An isohedron is aconvex polyhedron with symmetries acting transitively on its faces with respect to the center of gravity. Every isohedron has aneven number of faces (Grünbaum 1960). The isohedra make fairdice, and there are 30 of them (including finite solids and infinite classes of solids). AllPlatonic solids, regulardipyramids, and regulartrapezohedra are isohedra, as are someArchimedean duals.

The finite isohedra are the cube,disdyakis dodecahedron,deltoidal hexecontahedron,deltoidal icositetrahedron,disdyakis triacontahedron,dodecahedron, dyakis dodecahedron, hexakis tetrahedron,icosahedron, octahedral pentagonal dodecahedron,octahedron,pentagonal hexecontahedron,pentagonal icositetrahedron, pentakis dodecahedron,rhombic dodecahedron,rhombic triacontahedron,small triakis octahedron, tetragonal pentagonal dodecahedron,tetrahedron,tetrakis hexahedron, trapezoidal dodecahedron,great triakis octahedron, andtriakis tetrahedron.

Infinite families of isohedra where face shapes can be adjusted are given by the generalisosceles tetrahedra (includingisosceles tetrahedra with isosceles faces).

Infinite families of isohedra where the number of faces can be varied are given bydipyramids andtrapezohedra.

Infinite families of isohedra where the number of faces and face shapes can be varied are given by the in-out skewed dipyramids, up-down skewed dipyramids, and trapezohedra with asymmetric sides.

A two-dimensionallamina such as acoincan also be viewed as a degenerate case of a fair 2-sided solid.

See also

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• isohedron
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References

Grünbaum, B. "On Polyhedra in Having All Faces Congruent."Bull. Research Council Israel8F, 215-218, 1960.

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Cite this as:

Weisstein, Eric W. "Isohedron." FromMathWorld--A Wolfram Resource.https://mathworld.wolfram.com/Isohedron.html

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