Isohedral figure

Last updated
A set of isohedral dice Dice Set.jpg
A set of isohedral dice

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. [1]

Contents

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 [2] are isohedral. [3] For example, a rhombic icosahedron is an isozonohedron but not an isohedron. [4]

Examples

ConvexConcave
Hexagonale bipiramide.png
Hexagonal bipyramids, V4.4.6, are nonregular isohedral polyhedra.
Tiling Dual Semiregular V3-3-4-3-4 Cairo Pentagonal.svg
The Cairo pentagonal tiling, V3.3.4.3.4, is isohedral.
Rhombic dodecahedra.png
The rhombic dodecahedral honeycomb is isohedral (and isochoric, and space-filling).
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.
ClassNameSymmetryOrderConvexCoplanarNonconvex
4V33Platonic tetrahedron
tetragonal disphenoid
rhombic disphenoid
Td, [3,3], (*332)
D2d, [2+,2], (2*)
D2, [2,2]+, (222)
24
4
4
4
Tetrahedron.png Disphenoid tetrahedron.png Rhombic disphenoid.png
6V34Platonic cube
trigonal trapezohedron
asymmetric trigonal trapezohedron
Oh, [4,3], (*432)
D3d, [2+,6]
(2*3)
D3
[2,3]+, (223)
48
12
12
6
Hexahedron.png TrigonalTrapezohedron.svg Trigonal trapezohedron gyro-side.png
8V43Platonic 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
Octahedron.png Square bipyramid.png Rhombic bipyramid.png 4-scalenohedron-01.png 4-scalenohedron-025.png 4-scalenohedron-05.png 4-scalenohedron-15.png
12V35Platonic regular dodecahedron
pyritohedron
tetartoid
Ih, [5,3], (*532)
Th, [3+,4], (3*2)
T, [3,3]+, (*332)
120
24
12
Dodecahedron.png Pyritohedron.png Tetartoid.png Tetartoid cubic.png Tetartoid tetrahedral.png Concave pyritohedral dodecahedron.png Star pyritohedron-1.49.png
20V53Platonic regular icosahedron Ih, [5,3], (*532)120 Icosahedron.png
12V3.62Catalan triakis tetrahedron Td, [3,3], (*332)24 Triakis tetrahedron.png Triakis tetrahedron cubic.png Triakis tetrahedron tetrahedral.png 5-cell net.png
12V(3.4)2Catalan rhombic dodecahedron
deltoidal dodecahedron
Oh, [4,3], (*432)
Td, [3,3], (*332)
48
24
Rhombic dodecahedron.png Skew rhombic dodecahedron-116.png Skew rhombic dodecahedron-150.png Skew rhombic dodecahedron-200.png Skew rhombic dodecahedron-250.png Skew rhombic dodecahedron-450.png
24V3.82Catalan triakis octahedron Oh, [4,3], (*432)48 Triakis octahedron.png Stella octangula.svg Excavated octahedron.png
24V4.62Catalan tetrakis hexahedron Oh, [4,3], (*432)48 Disdyakis cube.png Pyramid augmented cube.png Tetrakis hexahedron cubic.png Tetrakis hexahedron tetrahedral.png Tetrahemihexacron.png Excavated cube.png
24V3.43Catalan deltoidal icositetrahedron Oh, [4,3], (*432)48 Strombic icositetrahedron.png Deltoidal icositetrahedron gyro.png Partial cubic honeycomb.png Deltoidal icositetrahedron octahedral.png Deltoidal icositetrahedron octahedral gyro.png Deltoidal icositetrahedron concave-gyro.png
48V4.6.8Catalan disdyakis dodecahedron Oh, [4,3], (*432)48 Disdyakis dodecahedron.png Disdyakis dodecahedron cubic.png Disdyakis dodecahedron octahedral.png Rhombic dodeca.png Hexahemioctacron.png DU20 great disdyakisdodecahedron.png
24V34.4Catalan pentagonal icositetrahedron O, [4,3]+, (432)24 Pentagonal icositetrahedron.png
30V(3.5)2Catalan rhombic triacontahedron Ih, [5,3], (*532)120 Rhombic triacontahedron.png
60V3.102Catalan triakis icosahedron Ih, [5,3], (*532)120 Triakis icosahedron.png Tetrahedra augmented icosahedron.png First stellation of icosahedron.png Great dodecahedron.png Pyramid excavated icosahedron.png
60V5.62Catalan pentakis dodecahedron Ih, [5,3], (*532)120 Pentakis dodecahedron.png Pyramid augmented dodecahedron.png Small stellated dodecahedron.png Great stellated dodecahedron.png DU58 great pentakisdodecahedron.png Third stellation of icosahedron.svg
60V3.4.5.4Catalan deltoidal hexecontahedron Ih, [5,3], (*532)120 Strombic hexecontahedron.png Deltoidal hexecontahedron on icosahedron dodecahedron.png Rhombic hexecontahedron.png
120V4.6.10Catalan disdyakis triacontahedron Ih, [5,3], (*532)120 Disdyakis triacontahedron.png Disdyakis triacontahedron dodecahedral.png Disdyakis triacontahedron icosahedral.png Disdyakis triacontahedron rhombic triacontahedral.png Small dodecahemidodecacron.png Compound of five octahedra.png Excavated rhombic triacontahedron.png
60V34.5Catalan pentagonal hexecontahedron I, [5,3]+, (532)60 Pentagonal hexecontahedron.png
2nV33.nPolar trapezohedron
asymmetric trapezohedron
Dnd, [2+,2n], (2*n)
Dn, [2,n]+, (22n)
4n
2n
TrigonalTrapezohedron.svg Tetragonal trapezohedron.png Pentagonal trapezohedron.png Hexagonal trapezohedron.png
Trigonal trapezohedron gyro-side.png Twisted hexagonal trapezohedron.png
2n
4n
V42.n
V42.2n
V42.2n
Polarregular n-bipyramid
isotoxal 2n-bipyramid
2n-scalenohedron
Dnh, [2,n], (*22n)
Dnh, [2,n], (*22n)
Dnd, [2+,2n], (2*n)
4n Triangular bipyramid.png Square bipyramid.png Pentagonal bipyramid.png Hexagonale bipiramide.png Pentagram Dipyramid.png 7-2 dipyramid.png 7-3 dipyramid.png 8-3 dipyramid.png 8-3-bipyramid zigzag.png 8-3-bipyramid-inout.png 8-3-dipyramid zigzag inout.png

k-isohedral figure

A polyhedron (or polytope in general) is k-isohedral if it contains k faces within its symmetry fundamental domains. [5] 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). [6] ("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). [7]

Here are some examples of k-isohedral polyhedra and tilings, with their faces colored by their k symmetry positions:

3-isohedral4-isohedralisohedral2-isohedral
2-hedral regular-faced polyhedraMonohedral polyhedra
Small rhombicuboctahedron.png Johnson solid 37.png Deltoidal icositetrahedron gyro.png 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-isohedral4-isohedralIsohedral3-isohedral
2-hedral regular-faced tilings Monohedral tilings
Distorted truncated square tiling.svg 3-uniform n57.png Herringbone bond.svg
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.

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. [8]

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.

See also

References

  1. McLean, K. Robin (1990), "Dungeons, dragons, and dice", The Mathematical Gazette, 74 (469): 243–256, doi:10.2307/3619822, JSTOR   3619822, S2CID   195047512 .
  2. Weisstein, Eric W. "Isozonohedron". mathworld.wolfram.com. Retrieved 2019-12-26.
  3. Weisstein, Eric W. "Isohedron". mathworld.wolfram.com. Retrieved 2019-12-21.
  4. Weisstein, Eric W. "Rhombic Icosahedron". mathworld.wolfram.com. Retrieved 2019-12-21.
  5. Socolar, Joshua E. S. (2007). "Hexagonal Parquet Tilings: k-Isohedral Monotiles with Arbitrarily Large k" (corrected PDF). The Mathematical Intelligencer. 29 (2): 33–38. arXiv: 0708.2663 . doi:10.1007/bf02986203. S2CID   119365079 . Retrieved 2007-09-09.
  6. Craig S. Kaplan, "Introductory Tiling Theory for Computer Graphics" Archived 2022-12-08 at the Wayback Machine , 2009, Chapter 5: "Isohedral Tilings", p. 35.
  7. Tilings and patterns, p. 20, 23.
  8. "Four Dimensional Dice up to Twenty Sides".