Motivation

Tiling the plane

Tiling patterns have fascinated humans for millenia, and can be observed in many different context all over the world. It can be shown that no regular polygon that has 7 or more sides can tile the plane by itself without gaps between the tiles.

Thus we only have the triangle, square, pentagon and hexagon to choose from. These all neatly tile, except for the regular pentagon, which does not.

Partitioning the Globe

A related problem is how to tile a sphere, with the obvious application of partitioning the globe into equal areas. Numerous methods for this have been devised in the past, which all take a planar tiling and warp it onto the surface of the sphere. In doing so, the polygons which started out as regular are warped, and are no longer regular. This presents the question, when looking to tile a sphere, does it make sense to restrict ourselves to regular polygons?

The Quest for a Pentagonal Tiling

The motivation for the A5 system was this. To see if a pentagonal tiling on a sphere could be found, perhaps even yielding characteristics superior to those of existing tilings.