The isoperimetric problem for pinwheel tilings
The pinwheel tiling is constructed by starting with a 1, 2, \(\sqrt{5}\) right triangle which is then divided into 5 similar triangles, expanded to get 5 triangles congruent to the original one. This arrangement is now moved, rotated, subdivided and expanded, in a certain way, and this process is repeated ad infinitum with the central fifth of each new pattern always coinciding with the entire previous pattern. The end result is a pinwheel tiling. In this aperiodic tiling there exist unions of tiles with ratio \(\text{(area)/(perimeter)}^2\) arbitrarily close to that of a circle. More precisely, the authors prove an isoperimetric property for this tiling: Given \(\varepsilon>0\) there is a distance \(R\) such that, for any disk \(C\) with radius \(r>R\) there is a region \(D\), whose boundary follows the edges of triangles in a tiling, that approximates \(C\) in the sense that \[ |(\text{perimeter of }D)-2\pi r|<\varepsilon r,\qquad \text{(area of }C\setminus D)+\text{(area of }D\setminus C)<\varepsilon r^2. \] In particular, \(\text{(area of }D)\setminus \text{(perimeter of }D)^2>(4\pi)^{-1}-\varepsilon\). The eixstence of such circles follows from the metric on pinwheel space being almost Euclidean at large distances: if \(P\), \(Q\) (points on the boundaries of the triangles) are separated by large Euclidean distance \(R\), then the shortest path from \(P\) to \(Q\) along the boundaries of triangles has length \(R+o(R)\).
scientific article; zbMATH DE number 906572