From Surf Wiki (app.surf) — the open knowledge base
Isohedral figure
Generalisation of dice with identical faces
Generalisation of dice with identical faces
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.
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 Platonic Solids, the bipyramids, and the trapezohedra are all isohedral. They are the duals of the (isogonal) Archimedean solids, Platonic 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 | |
|---|---|---|
| [[File:Hexagonale bipiramide.png | 180px]]Hexagonal bipyramids, V4.4.6, are nonregular isohedral polyhedra. |
Classes of isohedra by symmetry
| Faces | Faceconfig. | Class | Name | Symmetry | Order | Convex | Coplanar | Nonconvex | 4 | 6 | 8 | 12 | 20 | 12 | 12 | 24 | 24 | 24 | 48 | 24 | 30 | 60 | 60 | 60 | 120 | 60 | 2n | 2n4n | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| V33 | Platonic | tetrahedrontetragonal disphenoidrhombic disphenoid | Td, [3,3], (332)D2d, [2+,2], (2)D2, [2,2]+, (222) | 24 444 | [[Image:Tetrahedron.png | 60px | Tetrahedron]][[File:Disphenoid tetrahedron.png | 60px]][[File:Rhombic disphenoid.png | 60px]] | |||||||||||||||||||||||||||
| V34 | Platonic | cubetrigonal trapezohedronasymmetric trigonal trapezohedron | Oh, [4,3], (432)D3d, [2+,6](23)D3[2,3]+, (223) | 4812126 | [[Image:Hexahedron.png | 60px | Cube]][[File:TrigonalTrapezohedron.svg | 30px]][[File:Trigonal trapezohedron gyro-side.png | 60px]] | |||||||||||||||||||||||||||
| V43 | Platonic | octahedronsquare bipyramidrhombic bipyramidsquare scalenohedron | Oh, [4,3], (432)D4h,[2,4],(224)D2h,[2,2],(222)D2d,[2+,4],(22) | 481688 | [[Image:Octahedron.png | 60px | Octahedron]][[File:Square bipyramid.svg | 60px]][[File:Rhombic bipyramid.svg | 60px]][[File:4-scalenohedron-01.png | 60px]][[File:4-scalenohedron-025.png | 60px]][[File:4-scalenohedron-05.png | 60px]] | [[File:4-scalenohedron-15.png | 60px]] | ||||||||||||||||||||||
| V35 | Platonic | regular dodecahedronpyritohedrontetartoid | Ih, [5,3], (532)Th, [3+,4], (32)T, [3,3]+, (*332) | 1202412 | [[Image:Dodecahedron.png | 60px | Dodecahedron]][[File:Pyritohedron.png | 60px]][[File:Tetartoid.png | 60px]] | [[File:Tetartoid cubic.png | 60px]][[File:Tetartoid tetrahedral.png | 60px]] | [[File:Concave pyritohedral dodecahedron.png | 60px]][[File:Star_pyritohedron-1.49.png | 60px]] | |||||||||||||||||||||
| V53 | Platonic | regular icosahedron | Ih, [5,3], (*532) | 120 | [[Image:Icosahedron.png | 60px | Icosahedron]] | |||||||||||||||||||||||||||||
| V3.62 | Catalan | triakis tetrahedron | Td, [3,3], (*332) | 24 | [[Image:triakis tetrahedron.png | 60px | Triakis tetrahedron]] | [[File:Triakis tetrahedron cubic.png | 60px]][[File:Triakis tetrahedron tetrahedral.png | 60px]] | [[File:5-cell net.png | 60px]] | ||||||||||||||||||||||||
| V(3.4)2 | Catalan | rhombic dodecahedrondeltoidal dodecahedron | Oh, [4,3], (432)Td, [3,3], (332) | 4824 | [[Image:rhombic dodecahedron.png | 60px | Rhombic dodecahedron]][[File:Skew rhombic dodecahedron-116.png | 60px]][[File:Skew rhombic dodecahedron-150.png | 60px]] | [[File:Skew rhombic dodecahedron-200.png | 60px]] | [[File:Skew rhombic dodecahedron-250.png | 60px]][[File:Skew rhombic dodecahedron-450.png | 60px]] | ||||||||||||||||||||||
| V3.82 | Catalan | triakis octahedron | Oh, [4,3], (*432) | 48 | [[Image:triakis octahedron.png | 60px | Triakis octahedron]] | [[File:Stella octangula.svg | 60px]][[File:Excavated octahedron.png | 60px]] | ||||||||||||||||||||||||||
| V4.62 | Catalan | tetrakis hexahedron | Oh, [4,3], (*432) | 48 | [[File:Disdyakis cube.png | 60px | Tetrakis hexahedron]][[File:Pyramid augmented cube.png | 60px]] | [[File:Tetrakis hexahedron cubic.png | 60px]][[File:Tetrakis hexahedron tetrahedral.png | 60px]] | [[File:Tetrahemihexacron.png | 60px]][[File:Excavated cube.png | 60px]] | ||||||||||||||||||||||
| V3.43 | Catalan | deltoidal icositetrahedron | Oh, [4,3], (*432) | 48 | [[Image:Strombic icositetrahedron.png | 60px | Deltoidal icositetrahedron]][[File:Deltoidal icositetrahedron gyro.png | 60px]] | [[File:Partial cubic honeycomb.png | 60px]][[File:Deltoidal icositetrahedron octahedral.png | 60px]][[File:Deltoidal icositetrahedron octahedral gyro.png | 60px]] | [[File:Deltoidal icositetrahedron concave-gyro.png | 60px]] | ||||||||||||||||||||||
| V4.6.8 | Catalan | disdyakis dodecahedron | Oh, [4,3], (*432) | 48 | [[Image:disdyakis dodecahedron.png | 60px | Disdyakis dodecahedron]] | [[File:Disdyakis dodecahedron cubic.png | 60px]][[File:Disdyakis dodecahedron octahedral.png | 60px]][[File:Rhombic dodeca.png | 60px]] | [[File:Hexahemioctacron.png | 60px]][[File:DU20 great disdyakisdodecahedron.png | 60px]] | ||||||||||||||||||||||
| V34.4 | Catalan | pentagonal icositetrahedron | O, [4,3]+, (432) | 24 | [[Image:pentagonal icositetrahedron.png | 60px | Pentagonal icositetrahedron]] | |||||||||||||||||||||||||||||
| V(3.5)2 | Catalan | rhombic triacontahedron | Ih, [5,3], (*532) | 120 | [[Image:rhombic triacontahedron.png | 60px | Rhombic triacontahedron]] | |||||||||||||||||||||||||||||
| V3.102 | Catalan | triakis icosahedron | Ih, [5,3], (*532) | 120 | [[Image:triakis icosahedron.png | 60px | Triakis icosahedron]] | [[File:Tetrahedra augmented icosahedron.png | 60px]][[File:First stellation of icosahedron.png | 60px]][[File:Great dodecahedron.png | 60px]][[File:Pyramid excavated icosahedron.png | 60px]] | ||||||||||||||||||||||||
| V5.62 | Catalan | pentakis dodecahedron | Ih, [5,3], (*532) | 120 | [[Image:pentakis dodecahedron.png | 60px | Pentakis dodecahedron]] | [[File:Pyramid augmented dodecahedron.png | 60px]][[File:Small stellated dodecahedron.png | 60px]][[File:Great stellated dodecahedron.png | 60px]][[File:DU58 great pentakisdodecahedron.png | 60px]][[File:Third stellation of icosahedron.svg | 60px]] | |||||||||||||||||||||||
| V3.4.5.4 | Catalan | deltoidal hexecontahedron | Ih, [5,3], (*532) | 120 | [[File:Strombic hexecontahedron.png | 60px | Deltoidal hexecontahedron]] | [[File:Deltoidal hexecontahedron on icosahedron dodecahedron.png | 120px]] | [[File:Rhombic hexecontahedron.png | 60px]] | |||||||||||||||||||||||||
| V4.6.10 | Catalan | disdyakis triacontahedron | Ih, [5,3], (*532) | 120 | [[Image:disdyakis triacontahedron.png | 60px | Disdyakis triacontahedron]] | [[File:Disdyakis triacontahedron dodecahedral.png | 60px]][[File:Disdyakis triacontahedron icosahedral.png | 60px]][[File:Disdyakis triacontahedron rhombic triacontahedral.png | 60px]] | [[File:Small dodecahemidodecacron.png | 60px]][[File:Compound of five octahedra.png | 60px]][[File:Excavated rhombic triacontahedron.png | 60px]] | |||||||||||||||||||||
| V34.5 | Catalan | pentagonal hexecontahedron | I, [5,3]+, (532) | 60 | [[File:Pentagonal hexecontahedron.png | 60px | Pentagonal hexecontahedron]] | |||||||||||||||||||||||||||||
| V33.n | Polar | trapezohedronasymmetric trapezohedron | Dnd, [2+,2n], (2*n)Dn, [2,n]+, (22n) | 4n2n | [[File:TrigonalTrapezohedron.svg | 30px]][[File:Tetragonal trapezohedron.png | 60px]][[File:Pentagonal trapezohedron.png | 60px]][[File:Hexagonal trapezohedron.png | 60px]][[File:Trigonal trapezohedron gyro-side.png | 60px]][[File:Twisted hexagonal trapezohedron.png | 60px]] | |||||||||||||||||||||||||
| V42.nV42.2nV42.2n | Polar | regular n-bipyramidisotoxal 2n-bipyramid2n-scalenohedron | Dnh, [2,n], (22n)Dnh, [2,n], (22n)Dnd, [2+,2n], (2*n) | 4n | [[File:Triangular bipyramid.png | 60px]][[File:Square bipyramid.png | 60px]][[File:Pentagonal bipyramid.png | 60px]][[File:Hexagonale bipiramide.png | 60px]] | [[File:Pentagram Dipyramid.png | 60px]][[File:7-2 dipyramid.png | 60px]][[File:7-3 dipyramid.png | 60px]][[File:8-3 dipyramid.png | 60px]][[File:8-3-bipyramid zigzag.png | 60px]][[File:8-3-bipyramid-inout.png | 60px]][[File:8-3-dipyramid zigzag inout.png | 60px]] |
''k''-isohedral{{anchor|monohedral}} 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
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 | ||||
|---|---|---|---|---|---|---|---|---|---|
| [[File:Small rhombicuboctahedron.png | 160px]] | [[File:Johnson solid 37.png | 160px]] | [[File:Deltoidal icositetrahedron gyro.png | 160px]] | [[File:Pseudo-strombic icositetrahedron (2-isohedral).png | 160px]] | ||
| 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 | |||||
|---|---|---|---|---|---|---|---|---|---|---|
| [[File:Distorted truncated square tiling.svg | 160px]] | [[File:3-uniform_n57.svg | 160px]] | [[File:Herringbone bond.svg | 160px]] | [[File:P5-type10.svg | right | 160px]] | ||
| 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. |
References
References
- McLean, K. Robin. (1990). "Dungeons, dragons, and dice". The Mathematical Gazette.
- Weisstein, Eric W.. "Isozonohedron".
- Weisstein, Eric W.. "Isohedron".
- Weisstein, Eric W.. "Rhombic Icosahedron".
- Socolar, Joshua E. S.. (2007). "Hexagonal Parquet Tilings: ''k''-Isohedral Monotiles with Arbitrarily Large ''k''". The Mathematical Intelligencer.
- link. (2022-12-08 , 2009, Chapter 5: "Isohedral Tilings", p. 35.)
- [[Tilings and patterns]], p. 20, 23.
- "Four Dimensional Dice up to Twenty Sides".
This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.
Ask Mako anything about Isohedral figure — get instant answers, deeper analysis, and related topics.
Research with MakoFree with your Surf account
Create a free account to save articles, ask Mako questions, and organize your research.
Sign up freeThis content may have been generated or modified by AI. CloudSurf Software LLC is not responsible for the accuracy, completeness, or reliability of AI-generated content. Always verify important information from primary sources.
Report