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Projections of the world onto convex polyhedra. (English) Zbl 1037.52002

From the text: “In the chapter ‘Curious maps’ of M. Gardner’s [Time travel and other mathematical bewilderments (1988; Zbl 0641.00005)] many world maps are presented, some of them using a projection on a convex solid that is unfolded into a plane net.
The mathematician Charles Sanders Peirce designed a conformal map, namely a projection on eight isosceles right triangles that may be regarded as the faces of an octahedron, flattened until a space diagonal is zero. B. J. S. Cahill patented his butterfly map in 1913. The world is projected onto a regular octahedron. R. Buckminster Fuller’s first Dymaxion map was a projection of the world onto the fourteen faces of a cuboctahedron. In March 1, 1943 the map was completed by staff artists of Life. At about the same time, Irving Fisher designed Likaglobe, which folds into an icosahedron. In 1954 Fuller copyrighted his Dymaxion Skyocean Projection World Map, which slightly differed from Fisher’s Likaglobe.
Wolfram’s Mathematica package WorldPlot consists of various types of projections used in cartography. There is a large data block WorldData, where each country is represented by a set of points on its boundary.
We use the data to construct a gnomic projection of the world on any convex polyhedron. If necessary we translate the polyhedron in a position where its mass centre coincides with the origin of the co-ordinate system. Then points on the globe are projected onto the polyhedron. This means that we calculate the intercept of the half-straight line from the origin via given point on the globe by a face of the polyhedron.”

MSC:

52A10 Convex sets in \(2\) dimensions (including convex curves)

Citations:

Zbl 0641.00005

Software:

Mathematica
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