Via Sean Carroll at Cosmic Variance, a site devoted to discussions of different types of map projections:
“There is an endless variety of geographical maps for every kind of purpose. When looking at two different world maps one can wonder why the differences: do we draw the world as a rectangle, or an oval? Shouldn’t it be a circle? Should grid lines be parallel, straight or curved? Does South America’s ‘tail’ bend eastwards or westwards? What’s the ‘right’ way (or, more properly, is there one?) to draw our unique planet?
One important concern of cartography is solving how to project, i.e. transform or map points from an almost spherical lump of rock (our Earth) onto either flat sheets of paper or not-so flat phosphorus-coated glass.”
I was intrigued by polyhedral maps, printable cut-out forms of which are provided by the site.
“Several approaches were presented for reducing distortion when
transforming a spherical surface into a flat map, including:
- first mapping the sphere into an intermediate zero-Gaussian curvature surface like a cylinder or a cone, then converting the surface into a plane
- partially cutting the sphere and separately projecting each division in an interrupted map
Both techniques are combined in polyhedral maps:
- inscribe the sphere in a polyhedron, then separately project regions of the sphere onto each polyhedral face
- optionally, cut and disassemble the polyhedron into a flat map, called a “net” or fold-out
Intuitively, distortion in polyhedral maps is greater near vertices and edges, where the polyedron is farther from the inscribed sphere; also, increasing the number of faces is likely to reduce distortion (after all, a sphere is equivalent to a polyhedron with infinitely many faces). However, too many faces create additional gaps and direction changes in the unfolded map, greatly reducing its usefulness.”