Wallpaper Maps

A wallpaper map is a conformal projection of a spherical earth onto regular polygons with which the plane can be tiled continuously. A complete set of distinct wallpaper maps that satisfy certain natural symmetry conditions is derived and illustrated. Though all of the projections have been published before, the family had not been characterized as a whole. Some wallpaper maps generalize to one-parameter subfamilies in which the sphere is pre-transformed by a conformal automorphism. Within a decade of Schwarz’s publication of a formula for conformally mapping a circle onto a regular polygon, the noted American philosopher C. S. Peirce used it to calculate a conformal map of the world in a square (Figure 1), with which the plane can be tiled. The projection (more precisely, its inverse) is doubly periodic, and necessarily possesses isolated branch points, each of which is surrounded by multiple copies of a mapped neighborhood. Other doubly periodic projections were published sporadically, especially by O. S. Adams, until almost a century later L. P. Lee collected them and other polygonal maps in a summary monograph. No more have appeared since. This paper explains why, and in so doing gives a uniform account of this family of projections. Preliminaries For our purposes, a tiling is a covering of a surface by congruent, non-overlapping regular polygons, or tiles, fitted edge-to-edge and vertex-to-vertex. A tiling projection is a (one-to-many) conformal map from a tiling of the sphere to a tiling of the plane. Restricted to a single spherical tile and one of its planar image tiles, the map must (a) be conformal and bijective throughout the interior, (b) continue across each edge into an adjacent tile, conformally except at isolated singularities, (c) preserve the symmetry group of each spherical tile, and (d) be the same in all tiles. To state requirement (c) more precisely, let G and G′ be the (dihedral) symmetry groups of a spherical polygon, X , and one of its planar images, X ′, respectively; and let f : X → X ′ be the mapping function. Then for each g∈G there must exist g′∈G′ such that for every point x ∈X