Monohedrally knotted tilings of the 3-space
type: Item
instance of: scholarly article
title: Monohedrally knotted tilings of the 3-space
zbMATH Open document ID: 0891.52012
DOI: 10.1016/S0012-365X(96)00351-2
author: Joseph Zaks
published in: Discrete Mathematics
publication date: 1998-06-12T00:00:00Z
review text: A tiling by congruent copies of a single tile is called a monohedral tiling. The following theorem is from an unpublished example of P. McMullen. Theorem 1. For any polyhedral knot \(K\) in \({\mathbf E}^3\) and any \(\varepsilon > 0\) there exists a tile \(M\) which is isotropic to an \(\varepsilon\)-neighborhood of \(K\) such that \(M\) tiles \({\mathbf E}^3\) monohedrally. Zaks gives an elegent, one page construction that proves the following generalization. Theorem 2. Let \(G\) be a connected graph and \(f : G\rightarrow {\mathbf E}^3\) any piecewise linear embedding of \(G\) into \({\mathbf E}^3\). For any \(\varepsilon > 0\) there exists a tile \(M\) which is isotropic to an \(\varepsilon\)-neighborhood of a set \(H\), isotropic to \(f(G)\), such that \(M\) tiles \({\mathbf E}^3\) monohedrally.
Mathematics Subject Classification ID: 52C22
Mathematics Subject Classification ID: 05B45
zbMATH DE Number: 1110036
zbMATH Keywords: tiling
zbMATH Keywords: knot
reviewed by: Andrew Vince
MaRDI profile type: Publication
cites work: Knotted lattice-like space fillers
cites work: Q3735714
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description: scientific article; zbMATH DE number 1110036