Isohedron

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In geometry, an isohedron is a polyhedron with symmetries acting transitively on its faces. Their topology can be represented by a face configuration. All 5 Platonic solids and 13 Catalan solids are isohedra, as well as the infinite families of trapezohedra and bipyramids. Some forms allow geometric variations including concave and self-intersecting geometries

Convex isohedra make fair dice. An isohedron has an even number of faces.[1]

Examples[edit]

Faces Face
config.
Class Name Symmetry Image variations
4 V33 Platonic tetrahedron
tetragonal disphenoid
rhombic disphenoid
Td, [3,3], (*332)
D2d, [2+,2], (2*)
D2, [2,2]+, (222)
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)
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)
OctahedronSquare bipyramid.pngRhombic bipyramid.png4-scalenohedron-01.png4-scalenohedron-025.png4-scalenohedron-05.png4-scalenohedron-15.png
12 V53 Platonic dodecahedron
pyritohedron
tetartoid
Ih, [5,3], (*532)
Th, [3+,4], (3*2)
T, [3,3]+, (*332)
DodecahedronPyritohedron.pngTetartoid.pngTetartoid cubic.pngTetartoid tetrahedral.pngConcave pyritohedral dodecahedron.pngStar pyritohedron-1.49.png
20 V35 Platonic icosahedron Ih, [5,3], (*532) Icosahedron
12 V3.62 Catalan triakis tetrahedron Td, [3,3], (*332) Triakis tetrahedronTriakis tetrahedron cubic.pngTriakis tetrahedron tetrahedral.png5-cell net.png
12 V(3.4)2 Catalan rhombic dodecahedron
trapezoidal dodedecahedron
Oh, [4,3], (*432)
Td, [3,3], (*332)
Rhombic dodecahedronSkew rhombic dodecahedron-116.pngSkew rhombic dodecahedron-150.pngSkew rhombic dodecahedron-200.pngSkew rhombic dodecahedron-250.pngSkew rhombic dodecahedron-450.png
24 V3.82 Catalan triakis octahedron Oh, [4,3], (*432) Triakis octahedronStella octangula.svgExcavated octahedron.png
24 V4.62 Catalan tetrakis hexahedron Oh, [4,3], (*432) Tetrahemihexacron.pngTetrakis hexahedronTetrakis hexahedron cubic.pngTetrakis hexahedron tetrahedral.pngPyramid augmented cube.pngExcavated cube.png
24 V3.43 Catalan deltoidal icositetrahedron Oh, [4,3], (*432) Deltoidal icositetrahedronDeltoidal icositetrahedron gyro.pngPartial cubic honeycomb.pngDeltoidal icositetrahedron octahedral.pngDeltoidal icositetrahedron octahedral gyro.pngDeltoidal icositetrahedron concave-gyro.png
48 V4.6.8 Catalan disdyakis dodecahedron Oh, [4,3], (*432) Hexahemioctacron.pngDisdyakis dodecahedronDisdyakis dodecahedron cubic.pngDisdyakis dodecahedron octahedral.pngRhombic dodeca.pngDU20 great disdyakisdodecahedron.png
24 V34.4 Catalan pentagonal icositetrahedron O, [4,3]+, (432) Pentagonal icositetrahedron
30 V(3.5)2 Catalan rhombic triacontahedron Ih, [5,3], (*532) Rhombic triacontahedronRhombic hexecontahedron.png
60 V3.102 Catalan triakis icosahedron Ih, [5,3], (*532) Triakis icosahedronTetrahedra augmented icosahedron.pngFirst stellation of icosahedron.pngGreat dodecahedron.pngPyramid excavated icosahedron.png
60 V5.62 Catalan pentakis dodecahedron Ih, [5,3], (*532) Pentakis dodecahedronPyramid 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) Deltoidal hexecontahedron
120 V4.6.10 Catalan disdyakis triacontahedron Ih, [5,3], (*532) Small dodecahemidodecacron.pngDisdyakis triacontahedronDisdyakis triacontahedron dodecahedral.pngDisdyakis triacontahedron icosahedral.pngDisdyakis triacontahedron rhombic triacontahedral.pngCompound of five octahedra.pngExcavated rhombic triacontahedron.png
60 V34.5 Catalan pentagonal hexecontahedron I, [5,3]+, (532) Pentagonal hexecontahedron
2n V33.n Polar trapezohedron
asymmetric trapezohedron
Dnd, [2+,2n], (2*n)
Dn, [2,n]+, (22n)
TrigonalTrapezohedron.svgTrigonal trapezohedron gyro-side.pngTetragonal trapezohedron.pngPentagonal trapezohedron.pngHexagonal 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)
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

References[edit]

  1. ^ Grünbaum (1960)

External links[edit]

  • isohedra 25 classes of isohedra with a finite number of sides