Tessellating the Globe

This world jigsaw is very cool but feels a bit imperfect

The problem is that you can’t take multiple copies of the puzzle and tessellate them about the plane without visible seams showing up all over the place. This is because it’s based off an icosahedron and when it smushes flat the vertices of the triangles have to have a bit missing.

There are two possible geometries which could fix this. One is to divide the globe into two hemispheres, cut those out, and deform them both to be flat squares. Those squares can then be used to tesselate the plane via the hack of four of them going around their vertices instead of two. As a result the world is doubled around those vertices but topologically normal everywhere else.

The second geometry which works is to divide the world into triangles corresponding to the faces of a tetrahedron then deform all of those to be flat triangles. Those can the tessellate the plane by putting six triangles around each vertex instead of three, again doubling the world around the corners but leaving it topologically normal everywhere else.

There’s an interesting question of how to nicely align either of these geometries. For the first one I think it’s possible to make the squares aligned nicely with the equator and put two vertices in the Pacific Ocean and two in the Atlantic Ocean. The tetrahedral one may require the vertices be closer to land bodies and not align with latitude so well, although there’s no requirement that latitude and longitude lines be included on the map.