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THE TILED CIRCLE PROBLEM

Part of: Additive number theory; partitions Discrete geometry

Published online by Cambridge University Press:  31 March 2014

M. N. Huxley
M. N. Huxley*
Affiliation:
School of Mathematics, University of Cardiff, 23 Senghennydd Road, Cardiff CF24 4AG, U.K. email huxley@cardiff.ac.uk
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Abstract

How many square tiles are needed to tile a circular floor? Tiles are cut to fit the boundary. We give an algorithm for cutting, rotating and re-using the off-cut parts, so that a circular floor requires $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}} \pi R^2 + O(\delta R) + O(R^{2/3}) $ tiles, where $R$ is the radius and $\delta $ is the width of the cutting tool. The algorithm applies to any oval-shaped floor whose boundary has a continuous non-zero radius of curvature. The proof of the error estimate requires methods of analytic number theory.

MSC classification

Secondary: 11P21: Lattice points in specified regions 52C05: Lattices and convex bodies in $2$ dimensions
Type
Research Article
Information
Mathematika , Volume 60 , Issue 2 , July 2014 , pp. 321 - 346
Copyright
Copyright © University College London 2014 

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