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TILING WITH PUNCTURED INTERVALS

Part of: Designs and configurations Discrete geometry

Published online by Cambridge University Press:  29 October 2018

Harry Metrebian
Harry Metrebian*
Affiliation:
Trinity College Cambridge, CB2 1TQ, U.K. email rhkbm2@cam.ac.uk
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Abstract

It was shown by Gruslys, Leader and Tan that any finite subset of $\mathbb{Z}^{n}$ tiles $\mathbb{Z}^{d}$ for some $d$. The first non-trivial case is the punctured interval, which consists of the interval $\{-k,\ldots ,k\}\subset \mathbb{Z}$ with its middle point removed: they showed that this tiles $\mathbb{Z}^{d}$ for $d=2k^{2}$, and they asked if the dimension needed tends to infinity with $k$. In this note we answer this question: we show that, perhaps surprisingly, every punctured interval tiles $\mathbb{Z}^{4}$.

MSC classification

Primary: 05B45: Tessellation and tiling problems 05B50: Polyominoes 52C22: Tilings in $n$ dimensions
Type
Research Article
Information
Mathematika , Volume 65 , Issue 2 , 2019 , pp. 181 - 189
Copyright
Copyright © University College London 2018 

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