Rhombic polyhedra
# Projections of the world onto convex polyhedra

#### 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.