Realizations of the Abstract Regular H3 Polyhedra

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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 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. A realization R is a mapping from P to a Euclidean G space that is compatible with the associated real orthogonal representation of G. This work discusses an approach to thetheory of realizations of abstract regular polyhedra with an emphasis on the construction of a realization and its decomposition as a blend of subrealizations. To demonstrate the approach; it is applied to the polyhedra whose automorphism groups are abstractly isomorphic to the non-crystallographic Coxeter group H3.