Radin Charles, Sadun Lorenzo
THE ISOPERIMETRIC PROBLEM FOR PINWHEEL TILINGS
(389K, plain TeX)
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; if $P$ and $Q$ are points separated by large
Euclidean distance $R$, then the shortest path along tile edges from $P$
to $Q$ has length $R + o(R)$.