Peirce quincuncial projection

The Peirce quincuncial projection is the conformal map projection from the sphere to an unfolded square dihedron, developed by Charles Sanders Peirce in 1879. Each octant projects onto an isosceles right triangle, and these are arranged into a square. The name quincuncial refers to this arrangement: the north pole at the center and quarters of the south pole in the corners form a quincunx pattern like the pips on the five face of a traditional die. The projection has the distinctive property that it forms a seamless square tiling of the plane, conformal except at four singular points along the equator. Typically the projection is square and oriented such that the north pole lies at the center, but an oblique aspect in a rectangle was proposed by Émile Guyou in 1887, and a transverse aspect was proposed by Oscar Adams in 1925. The projection has seen use in digital photography for portraying spherical panoramas. The maturation of complex analysis led to general techniques for conformal mapping, where points of a flat surface are handled as numbers on the complex plane. While working at the U.S. Coast and Geodetic Survey, the American philosopher Charles Sanders Peirce published his projection in 1879, having been inspired by H. A. Schwarz's 1869 conformal transformation of a circle onto a polygon of n sides (known as the Schwarz–Christoffel mapping). In the normal aspect, Peirce's projection presents the Northern Hemisphere in a square; the Southern Hemisphere is split into four isosceles triangles symmetrically surrounding the first one, akin to star-like projections. In effect, the whole map is a square, inspiring Peirce to call his projection quincuncial, after the arrangement of five items in a quincunx. After Peirce presented his projection, two other cartographers developed similar projections of the hemisphere (or the whole sphere, after a suitable rearrangement) on a square: Guyou in 1887 and Adams in 1925. The three projections are transversal versions of each other (see related projections below).