The isoperimetric problem for pinwheel tilings

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).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.