A quincuncial map is a conformal map projection that is conformal everywhere except at the corners of the inner hemisphere. In mathematics, a conformal map is a function which preserves angles. ... The Mercator projection shows courses of constant bearing as straight lines. ...

Peirce Quincuncial map

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. In particular, three notable cartographers developed aspects of a conformal projection of one hemisphere (or the whole world, after a suitable rearrangement) on a square. All three approaches require evaluation of elliptic integrals of the first kind. Complex analysis is the branch of mathematics investigating functions of complex numbers, and is of enormous practical use in many branches of mathematics, including applied mathematics. ... In mathematics, a mapping w = f(z) is angle-preserving or (more usually) conformal at a point z0, if it preserves oriented angles between curves through z0, as well as their orientation, i. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... Cartography is the study of map making and cartographers are map makers. ... In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse and were first studied by Giulio Fagnano and Leonhard Euler. ...

Peirce's Quincuncial Projection

While working at the U.S. Coast and Geodetic Survey, the American philosopher (actually polymath) Charles Sanders Peirce disclosed his projection in 1879. In the normal aspect, it presents the northern hemisphere in a square; the other hemisphere is split into four 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 cross. The U.S. Coast and Geodetic Survey was established by President Thomas Jefferson in 1807 as the Survey of the Coast. ... Leonardo da Vinci A polymath (from greek polys(πολύς) meaning much, great in quantity, and mathisi(μάθηση) meaning learning), is a person who excels in multiple fields, particularly in both arts and sciences. ... Charles Sanders Peirce Charles Sanders Peirce (September 10, 1839 – April 19, 1914) was an American logician, philosopher, scientist, and mathematician. ... 1879 (MDCCCLXXIX) was a common year starting on Wednesday (see link for calendar). ... Insert non-formatted text here The Northern Hemisphere is the half of a planets surface (or celestial sphere) that is north of the equator (the word hemisphere literally means half ball). On the Earth, the Northern Hemisphere contains most of the land and population. ... A Greek cross (all arms of equal length) above a saltire, a cross rotated by 45 degrees For other uses, see Cross (disambiguation). ...

Tiled Peirce Quincuncial maps

Peirce's projection is conformal everywhere except at the corners of the inner hemisphere (thus the midpoints of edges in the whole map), where the Equator breaks abruptly. Scale is highly stretched near those four points; conversely, polar regions are rather compressed. The Equator and four meridians are straight but broken lines; all other graticule lines are complex curves.

Pieces of a quincuncial map can evidently be rearranged as a 2:1 rectangle. Also, the map trivially tessellates the plane; i.e., repeated copies can completely cover (tile) an arbitrary area, each copy's features exactly matching those of its neighbors. A tessellated plane A tessellation of the plane is a collection of plane figures that fill the plane with no overlaps and no gaps. ...

References

• Map Projections:Conformal Projections