiregular dual. Some theorists regard these figures as truly quasiregular because they share the same symmetries, but this is not generally accepted.

A polyhedron which is isohedral and isogonal is said to be noble.

Not all isozonohedra[2] are isohedral.[3] For example, a rhombic icosahedron is an isozonohedron but not an isohedron.[4]

## Contents

* 1 Examples
* 2 Classes of isohedra by symmetry
* 3 k-isohedral figure
* 4 Related terms
* 5 See also
* 6 References
* 7 External links

## Examples

Convex | Concave

|

Hexagonal bipyramids, V4.4.6, are nonregular isohedral polyhedra.  |
The Cairo pentagonal tiling, V3.3.4.3.4, is isohedral.  |
The rhombic dodecahedral honeycomb is isohedral (and isochoric, and space-filling).  |
A square tiling distorted into a spiraling H tiling (topologically equivalent) is still isohedral.

## Classes of isohedra by symmetry

Faces | Face
config. | Class | Name | Symmetry | Order | Convex | Coplanar | Nonconvex

|
|
|
|
|
|
|
|

4  | V33 | Platonic  | tetrahedron
tetragonal disphenoid
rhombic disphenoid  | Td, [3,3], (*332)
D2d, [2+,2], (2*)
D2, [2,2]+, (222)  | 24

4  | 60px |  |
6  | V34 | Platonic  | cube
trigonal trapezohedron
asymmetric trigonal trapezohedron  | Oh, [4,3], (*432)
D3d, [2+,6]
(2*3)

[2,3]+, (223)  | 48

6  | 30px |  |
8  | V43 | Platonic  | octahedron
square bipyramid
rhombic bipyramid
square scalenohedron  | Oh, [4,3], (*432)
D4h,[2,4],(*224)
D2h,[2,2],(*222)
D2d,[2+,4],(2*2)  | 48

8  | 60px60px60px60px60px |  |
12  | V35 | Platonic  | regular dodecahedron
pyritohedron
tetartoid  | Ih, [5,3], (*532)
Th, [3+,4], (3*2)
T, [3,3]+, (*332)  | 120

12  | 60px60px | 60px60px | 60px
20  | V53 | Platonic  | regular icosahedron | Ih, [5,3], (*532)  | 120  |  |  |
12  | V3.62 | Catalan  | triakis tetrahedron | Td, [3,3], (*332)  | 24  |  | 60px60px |
12  | V(3.4)2 | Catalan  | rhombic dodecahedron
deltoidal dodecahedron  | Oh, [4,3], (*432)
Td, [3,3], (*332)  | 48
24  | 60px60px | 60px | 60px
24  | V3.82 | Catalan  | triakis octahedron | Oh, [4,3], (*432)  | 48  |  |  | 60px
24  | V4.62 | Catalan  | tetrakis hexahedron | Oh, [4,3], (*432)  | 48  | 60px | 60px60px | 60px
24  | V3.43 | Catalan  | deltoidal icositetrahedron | Oh, [4,3], (*432)  | 48  | 60px | 60px60px60px |
48  | V4.6.8  | Catalan  | disdyakis dodecahedron | Oh, [4,3], (*432)  | 48  |  | 60px60px60px | 60px
24  | V34.4  | Catalan  | pentagonal icositetrahedron | O, [4,3]+, (432)  | 24  |  |  |
30  | V(3.5)2 | Catalan  | rhombic triacontahedron | Ih, [5,3], (*532)  | 120  |  |  |
60  | V3.102 | Catalan  | triakis icosahedron | Ih, [5,3], (*532)  | 120  |  |  | 60px60px60px
60  | V5.62 | Catalan  | pentakis dodecahedron | Ih, [5,3], (*532)  | 120  |  |  | 60px60px60px60px
60  | V3.4.5.4  | Catalan  | deltoidal hexecontahedron | Ih, [5,3], (*532)  | 120  |  | 120px |
120  | V4.6.10  | Catalan  | disdyakis triacontahedron | Ih, [5,3], (*532)  | 120  |  | 60px60px60px | 60px60px
60  | V34.5  | Catalan  | pentagonal hexecontahedron | I, [5,3]+, (532)  | 60  |  |  |
2n | V33.n | Polar  | trapezohedron
asymmetric trapezohedron  | Dnd, [2+,2n], (2*n)
Dn, [2,n]+, (22n)  | 4n
2n | 60px60px60px
60px |  |

4n | V42.n
V42.2n
V42.2n | Polar  | regular n-bipyramid
isotoxal 2n-bipyramid
2n-scalenohedron  | Dnh, [2,n], (*22n)
Dnh, [2,n], (*22n)
Dnd, [2+,2n], (2*n)  | 4n | 60px60px60px |  | 60px60px60px60px60px60px

## k-isohedral figure

A polyhedron (or polytope in general) is k-isohedral if it contains k faces within its symmetry fundamental domains.[5] Similarly, a k-isohedral tiling has k separate symmetry orbits (it may contain m different face shapes, for m = k, or only for some m < k).[6] ("1-isohedral" is the same as "isohedral".)

A monohedral polyhedron or monohedral tiling (m = 1) has congruent faces, either directly or reflectively, which occur in one or more symmetry positions. An m-hedral polyhedron or tiling has m different face shapes ("dihedral", "trihedral"... are the same as "2-hedral", "3-hedral"... respectively).[7]

Here are some examples of k-isohedral polyhedra and tilings, with their faces colored by their k symmetry positions:

3-isohedral  | 4-isohedral  | isohedral  | 2-isohedral

|
|
|

2-hedral regular-faced polyhedra  | Monohedral polyhedra
|  |  |
The rhombicuboctahedron has 1 triangle type and 2 square types.  | The pseudo-rhombicuboctahedron has 1 triangle type and 3 square types.  | The deltoidal icositetrahedron has 1 face type.  | The pseudo-deltoidal icositetrahedron has 2 face types, with same shape.
2-isohedral  | 4-isohedral  | Isohedral  | 3-isohedral

|
|
|

2-hedral regular-faced tilings  | Monohedral tilings
|  |  |
The Pythagorean tiling has 2 square types (sizes).  | This 3-uniform tiling has 3 triangle types, with same shape, and 1 square type.  | The herringbone pattern has 1 rectangle type.  | This pentagonal tiling has 3 irregular pentagon types, with same shape.

## Related terms

A cell-transitive or isochoric figure is an n-polytope (n ≥ 4) or n-honeycomb (n ≥ 3) that has its cells congruent and transitive with each others. In 3 dimensions, the catoptric honeycombs, duals to the uniform honeycombs, are isochoric. In 4 dimensions, isochoric polytopes have been enumerated up to 20 cells.[8]

A facet-transitive or isotopic figure is an n-dimensional polytope or honeycomb with its facets ((n−1)-faces) congruent and transitive. The dual of an isotope is an isogonal polytope. By definition, this isotopic property is common to the duals of the uniform polytopes.

* An isotopic 2-dimensional figure is isotoxal, i.e. edge-transitive.
* An isotopic 3-dimensional figure is isohedral, i.e. face-transitive.
* An isotopic 4-dimensional figure is isochoric, i.e. cell-transitive.

## See also

* Edge-transitive
* Anisohedral tiling

## References

1. ↑ McLean, K. Robin (1990), "Dungeons, dragons, and dice", The Mathematical Gazette 74 (469): 243–256, doi:10.2307/3619822 .
2. ↑ Weisstein, Eric W.. "Isozonohedron" (in en). http://mathworld.wolfram.com/Isozonohedron.html.
3. ↑ Weisstein, Eric W.. "Isohedron" (in en). http://mathworld.wolfram.com/Isohedron.html.
4. ↑ Weisstein, Eric W.. "Rhombic Icosahedron" (in en). http://mathworld.wolfram.com/RhombicIcosahedron.html.
5. ↑ Socolar, Joshua E. S. (2007). "Hexagonal Parquet Tilings: k-Isohedral Monotiles with Arbitrarily Large k" (corrected PDF). The Mathematical Intelligencer 29: 33–38. doi:10.1007/bf02986203. http://www.phy.duke.edu/~socolar/hexparquet.pdf. Retrieved 2007-09-09.
6. ↑ Craig S. Kaplan, "Introductory Tiling Theory for Computer Graphics" , 2009, Chapter 5: "Isohedral Tilings", p. 35.
7. ↑ Tilings and Patterns, p. 20, 23.
8. ↑ "Four Dimensional Dice up to Twenty Sides". http://www.polytope.net/hedrondude/dice4.htm.

## External links

* Olshevsky, George. "Isotope". Glossary for Hyperspace. Archived from the original on 4 February 2007. https://web.archive.org/web/20070204075028/members.aol.com/Polycell/glossary.html#Isotope.
* Weisstein, Eric W.. "Isohedral tiling". http://mathworld.wolfram.com/IsohedralTiling.html.
* Weisstein, Eric W.. "Isohedron". http://mathworld.wolfram.com/Isohedron.html.
* isohedra 25 classes of isohedra with a finite number of sides
* Dice Design at The Dice Lab

* 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

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