Isogons and isohedra

Convex three-dimensional polyhedra (cf. Polyhedron) whose polyhedral angles are equal (isogons), or all faces of which are the same (isohedra); the group of rotations (with reflections) of an isogon (isohedron) about the centre of gravity takes any vertex (face) to any other vertex (face). To each isogon corresponds a dual isohedron, and conversely. If a convex polyhedron is an isogon and an isohedron, then it is a regular polyhedron. There are 13 special cases and two infinite series of combinatorially-different isogons, which can be realized as semi-regular polyhedra.

Comments

The 13 special isogons are called Archimedean solids, and their duals Catalan solids. Some of them play a role in the geometry of numbers and crystallography. For non-convex isogons and isohedra see [a2].

References

[a1]	L. Féjes Toth, "Reguläre Figuren" , Ungar. Akad. Wissenschaft. (1965)
[a2]	B. Grünbaum, G. Shephard, "Polyhedra with transitivity properties" C.R. Math. Rep. Acad. Sci. Canada , 6 (1984) pp. 61–66