Isohedral figure

"isohedron" redirects here. Not to be confused with icosahedron.

For the related Isohedral numbers, see Anisohedral tiling.

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]

A set of isohedral dice

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]

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 4 4
6	V34	Platonic	cube trigonal trapezohedron asymmetric trigonal trapezohedron	Oh, [4,3], (*432) D3d, [2+,6] (2*3) D3 [2,3]+, (223)	48 12 12 6
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 16 8 8
12	V35	Platonic	regular dodecahedron pyritohedron tetartoid	Ih, [5,3], (*532) Th, [3+,4], (3*2) T, [3,3]+, (*332)	120 24 12
20	V53	Platonic	regular icosahedron	Ih, [5,3], (*532)	120
12	V3.62	Catalan	triakis tetrahedron	Td, [3,3], (*332)	24
12	V(3.4)2	Catalan	rhombic dodecahedron deltoidal dodecahedron	Oh, [4,3], (*432) Td, [3,3], (*332)	48 24
24	V3.82	Catalan	triakis octahedron	Oh, [4,3], (*432)	48
24	V4.62	Catalan	tetrakis hexahedron	Oh, [4,3], (*432)	48
24	V3.43	Catalan	deltoidal icositetrahedron	Oh, [4,3], (*432)	48
48	V4.6.8	Catalan	disdyakis dodecahedron	Oh, [4,3], (*432)	48
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
60	V5.62	Catalan	pentakis dodecahedron	Ih, [5,3], (*532)	120
60	V3.4.5.4	Catalan	deltoidal hexecontahedron	Ih, [5,3], (*532)	120
120	V4.6.10	Catalan	disdyakis triacontahedron	Ih, [5,3], (*532)	120
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
2n 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

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, JSTOR 3619822, S2CID 195047512.
2. Weisstein, Eric W. "Isozonohedron". mathworld.wolfram.com. Retrieved 2019-12-26.
3. Weisstein, Eric W. "Isohedron". mathworld.wolfram.com. Retrieved 2019-12-21.
4. Weisstein, Eric W. "Rhombic Icosahedron". mathworld.wolfram.com. Retrieved 2019-12-21.
5. Socolar, Joshua E. S. (2007). "Hexagonal Parquet Tilings: k-Isohedral Monotiles with Arbitrarily Large k" (corrected PDF). The Mathematical Intelligencer. 29 (2): 33–38. arXiv:0708.2663. doi:10.1007/bf02986203. S2CID 119365079. Retrieved 2007-09-09.
6. Craig S. Kaplan, "Introductory Tiling Theory for Computer Graphics" Archived 2022-12-08 at the Wayback Machine, 2009, Chapter 5: "Isohedral Tilings", p. 35.
7. Tilings and patterns, p. 20, 23.
8. "Four Dimensional Dice up to Twenty Sides".

External links

• Olshevsky, George. "Isotope". Glossary for Hyperspace. Archived from the original on 4 February 2007.
• Weisstein, Eric W. "Isohedral tiling". MathWorld.
• Weisstein, Eric W. "Isohedron". MathWorld.
• isohedra 25 classes of isohedra with a finite number of sides
• Dice Design at The Dice Lab