Regular projective polyhedra with planar faces. I
type: Item
instance of: scholarly article
title: Regular projective polyhedra with planar faces. I
zbMATH Open document ID: 0952.52011
DOI: 10.1007/PL00000128
author: Jorge L. Arocha
author: L. Montejano
author: Javier Bracho
published in: Aequationes Mathematicae
publication date: 2001-01-17T00:00:00Z
review text: A regular projective polyhedron is defined as a faithful realization in \(\mathbb{P}^3\) of a regular combinatorial polytope of rank 3. The study of regular projective polyhedra in \(\mathbb{P}^3\) is equivalent to the study of finite regular polyhedra in \(\mathbb{R}^4\), but the authors choose to study these polyhedra directly in \(\mathbb{P}^3\). More precisely, this paper deals only with Platonic projective polyhedra, i.e. the projective polyhedra which are realized as an embedded surface. Moreover it is assumed that faces are planar and the vertex figures are skew. After recalling some of the terminology of polyhedra in the projective context, the authors classify these polyhedra in two steps: \(\bullet\) A list of the Platonic polyhedra is given: an infinite family of tori \(4,4;r\), two polyhedra \(4,6;\pi/4\) and \(6,4;\pi/4\) related to the skew \(\{4,6\}\) in \(\mathbb{R}^4\), two polyhedra \(4,8;\pi/8\) and \(8,4;\pi/8\) related to the 12-cell. \(\bullet\) A trigonometric equation, the waist equation, similar to the one used by Coxeter in his work on regular skew polyhedra in 3 and 4 dimensions, is used to establish that the list is complete.
reviewed by: Charles Leytem
Mathematics Subject Classification ID: 52B11
Mathematics Subject Classification ID: 51M20
Mathematics Subject Classification ID: 51F15
zbMATH DE Number: 1436165
zbMATH Keywords: embedded polyhedron
zbMATH Keywords: regular projective polyhedron
zbMATH Keywords: Platonic projective polyhedra
zbMATH Keywords: vertex figures
zbMATH Keywords: waist equation
Wikidata QID: Q56580548
MaRDI profile type: Publication
full work available at URL: pl00000128
OpenAlex ID: W2140504886
description: scientific article; zbMATH DE number 1436165