Isohedral figure

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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]

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.

Examples[edit]

Convex Concave
Hexagonale bipiramide.png
The hexagonal bipyramid, V4.4.6 is a nonregular example of an isohedral polyhedron.
Tiling Dual Semiregular V3-3-4-3-4 Cairo Pentagonal.svg
The isohedral Cairo pentagonal tiling, V3.3.4.3.4
Rhombic dodecahedra.png
The rhombic dodecahedral honeycomb is an example of an isohedral (and isochoric) space-filling honeycomb.
Capital I4 tiling-4color.svg
Topological square tiling distorted into spiraling I shapes.

Classes of isohedra by symmetry[edit]

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
TetrahedronDisphenoid tetrahedron.pngRhombic 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
CubeTrigonalTrapezohedron.svgTrigonal 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
OctahedronSquare bipyramid.pngRhombic bipyramid.png4-scalenohedron-01.png4-scalenohedron-025.png4-scalenohedron-05.png 4-scalenohedron-15.png
12 V53 Platonic regular dodecahedron
pyritohedron
tetartoid
Ih, [5,3], (*532)
Th, [3+,4], (3*2)
T, [3,3]+, (*332)
120
24
24
DodecahedronPyritohedron.pngTetartoid.png Tetartoid cubic.pngTetartoid tetrahedral.png Concave pyritohedral dodecahedron.pngStar pyritohedron-1.49.png
20 V35 Platonic regular icosahedron Ih, [5,3], (*532) 120 Icosahedron
12 V3.62 Catalan triakis tetrahedron Td, [3,3], (*332) 24 Triakis tetrahedron Triakis tetrahedron cubic.pngTriakis tetrahedron tetrahedral.png 5-cell net.png
12 V(3.4)2 Catalan rhombic dodecahedron
trapezoidal dodedecahedron
Oh, [4,3], (*432)
Td, [3,3], (*332)
48
24
Rhombic dodecahedronSkew rhombic dodecahedron-116.pngSkew rhombic dodecahedron-150.png Skew rhombic dodecahedron-200.png Skew rhombic dodecahedron-250.pngSkew rhombic dodecahedron-450.png
24 V3.82 Catalan triakis octahedron Oh, [4,3], (*432) 48 Triakis octahedron Stella octangula.svgExcavated octahedron.png
24 V4.62 Catalan tetrakis hexahedron Oh, [4,3], (*432) 48 Tetrakis hexahedronPyramid augmented cube.png Tetrakis hexahedron cubic.pngTetrakis hexahedron tetrahedral.png Tetrahemihexacron.pngExcavated cube.png
24 V3.43 Catalan deltoidal icositetrahedron Oh, [4,3], (*432) 48 Deltoidal icositetrahedronDeltoidal icositetrahedron gyro.png Partial cubic honeycomb.pngDeltoidal icositetrahedron octahedral.pngDeltoidal icositetrahedron octahedral gyro.png Deltoidal icositetrahedron concave-gyro.png
48 V4.6.8 Catalan disdyakis dodecahedron Oh, [4,3], (*432) 48 Disdyakis dodecahedron Disdyakis dodecahedron cubic.pngDisdyakis dodecahedron octahedral.pngRhombic dodeca.png Hexahemioctacron.pngDU20 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 Tetrahedra augmented icosahedron.pngFirst stellation of icosahedron.pngGreat dodecahedron.pngPyramid excavated icosahedron.png
60 V5.62 Catalan pentakis dodecahedron Ih, [5,3], (*532) 120 Pentakis dodecahedron Pyramid augmented dodecahedron.pngSmall stellated dodecahedron.pngGreat stellated dodecahedron.pngDU58 great pentakisdodecahedron.pngThird stellation of icosahedron.png
60 V3.4.5.4 Catalan deltoidal hexecontahedron Ih, [5,3], (*532) 120 Deltoidal hexecontahedron Deltoidal hexecontahedron on icosahedron dodecahedron.png Rhombic hexecontahedron.png
120 V4.6.10 Catalan disdyakis triacontahedron Ih, [5,3], (*532) 120 Disdyakis triacontahedron Disdyakis triacontahedron dodecahedral.pngDisdyakis triacontahedron icosahedral.pngDisdyakis triacontahedron rhombic triacontahedral.png Small dodecahemidodecacron.pngCompound of five octahedra.pngExcavated 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
TrigonalTrapezohedron.svgTetragonal trapezohedron.pngPentagonal trapezohedron.pngHexagonal trapezohedron.png
Trigonal trapezohedron gyro-side.pngTwisted 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 Triangular bipyramid.pngSquare bipyramid.pngPentagonale bipiramide.pngHexagonale bipiramide.png Pentagram Dipyramid.png7-2 dipyramid.png7-3 dipyramid.png8-3 dipyramid.png8-3-bipyramid zigzag.png8-3-bipyramid-inout.png8-3-dipyramid zigzag inout.png

k-isohedral figure[edit]

A polyhedron (or polytope in general) is k-isohedral if it contains k faces within its symmetry fundamental domain.[3]

Similarly a k-isohedral tiling has k separate symmetry orbits (and may contain m different shaped faces for some m < k).[4]

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).[5]

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

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.[6]

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[edit]

Notes[edit]

  1. ^ McLean, K. Robin (1990), "Dungeons, dragons, and dice", The Mathematical Gazette, 74 (469): 243–256, JSTOR 3619822.
  2. ^ Grünbaum (1960)
  3. ^ 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.
  4. ^ Craig S. Kaplan. "Introductory Tiling Theory for Computer Graphics". 2009. Chapter 5 "Isohedral Tilings". p. 35.
  5. ^ Tilings and Patterns, p.20, 23
  6. ^ http://www.polytope.net/hedrondude/dice4.htm

References[edit]

  • Peter R. Cromwell, Polyhedra, Cambridge University Press 1997, ISBN 0-521-55432-2, p. 367 Transitivity

External links[edit]