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

TL;DR

This paper presents a method to realize regular abstract polyhedra in three-dimensional space using their automorphism groups, demonstrated on polyhedra related to the non-crystallographic Coxeter group H3.

Contribution

It introduces a Wythoff construction-based approach to generate geometric realizations from automorphism groups, specifically applied to H3-related polyhedra.

Findings

Successfully constructed realizations of polyhedra with automorphism group H3.

Demonstrated the method's effectiveness for non-crystallographic Coxeter groups.

Provided explicit geometric models for abstract polyhedra in Euclidean space.

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 $H_3$.