Isohedral figure

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]

"isohedron" redirects here. Not to be confused with icosahedron.
For the related Isohedral numbers, see Anisohedral tiling.
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
ConvexConcave

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
FacesFace
config.		ClassNameSymmetryOrderConvexCoplanarNonconvex
4 V33 Platonic tetrahedron
tetragonal disphenoid
rhombic disphenoid		 Td, [3,3], (*332)
D2d, [2+,2], (2*)
D2, [2,2]+, (222)		 24
4
4
4		 Tetrahedron
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		 Cube
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		 Octahedron
12 V35 Platonic regular dodecahedron
pyritohedron
tetartoid		 Ih, [5,3], (*532)
Th, [3+,4], (3*2)
T, [3,3]+, (*332)		 120
24
12		 Dodecahedron
20 V53 Platonic regular icosahedron Ih, [5,3], (*532) 120 Icosahedron
12 V3.62 Catalan triakis tetrahedron Td, [3,3], (*332) 24 Triakis tetrahedron
12 V(3.4)2 Catalan rhombic dodecahedron
deltoidal dodecahedron		 Oh, [4,3], (*432)
Td, [3,3], (*332)		 48
24		 Rhombic dodecahedron
24 V3.82 Catalan triakis octahedron Oh, [4,3], (*432) 48 Triakis octahedron
24 V4.62 Catalan tetrakis hexahedron Oh, [4,3], (*432) 48 Tetrakis hexahedron
24 V3.43 Catalan deltoidal icositetrahedron Oh, [4,3], (*432) 48 Deltoidal icositetrahedron
48 V4.6.8 Catalan disdyakis dodecahedron Oh, [4,3], (*432) 48 Disdyakis dodecahedron
24 V34.4 Catalan pentagonal icositetrahedron O, [4,3]+, (432) 24 Pentagonal icositetrahedron
30 V(3.5)2 Catalan rhombic triacontahedron Ih, [5,3], (*532) 120 Rhombic triacontahedron
60 V3.102 Catalan triakis icosahedron Ih, [5,3], (*532) 120 Triakis icosahedron
60 V5.62 Catalan pentakis dodecahedron Ih, [5,3], (*532) 120 Pentakis dodecahedron
60 V3.4.5.4 Catalan deltoidal hexecontahedron Ih, [5,3], (*532) 120 Deltoidal hexecontahedron
120 V4.6.10 Catalan disdyakis triacontahedron Ih, [5,3], (*532) 120 Disdyakis triacontahedron
60 V34.5 Catalan pentagonal hexecontahedron I, [5,3]+, (532) 60 Pentagonal hexecontahedron
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.

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

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: 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".
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