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Tiling Three‐Space by Combinatorially Equivalent Convex Polytopes
Author(s) -
Schulte Egon
Publication year - 1984
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-49.1.128
Subject(s) - polytope , mathematics , combinatorics , convex polytope , regular polygon , euclidean space , face (sociological concept) , space (punctuation) , convex body , convex hull , convex set , discrete mathematics , geometry , convex optimization , computer science , social science , sociology , operating system
The paper settles a problem of Danzer, Grünbaum, and Shephard on tilings by convex polytopes. We prove that, for a given three‐dimensional convex polytope P , there is a locally finite tiling of the Euclidean three‐space by convex polytopes each combinatorially equivalent to P . In general, face‐to‐face tilings will not exist.