Geometric realizations of regular abstract polyhedra with automorphism group $H_3$

Abstract

A \textit{geometric realization} of an abstract polyhedron $\mathcal{P}$ is a mapping $\rho : \mathcal{P} \to \mathbb{E}^3$ 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 a regular abstract polyhedron from its automorphism group $\Gamma$. The method entails finding a real orthogonal representation of $\Gamma$ of degree 3 and applying its image to suitably chosen open sets in space. To demonstrate the use of the method, we apply it to the abstract polyhedra whose automorphism groups are isomorphic to the non-crystallographic Coxeter group $H_3$.