Platonic bodies, sphere packings and hyperbolic geometry
type: Item
instance of: scholarly article
title: Platonic bodies, sphere packings and hyperbolic geometry
zbMATH Open document ID: 0999.52014
published in: Mathematische Semesterberichte
publication date: 2001-12-05T00:00:00Z
review text: In the first part of the paper, the author describes more or less the content of his Habilitationsvortrag from 1994, reviewing the theory of polygonal tilings of the sphere and of the hyperbolic plane. The second half introduces to some more recent developments which are closely related to this, in particular, there is a description of some punctured hyperbolic surfaces which are extreme in their moduli space with respect to, e.g., length of the systoles. As a general phenomenon, these extreme surfaces are quotients of the hyperbolic plane by some arithmetically defined group. The fact is stressed that the hyperbolic world is richer than the euclidean one. The Gauss conjecture on the infinity of real quadratic fields with class number one is formulated in a geometric way. Reviewer's remark: The author mainly summarizes results. For proofs the reader is referred to the literature.
reviewed by: Stefan Kühnlein
Mathematics Subject Classification ID: 52C20
Mathematics Subject Classification ID: 11F99
Mathematics Subject Classification ID: 32G15
Mathematics Subject Classification ID: 53C22
zbMATH DE Number: 1508091
zbMATH Keywords: regular bodies
zbMATH Keywords: systoles
zbMATH Keywords: arithmetic groups
zbMATH Keywords: sphere packings
author: Paul Schmutz Schaller
MaRDI profile type: Publication
full work available at URL: s005910070012
OpenAlex ID: W2074390696
DOI: 10.1007/S005910070012
description: scientific article; zbMATH DE number 1508091