A mathematically-impossible, hexagonally-tiled world map

In interior decorating, hexagonal tiles are really hot right now. For instance, here’s three examples in my workplace alone:

Let me tell you some thoughts I have about placing the continents of the world on reconfigurable hexagonal tiles.

Reconfigurable Worldmaps

I’ve talked about reconfigurable world maps before. It’s a theme. I’ve written how it’s my favourite part of the Authagraph map. In last year’s TEDx talk, I mentioned how the Guyou projection (like its famous cousin, the Peirce quincunxial projection) is tileable. Going back further in 2011, made an infinite, never-completed puzzle in the shape of a set of Tangram pieces.

Jessica Rosenkrantz is an artist whose work I love and that I follow on twitter. Recently, she has been working on wonderful intricate jigsaw puzzles that do this. In this case, she has taken the idea of a reconfigurable earth to a stunning, beautiful conclusion with her earth puzzle:

And you can order one for yourself!

Icosahedral projections

Rosenkrantz’s puzzle is based on an icosahedral projection. Those familiar with Buckminster Fuller’s dymaxion projection can see how the globe is projected onto the faces of an icosahedron, and then strategically cut open and flattened on the table into a map of twenty triangles:

Here, Fuller has placed the triangles of his icosahedron so that his particular arrangement avoids interrupting major landmasses as much as possible – and then when not possible, he cut a couple of the triangles up. Why not? That’s a great kind of ingenuity.

Icosahedral projections cannot fully tile the plane the way the Authagraph, Guyou, or Peirce Quincunxial projections can. With multiple copies of itself, an icosahedral projection would necessarily have gaps or overlaps. But maybe this can add to the fun where reconfigurable maps are concerned.

Hexagonal tiling of the sphere

How can we get from an icosahedral projection, which is a triangular tiling of the sphere, to a hexagonal tiling?

Here we run into a bit of a problem since there’s a mathematical proof that this is impossible – assuming 3 edges / faces to a vertex:

For n hexagons, there will be e = 6n / 2 vertices, f = n faces, and v = 6n / 3 vertices. The Euler characteristic v – e + f of a convex polyhedron should always be 2. But instead we get 0n.

So it’s just not possible to tile the sphere with only hexagons. Matt Parker explains this very well in his campaign last year to change geometrically incorrect soccer ball signage. Incidentally, correctly drawn soccer balls is also a cause that is near and dear to my heart.

Therefore we must cheat.

Cheating

If it’s for artistic purposes or decoration, I don’t have any problems whatsoever with this kind of cheating. Far worse cartographic sins have been committed in the name of interior decoration – at least we haven’t omitted New Zealand. And besides, I get a kick out of performing this kind of ‘magic trick’ for those that are mathematically in the know.

Using Dymaxion’s hard work of rearranging the globe with respect to the icosahedron – and admittedly with zero finesse – let’s just slap a hexagonal grid over top to see what happens:

That’s promising. I have removed any hexagons that were mostly ocean (sorry Hawaii). A tile maker can always add some number of blank hexagon tiles as necessary.

But! Notice the hexagons that are at the corners of the blue triangles: they all have at least one sixth of their space empty. This is where the cheating comes in. In the Dymaxion map, each one of the corners of the icosahedron is a pentagonal pyramid that has been cut and flattened out:

For our map, if we ignore this gap by filling it in with ocean that doesn’t exist (not the first time such liberties were taken), then I think we can achieve our mathematically impossible, hexagonally tiled world. Tada:

What now?

For fun, I think I’ll print out my own version on cardstock – with some number of blank hexagons to fill any gaps where needed – and see what happens. I’ll be interested to see if there are any serious repercussions to the cheating. I know for instance, the distance between Iceland and Scotland can never be shortened.

If any adventurous reader out there feels like it, they can get some unbaked hexagonal tiles, paint them and then turn them into a coffee table or something.  Who knows.  Let me know, I’d be excited to see what people come up with.