# Projections of the world onto convex polyhedra

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Izidor Hafner

Tomislav Zitko

Faculty of Electrical Engineering, University of Ljubljana

Trzaska 25
, 1000 Ljubljana
, Slovenia

e-mail: izidor.hafner@fe.uni-lj.si

In the chapter Courious Maps of M. Gardner's Time Travel (W.H. Freeman and Company, New York, 1988),
many world's maps are presented, some of them use 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.

The following animations use LiveGraphics3D animation applet. To invoke an animation click on a figure.
Then drag to rotate, press Shift and drag vertical to change size, or horizontal to rotate. Press S to get stereo
view.

The next figure shows a cylindrical projection with a maze on it. The task is to join the black and the grey dot on
it or for instance to find a path from Belgrade to New York. A set of 19 figures of this sort can be
downloaded from
here.