Isohedron

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.

Examples

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)
6	V34	Platonic	cube trigonal trapezohedron asymmetric trigonal trapezohedron	Oh, [4,3], (*432) D3d, [2+,6], (2*3) D3, [2,3]+, (223)
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)
12	V53	Platonic	dodecahedron pyritohedron tetartoid	Ih, [5,3], (*532) Th, [3+,4], (3*2) T, [3,3]+, (*332)	60px
20	V35	Platonic	icosahedron	Ih, [5,3], (*532)
12	V3.62	Catalan	triakis tetrahedron	Td, [3,3], (*332)	60px60px
12	V(3.4)2	Catalan	rhombic dodecahedron trapezoidal dodedecahedron	Oh, [4,3], (*432) Td, [3,3], (*332)
24	V3.82	Catalan	triakis octahedron	Oh, [4,3], (*432)	60px60px
24	V4.62	Catalan	tetrakis hexahedron	Oh, [4,3], (*432)	Tetrakis hexahedron60px60px60px
24	V3.43	Catalan	deltoidal icositetrahedron	Oh, [4,3], (*432)	60px60px60px
48	V4.6.8	Catalan	disdyakis dodecahedron	Oh, [4,3], (*432)	60px60px60px60px
24	V34.4	Catalan	pentagonal icositetrahedron	O, [4,3]+, (432)
30	V(3.5)2	Catalan	rhombic triacontahedron	Ih, [5,3], (*532)	60px
60	V3.102	Catalan	triakis icosahedron	Ih, [5,3], (*532)	60px
60	V5.62	Catalan	pentakis dodecahedron	Ih, [5,3], (*532)	60px
60	V3.4.5.4	Catalan	deltoidal hexecontahedron	Ih, [5,3], (*532)
120	V4.6.10	Catalan	disdyakis triacontahedron	Ih, [5,3], (*532)	60px60px60px60px
60	V34.5	Catalan	pentagonal hexecontahedron	I, [5,3]+, (532)
2n	V33.n	Polar	trapezohedron asymmetric trapezohedron	Dnd, [2+,2n], (2*n) Dn, [2,n]+, (22n)	60px
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)	60px

References

• Weisstein, Eric W., "Isohedron", MathWorld.
• Branko Grünbaum On Polyhedra in E³ having all faces congruent. Bull. Research Council Israel 8F, 215-218, 1960.

External links

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