Duality, sections and projections of certain euclidean tilings
type: Item
instance of: scholarly article
title: Duality, sections and projections of certain euclidean tilings
zbMATH Open document ID: 0796.52004
DOI: 10.1007/BF01610620
published in: Geometriae Dedicata
publication date: 1994-09-20T00:00:00Z
review text: The author introduces the notion of a pegged tiling, that is, a tiling (of part of Euclidean \(n\)-space) by polyhedra, with a peg (i.e., a point) assigned to each tile in such a way that for every two tiles which meet on a facet the difference of the pegs (the connecting line) is an outer normal vector of that facet. In the first part of the paper fundamental properties of pegged tilings are established. In particular, they admit a natural duality (pegs become vertices, and vice versa). The main motivation for the study of pegged tilings is the fact that they are a natural framework to describe the projection method used in the construction of quasi-periodic tilings: Dualizing a pegged tiling, taking a general section of the result, and then dualizing again, yields the projection. In the second part an interesting connection to convex functions is investigated, namely, that a pegged tiling can be lifted to (the graph of) a strongly locally polyhedral convex function, i.e., there is a convex function such that the maximal regions on which it is affine coincide with the tiles. As a consequence, pegged tilings are (slightly) generalized Laguerre (and therefore generalized Voronovi) tilings. Some fundamental operations on tilings (including the projection method) have natural counterparts in terms of operations on convex functions applied to their lifts.
Mathematics Subject Classification ID: 52C22
Mathematics Subject Classification ID: 51M20
Mathematics Subject Classification ID: 82D25
Mathematics Subject Classification ID: 52A41
zbMATH DE Number: 569281
zbMATH Keywords: Laguerre tilings
zbMATH Keywords: Voronovi tilings
zbMATH Keywords: dual tiling
zbMATH Keywords: projection method
zbMATH Keywords: pegged tilings
zbMATH Keywords: quasi-periodic tilings
zbMATH Keywords: convex functions
author: Peter McMullen
MaRDI profile type: Publication
cites work: Power Diagrams: Properties, Algorithms and Applications
cites work: A criterion for the affine equivalence of cell complexes in \(R^ d\) and convex polyhedra in \(R^{d+1}\)
cites work: Q4346774
cites work: Q3905310
cites work: Equivalence of the generalised grid and projection methods for the construction of quasiperiodic tilings
cites work: Q5547252
cites work: Q3735714
cites work: Dualisation of Voronoi domains and Klotz construction: a general method for the generation of proper space fillings
cites work: QUASI-PERIODIC TILINGS OF ORDINARY SPACE
cites work: A classification of two-dimensional quasiperiodic tilings obtained with the grid method
cites work: Self-similarity of quasilattices in two dimensions. I. The n-gonal quasilattice
cites work: An n-gonal quasilattice in two dimensions as a dual lattice to a multiple periodic grid
cites work: Convex Analysis
cites work: PERIODIC AND QUASI-PERIODIC LAGUERRE TILINGS
description: scientific article; zbMATH DE number 569281