An isohedron is a convex polyhedron with symmetries acting transitively on its faces with respect to the center of gravity. Every isohedron has an even number of faces (Grünbaum 1960). The isohedra make fair dice, and there are 30 of them (including finite solids and infinite classes of solids). All Platonic solids, regular dipyramids, and regular trapezohedra are isohedra, as are some Archimedean 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, and triakis tetrahedron.
Infinite families of isohedra where face shapes can be adjusted are given by the general isosceles tetrahedra (including isosceles tetrahedra with isosceles faces).
Infinite families of isohedra where the number of faces can be varied are given by dipyramids and trapezohedra.
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-dimensional lamina such as a coin can also be viewed as a degenerate case of a fair 2-sided solid.
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Israel 8F, 215-218, 1960.Grünbaum, B. and Shepard, G. C.
"Spherical Tilings with Transitivity Properties." In