September  2008, 22(3): 791-806. doi: 10.3934/dcds.2008.22.791

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, United Kingdom, United Kingdom, United Kingdom

Received  May 2007 Revised  May 2008 Published  August 2008

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.
Citation: Marcello Trovati, Peter Ashwin, Nigel Byott. Packings induced by piecewise isometries cannot contain the arbelos. Discrete & Continuous Dynamical Systems, 2008, 22 (3) : 791-806. doi: 10.3934/dcds.2008.22.791
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