Sommerville's missing tetrahedra
type: Item
instance of: scholarly article
title: Sommerville's missing tetrahedra
zbMATH Open document ID: 1120.52007
author: Allan L. Edmonds
published in: Discrete \& Computational Geometry
publication date: 2007-04-26T00:00:00Z
review text: In 1923 \textit{D. M. Y. Sommerville} [Edinb. R. S. Proc. 43, 85--116 (1923; JFM 49.0411.03)] determined a list of four tetrahedra such that there exist face-to-face tilings of the Euclidean space by properly congruent copies of a tetrahedron. The paper under review considers the case which was not examined by Sommerville. It is proved that the missing tetrahedron does not admit any face-to-face tiling of the Euclidean space if the copies of tetrahedron are properly congruent, i.e. congruent by an orientation-preserving isometry. Therefore, the Sommerville's classification is complete. However this tetrahedron admits tilings that are either non-proper face-to-face or proper non-face-to-face. Remark that no assumptions about a symmetry group of the tiling are made.
reviewed by: Elizaveta Zamorzaeva
Mathematics Subject Classification ID: 52C22
Mathematics Subject Classification ID: 05B45
zbMATH DE Number: 5146121
zbMATH Keywords: tilings of the Euclidean space
zbMATH Keywords: tilings by tetrahedra
zbMATH Keywords: face-to-face tilings
MaRDI profile type: Publication
full work available at URL: s00454-006-1293-4
OpenAlex ID: W2032167602
DOI: 10.1007/S00454-006-1293-4
description: scientific article; zbMATH DE number 5146121