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]

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.

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

Examples

Convex			Concave
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

Faces	Face config.	Class	Name	Symmetry	Order	Convex	Coplanar	Nonconvex
4	V3^3	Platonic	tetrahedron tetragonal disphenoid rhombic disphenoid	Td, [3,3], (*332) D2d, [2^+,2], (2*) D2, [2,2]^+, (222)	24 4 4 4
6	V3^4	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	V4^3	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	V3^5	Platonic	regular dodecahedron pyritohedron tetartoid	Ih, [5,3], (*532) Th, [3^+,4], (3*2) T, [3,3]^+, (*332)	120 24 12
20	V5^3	Platonic	regular icosahedron	Ih, [5,3], (*532)	120
12	V3.6^2	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	V3.8^2	Catalan	triakis octahedron	Oh, [4,3], (*432)	48
24	V4.6^2	Catalan	tetrakis hexahedron	Oh, [4,3], (*432)	48
24	V3.4^3	Catalan	deltoidal icositetrahedron	Oh, [4,3], (*432)	48
48	V4.6.8	Catalan	disdyakis dodecahedron	Oh, [4,3], (*432)	48
24	V3^4.4	Catalan	pentagonal icositetrahedron	O, [4,3]^+, (432)	24
30	V(3.5)^2	Catalan	rhombic triacontahedron	Ih, [5,3], (*532)	120
60	V3.10^2	Catalan	triakis icosahedron	Ih, [5,3], (*532)	120
60	V5.6^2	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	V3^4.5	Catalan	pentagonal hexecontahedron	I, [5,3]^+, (532)	60
2n	V3^3.n	Polar	trapezohedron asymmetric trapezohedron	Dnd, [2^+,2n], (2*n) Dn, [2,n]^+, (22n)	4n 2n
2n 4n	V4^2.n V4^2.2n V4^2.2n	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.

Related terms

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).

See also

• Edge-transitive
• Anisohedral tiling

Notes

1. McLean, K. Robin (1990), "Dungeons, dragons, and dice", The Mathematical Gazette, 74 (469): 243–256, doi:10.2307/3619822, 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. arXiv:0708.2663. doi:10.1007/bf02986203. S2CID 119365079. 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

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