The World of Ladon is starting to take shape. It’s a spherical world built from geometric shapes, mostly hexagons. Hexagons are ubiquitous in Ladon.

I was experimenting with constructing the sphere during a flight. I ran into a problem that I should have seen coming, but didn’t — there is no way, mathematically, to tile a sphere with regular hexagons. There are two approaches that come close, and each has advantages and disadvantages.

The first approach is to start with regular hexagons, tiled as they would be on a flat surface. This is the common “honeycomb” pattern. You can wrap this tiling around a sphere, but the hexagons will begin to overlap. You can correct for this by changing the size of the hexagons as you go, but, depending on which way you choose to do the wrapping, you will run into “poles” at some points on the sphere. At these points, your hexagons will essentially vanish to zero. This is exactly the same problem that mapmakers encounter — the reason why Greenland appears to be as big as Africa on a map, even though Africa is about 14 times larger in reality. By breaking the map up into sections, as mapmakers have done in different ways for a very long time, this approach can be fudged enough to work. If your sphere is large enough, the viewer will probably never notice the difference. Some of your hexagons just have to be slightly distorted.

The other approach is more mathematical. A sphere can be “tessellated” into triangles. You can then group these triangles into hexagons. The big drawback to this approach is that none of the hexagons are regular. They’re all distorted pretty heavily. And worse, there are exactly twelve locations on the sphere that aren’t hexagons at all, they’re pentagons. The most familiar way to visualize this is a soccer ball. No matter how small you make the tiles, those pentagons don’t go away.

The above image shows the two approaches side by side. The regular hexagons on the left begin to overlap as they bend around the sphere. The tessellation on the right fits together perfectly, but is made up of very distorted hexagons with a pentagon in the middle.

The problem with the distortion, in either approach, is that our 3D modeler needs to be able to crank out these hexagonal tiles. If they aren’t all the same shape, roughly, then he’ll have to model each one individually! I don’t think we pay him enough for that…