Английская Википедия:Isohedral figure

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Файл:Dice Set.jpg

A set of isohedral dice

In geometry, a tessellation of dimension Шаблон:Math (a plane tiling) or higher, or a polytope of dimension Шаблон:Math (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 Шаблон:Mvar and Шаблон:Mvar, there must be a symmetry of the entire figure by translations, rotations, and/or reflections that maps Шаблон:Mvar onto Шаблон:Mvar. 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. 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 are isohedral. For example, a rhombic icosahedron is an isozonohedron but not an isohedron.

Examples

Convex			Concave
Файл:Hexagonale bipiramide.png Hexagonal bipyramids, V4.4.6, are nonregular isohedral polyhedra.	Файл:Tiling Dual Semiregular V3-3-4-3-4 Cairo Pentagonal.svg The Cairo pentagonal tiling, V3.3.4.3.4, is isohedral.	Файл:Rhombic dodecahedra.png The rhombic dodecahedral honeycomb is isohedral (and isochoric, and space-filling).	Файл:Capital I4 tiling-4color.svg 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	TetrahedronФайл:Disphenoid tetrahedron.pngФайл:Rhombic disphenoid.png
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	CubeФайл:TrigonalTrapezohedron.svgФайл:Trigonal trapezohedron gyro-side.png
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	OctahedronФайл:Square bipyramid.pngФайл:Rhombic bipyramid.pngФайл:4-scalenohedron-01.pngФайл:4-scalenohedron-025.pngФайл:4-scalenohedron-05.png		Файл:4-scalenohedron-15.png
12	V35	Platonic	regular dodecahedron pyritohedron tetartoid	Ih, [5,3], (*532) Th, [3+,4], (3*2) T, [3,3]+, (*332)	120 24 12	DodecahedronФайл:Pyritohedron.pngФайл:Tetartoid.png	Файл:Tetartoid cubic.pngФайл:Tetartoid tetrahedral.png	Файл:Concave pyritohedral dodecahedron.pngФайл:Star pyritohedron-1.49.png
20	V53	Platonic	regular icosahedron	Ih, [5,3], (*532)	120	Icosahedron
12	V3.62	Catalan	triakis tetrahedron	Td, [3,3], (*332)	24	Triakis tetrahedron	Файл:Triakis tetrahedron cubic.pngФайл:Triakis tetrahedron tetrahedral.png	Файл:5-cell net.png
12	V(3.4)2	Catalan	rhombic dodecahedron deltoidal dodecahedron	Oh, [4,3], (*432) Td, [3,3], (*332)	48 24	Rhombic dodecahedronФайл:Skew rhombic dodecahedron-116.pngФайл:Skew rhombic dodecahedron-150.png	Файл:Skew rhombic dodecahedron-200.png	Файл:Skew rhombic dodecahedron-250.pngФайл:Skew rhombic dodecahedron-450.png
24	V3.82	Catalan	triakis octahedron	Oh, [4,3], (*432)	48	Triakis octahedron		Файл:Stella octangula.svgФайл:Excavated octahedron.png
24	V4.62	Catalan	tetrakis hexahedron	Oh, [4,3], (*432)	48	Tetrakis hexahedronФайл:Pyramid augmented cube.png	Файл:Tetrakis hexahedron cubic.pngФайл:Tetrakis hexahedron tetrahedral.png	Файл:Tetrahemihexacron.pngФайл:Excavated cube.png
24	V3.43	Catalan	deltoidal icositetrahedron	Oh, [4,3], (*432)	48	Deltoidal icositetrahedronФайл:Deltoidal icositetrahedron gyro.png	Файл:Partial cubic honeycomb.pngФайл:Deltoidal icositetrahedron octahedral.pngФайл:Deltoidal icositetrahedron octahedral gyro.png	Файл:Deltoidal icositetrahedron concave-gyro.png
48	V4.6.8	Catalan	disdyakis dodecahedron	Oh, [4,3], (*432)	48	Disdyakis dodecahedron	Файл:Disdyakis dodecahedron cubic.pngФайл:Disdyakis dodecahedron octahedral.pngФайл:Rhombic dodeca.png	Файл:Hexahemioctacron.pngФайл:DU20 great disdyakisdodecahedron.png
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		Файл:Tetrahedra augmented icosahedron.pngФайл:First stellation of icosahedron.pngФайл:Great dodecahedron.pngФайл:Pyramid excavated icosahedron.png
60	V5.62	Catalan	pentakis dodecahedron	Ih, [5,3], (*532)	120	Pentakis dodecahedron		Файл:Pyramid augmented dodecahedron.pngФайл:Small stellated dodecahedron.pngФайл:Great stellated dodecahedron.pngФайл:DU58 great pentakisdodecahedron.pngФайл:Third stellation of icosahedron.svg
60	V3.4.5.4	Catalan	deltoidal hexecontahedron	Ih, [5,3], (*532)	120	Deltoidal hexecontahedron	Файл:Deltoidal hexecontahedron on icosahedron dodecahedron.png	Файл:Rhombic hexecontahedron.png
120	V4.6.10	Catalan	disdyakis triacontahedron	Ih, [5,3], (*532)	120	Disdyakis triacontahedron	Файл:Disdyakis triacontahedron dodecahedral.pngФайл:Disdyakis triacontahedron icosahedral.pngФайл:Disdyakis triacontahedron rhombic triacontahedral.png	Файл:Small dodecahemidodecacron.pngФайл:Compound of five octahedra.pngФайл:Excavated rhombic triacontahedron.png
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	Файл:TrigonalTrapezohedron.svgФайл:Tetragonal trapezohedron.pngФайл:Pentagonal trapezohedron.pngФайл:Hexagonal trapezohedron.png Файл:Trigonal trapezohedron gyro-side.pngФайл:Twisted hexagonal trapezohedron.png
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	Файл:Triangular bipyramid.pngФайл:Square bipyramid.pngФайл:Pentagonal bipyramid.pngФайл:Hexagonale bipiramide.png		Файл:Pentagram Dipyramid.pngФайл:7-2 dipyramid.pngФайл:7-3 dipyramid.pngФайл:8-3 dipyramid.pngФайл:8-3-bipyramid zigzag.pngФайл:8-3-bipyramid-inout.pngФайл:8-3-dipyramid zigzag inout.png

k-isohedralШаблон:Anchor figure

A polyhedron (or polytope in general) is k-isohedral if it contains k faces within its symmetry fundamental domains. 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). ("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).

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
Файл:Small rhombicuboctahedron.png	Файл:Johnson solid 37.png	Файл:Deltoidal icositetrahedron gyro.png	Файл:Pseudo-strombic icositetrahedron (2-isohedral).png
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
Файл:Distorted truncated square tiling.png	Файл:3-uniform n57.png	Файл:Herringbone bond.svg	Файл:P5-type10.png
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.

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

External links

• Шаблон:GlossaryForHyperspace
• Шаблон:MathWorld
• Шаблон:MathWorld
• isohedra 25 classes of isohedra with a finite number of sides
• Dice Design at The Dice Lab