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Packings induced by piecewise isometries cannot contain the arbelos
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.