Geometric realizations of abstract regular polyhedra with automorphism group H3

• Main Authors: Loyola, Mark L, Aranas, Jonn Angel L
• Format: text
• Published: Archīum Ateneo 2020
• Subjects: abstract regular polyhedra; geometric realizations; non-crystallographic Coxeter group H3; string C-groups; Geometry and Topology; Mathematics
• Online Access: https://archium.ateneo.edu/mathematics-faculty-pubs/143; https://archium.ateneo.edu/cgi/viewcontent.cgi?article=1142&context=mathematics-faculty-pubs
• Institution: Ateneo De Manila University

A geometric realization of an abstract polyhedron 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 Gamma. The method entails finding a real orthogonal representation of Gamma of degree 3 and applying its image to suitably chosen (not necessarily connected) open sets in space. To demonstrate the use of the method, it is applied to the abstract polyhedra whose automorphism groups are isomorphic to the non-crystallographic Coxeter group H3.