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.
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.
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.
The hexagonal bipyramid, V4.4.6 is a nonregular example of an isohedral polyhedron.
The isohedral Cairo pentagonal tiling, V22.214.171.124.4
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
|Td, [3,3], (*332)
D2d, [2+,2], (2*)
D2, [2,2]+, (222)
asymmetric trigonal trapezohedron
|Oh, [4,3], (*432)
|Oh, [4,3], (*432)
|Ih, [5,3], (*532)
Th, [3+,4], (3*2)
T, [3,3]+, (*332)
|20||V53||Platonic||regular icosahedron||Ih, [5,3], (*532)||120|
|12||V3.62||Catalan||triakis tetrahedron||Td, [3,3], (*332)||24|
|Oh, [4,3], (*432)
Td, [3,3], (*332)
|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||V126.96.36.199||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|
|Dnd, [2+,2n], (2*n)
Dn, [2,n]+, (22n)
|Dnh, [2,n], (*22n)
Dnh, [2,n], (*22n)
Dnd, [2+,2n], (2*n)
Similarly a k-isohedral tiling has k separate symmetry orbits (and may contain m different shaped faces for some m < k).
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).
Here are some example k-isohedral polyhedra and tilings, with their faces colored by their k symmetry positions:
|(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-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.
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).
- McLean, K. Robin (1990), "Dungeons, dragons, and dice", The Mathematical Gazette, 74 (469): 243–256, JSTOR 3619822.
- Grünbaum (1960)
- Weisstein, Eric W. "Isozonohedron". mathworld.wolfram.com. Retrieved 2019-12-26.
- Weisstein, Eric W. "Isohedron". mathworld.wolfram.com. Retrieved 2019-12-21.
- Weisstein, Eric W. "Rhombic Icosahedron". mathworld.wolfram.com. Retrieved 2019-12-21.
- 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.
- Craig S. Kaplan. "Introductory Tiling Theory for Computer Graphics". 2009. Chapter 5 "Isohedral Tilings". p. 35.
- Tilings and Patterns, p.20, 23
- Peter R. Cromwell, Polyhedra, Cambridge University Press 1997, ISBN 0-521-55432-2, p. 367 Transitivity