Isohedral figure

In geometry, a polytope of dimension 3 (a polyhedron) or higher is isohedral or face-transitive when 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 faces A and B, there must be a symmetry of the entire solid by rotations and reflections that maps A onto B. For this reason, convex isohedral polyhedra are the shapes that will make fair dice.[1]

"isohedron" redirects here. It is 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. A form that is isohedral and has regular vertices is also edge-transitive (isotoxal) and 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. An isohedron has an even number of faces.[2]

A polyhedron which is isohedral has a dual polyhedron that is vertex-transitive (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 (isogonal, isotoxal, and isohedral). A polyhedron which is isohedral and isogonal is said to be noble.

Note: not all isozonohedra[3] are isohedral.[4] Example: a rhombic icosahedron is an isozonohedron but not an isohedron.[5]

Examples
ConvexConcave

The hexagonal bipyramid, V4.4.6 is a nonregular example of an isohedral polyhedron.
The isohedral Cairo pentagonal tiling, V3.3.4.3.4
The rhombic dodecahedral honeycomb is an example of an isohedral (and isochoric) space-filling honeycomb.
Topological square tiling distorted into spiraling H shapes.

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
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 domain.[6]

Similarly a k-isohedral tiling has k separate symmetry orbits (and may contain m different shaped faces for some m < k).[7]

A monohedral polyhedron or monohedral tiling (m=1) has congruent faces, as either direct or reflectively, which occur in one or more symmetry positions. An r-hedral polyhedra or tiling has r types of faces (also called dihedral, trihedral for 2 or 3 respectively).[8]

Here are some example 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 type of triangle and 2 types of squares The pseudo-rhombicuboctahedron has 1 type of triangle and 3 types of squares. The deltoidal icositetrahedron has with 1 type of face. The pseudo-deltoidal icositetrahedron has 2 types of identical-shaped faces.
2-isohedral 4-isohedral Isohedral 3-isohedral
(2-hedral) regular-faced tilings Monohedral tilings
The Pythagorean tiling has 2 sizes of squares. This 3-uniform tiling has 3 types identical-shaped triangles and 1 type of square. The herringbone pattern has 1 type of rectangular face. This pentagonal tiling has 3 types of identical-shaped irregular pentagon faces.

A cell-transitive or isochoric figure is an n-polytope (n>3) or honeycomb that has its cells congruent and transitive with each other. In 3-dimensional honeycombs, the catoptric honeycombs, duals to the uniform honeycombs are isochoric. In 4-dimensions, isochoric polytopes have been enumerated up to 20 cells.[9]

A facet-transitive or isotopic figure is a n-dimensional polytopes 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 (edge-transitive).
  • An isotopic 3-dimensional figure is isohedral (face-transitive).
  • An isotopic 4-dimensional figure is isochoric (cell-transitive).
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See also

Notes
  1. McLean, K. Robin (1990), "Dungeons, dragons, and dice", The Mathematical Gazette, 74 (469): 243–256, JSTOR 3619822.
  2. Grünbaum (1960)
  3. Weisstein, Eric W. "Isozonohedron". mathworld.wolfram.com. Retrieved 2019-12-26.
  4. Weisstein, Eric W. "Isohedron". mathworld.wolfram.com. Retrieved 2019-12-21.
  5. Weisstein, Eric W. "Rhombic Icosahedron". mathworld.wolfram.com. Retrieved 2019-12-21.
  6. 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. Retrieved 2007-09-09.
  7. Craig S. Kaplan. "Introductory Tiling Theory for Computer Graphics". 2009. Chapter 5 "Isohedral Tilings". p. 35.
  8. Tilings and Patterns, p.20, 23
  9. http://www.polytope.net/hedrondude/dice4.htm

References
  • Peter R. Cromwell, Polyhedra, Cambridge University Press 1997, ISBN 0-521-55432-2, p. 367 Transitivity
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