Realizations of the abstract regular H polyhedra.

Regular polyhedra and related structures such as complexes and nets play a prominent role in the study of materials such as crystals, nanotubes and viruses. An abstract regular polyhedron {\cal P} is the combinatorial analog of a classical regular geometric polyhedron. It is a partially ordered set of elements called faces that are completely characterized by a string C-group (G, T), which consists of a group G generated by a set T of involutions.

Geometric realizations of abstract regular polyhedra with automorphism group H.

A geometric realization of an abstract polyhedron {\cal P} is a mapping that sends an i-face to an open set of dimension i. This work adapts a method based on Wythoff construction to generate a full rank realization of an abstract regular polyhedron from its automorphism group Γ. The method entails finding a real orthogonal representation of Γ of degree 3 and applying its image to suitably chosen (not necessarily connected) open sets in space.