Self-Affine Tiling of Polyhedra

We obtain a complete classification of polyhedral sets (unions of finitely many convex polyhedra) that admit self-affine tilings, i.e., partitions into parallel shifts of one set that is affinely similar to the initial one. In every dimension, there exist infinitely many nonequivalent polyhedral set...

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Bibliographic Details
Published inDoklady. Mathematics Vol. 104; no. 2; pp. 267 - 272
Main Authors Protasov, V. Yu, Zaitseva, T. I.
Format Journal Article
LanguageEnglish
Published Moscow Pleiades Publishing 01.09.2021
Springer Nature B.V
Subjects
Integers
Mathematics
Mathematics and Statistics
Parallelepipeds
Polyhedra
Quotients
Tiling
polyhedron
integer attractor
self-affinity
tiling
tile
Haar system
cone
Online AccessGet full text
ISSN1064-5624
1531-8362
DOI10.1134/S1064562421050112

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Summary:We obtain a complete classification of polyhedral sets (unions of finitely many convex polyhedra) that admit self-affine tilings, i.e., partitions into parallel shifts of one set that is affinely similar to the initial one. In every dimension, there exist infinitely many nonequivalent polyhedral sets possessing this property. Under an additional assumption that the affine similarity is defined by an integer matrix and by integer shifts (“digits”) from different quotient classes with respect to this matrix, the only polyhedral set of this kind is a parallelepiped. Applications to multivariate wavelets and to Haar systems are discussed.
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ISSN:1064-5624
1531-8362
DOI:10.1134/S1064562421050112