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The isoperimetric problem for pinwheel tilings

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Abstract

In aperiodic “pinwheel” tilings of the plane there exist unions of tiles with ratio (area)/(perimeter)2 arbitrarily close to that of a circle. Such approximate circles can be constructed with arbitrary center and any sufficiently large radius. The existence of such circles follows from the metric on pinwheel space being almost Euclidean at large distances; ifP andQ are points separated by large Euclidean distanceR, then the shortest path along tile edges fromP toQ has lengthR+o(R).

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Authors and Affiliations

  1. Mathematics Department, University of Texas, 78712, Austin, TX, USA

    Charles Radin & Lorenzo Sadun

Authors
  1. Charles Radin
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  2. Lorenzo Sadun
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Communicated by Ya. G. Sinai

Research supported in part by NSF Grant No. DMS-9304269 and Texas ARP Grant 003658-113.

Research supported in part by an NSF Mathematical Sciences Postdoctoral Fellowship and Texas ARP Grant 003658-037.

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Radin, C., Sadun, L. The isoperimetric problem for pinwheel tilings. Commun.Math. Phys. 177, 255–263 (1996). https://doi.org/10.1007/BF02102438

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  • DOI: https://doi.org/10.1007/BF02102438

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