A "Regular" pentagonal tiling of the plane

Abstract

The paper introduces conformal tilings, wherein tiles have spec- ified conformal shapes. The principal example involves conformally regular pentagons which tile the plane in a pattern generated by a subdivision rule. Combinatorial symmetries imply rigid conformal symmetries, which in turn illustrate a new type of tiling self-similarity. In parallel with the conformal tilings, the paper develops discrete tilings based on circle packings. These faithfully reflect the key features of the theory and provide the tiling illustra-tions of the paper. Moreover, it is shown that under refinement the discrete tiles converge to their true conformal shapes, shapes for which no other ap-proximation techniques are known. The paper concludes with some further examples which may contribute to the study of tilings and shinglings being carried forward by Cannon, Floyd, and Parry. © 1997 American Mathematical Society.