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2008, Volume 22, Issue 3: 791-806. Doi: 10.3934/dcds.2008.22.791
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Packings induced by piecewise isometries cannot contain the arbelos

  • 1.

    Mathematics Research Institute, School of Engineering, Computing and Mathematics, Harrison Building, University of Exeter, Exeter, EX4 4QF

Received:  April 30, 2007
Revised:  April 30, 2008
Published:  August 2008
Abstract Related Papers Cited by
    Abstract
  • Planar piecewise isometries with convex polygonal atoms that are piecewise irrational rotations can naturally generate a packing of phase space given by periodic cells that are discs. We show that such packings cannot contain certain subpackings of Apollonian packings, namely those belonging to a family of Arbelos subpackings. We do this by showing that the unit complex numbers giving the directions of tangency within such an isometric-generated packing lie in a finitely generated subgroup of the circle group, whereas this is not the case for the Arbelos subpackings. In the opposite direction, we show that, given an arbitrary disc packing of a polygonal region, there is a piecewise isometry whose regular cells approximate the given packing to any specified precision.
    Mathematics Subject Classification: Primary: 37E05; Secondary: 52C26.

    Citation:
    shu

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