Mapping the earth is a classic problem. For thousands of years cartographers, mathematicians, and inventors have come up with methods to map the curved surface of the earth to a flat plane. The main problem is that you cannot do this perfectly, such that both the shape and size of the surface are depicted properly everywhere. This has intrigued me for a long time. Why not just take a map of a small part of the earth, which is almost perfect, glue neighboring maps to it, and repeat this until the whole earth is shown? Of course you get interrupts, but does this matter? What does such a map look like?

To check this out, we developed myriahedral projections.

For a quick impression of myriahedral projections, see the TU/e-animation or the YouTube-movie. 

The method

The method used is the same for each type of myriahedral projection:

• Project the globe on a myriahedron

• Label the edges as cuts or folds

• Unfold the globe

A myriahedron is a polyhedron with a very large number of faces. For this reason, we call the results myriahedral projections. In step 2 and 3, this myriahedron is cut open and unfolded. The resulting maps have a large number of interrupts, but are (almost) conformal and conserve areas.

Now, dependent on which mesh is used and which strategy for labeling the edges, different maps are obtained. First, one can cut along parallels or meridians. We call these graticular projections, since they are based on the graticule. The resulting images resemble familiar map projections.

Second, one can take a regular polyhedron, subdivide the faces, and fold it out.

One can take the shapes of the continents into account, when defining the cuts. By changing priorities, one can obtain maps where most cuts are through oceans or continents.

Traditional maps are often Euro-centric. Focussing on for instance Australia gives different results (thanks to David Mayes for this suggestion).