You Can Play Chess in Hyperbolic Space Now, and It Sucks

For a game whose rules can be taught to a small child, chess is bewilderingly complicated—there are more possible games of chess than there are atoms in the observable universe, and despite the predictions of the world’s foremost brain genius, it seems unlikely that the game will ever be “solved.”

But! If normal, two-dimensional chess (or its variations) isn’t enough for you, then rejoice, because thanks to the efforts of YouTuber mannymakes, who maintains a site of mind-bending chess variants, you can now play chess on the hyperbolic plane!

If you’re not familiar with the concept of hyperbolic space, it’s one of the two possible two-dimensional spaces in which the rules of Euclidean geometry do not apply. OK, but what does this actually mean?

Euclidean geometry is the kind you were taught at school, the world of sines and cosines, and the square on the hypotenuse being equal to the squares on the other two sides. Its laws—first set out by ancient Greek mathematician Euclid nearly 3000 years ago in his treatise Elements—can be reduced to five simple axioms.

Four of these are easy to prove, but the fifth is not. In fact, the question of how to prove it stumped mathematicians for millennia. The postulate itself is simple enough: it states that two parallel lines will never meet. On face value, the statement seems self-evident—but the reason it resisted proof for so long is that it’s not actually true.

Consider, for example, the surface of a sphere. If you draw two parallel lines directly north from the equator, those lines will converge. They come together at the sphere’s north pole. The surface of the sphere is two-dimensional, but it is also curved, forming a closed loop. As well as allowing parallel lines to converge, this also means that if you move far enough in any direction, you’ll always end up exactly where you started.

So what’s hyperbolic geometry? It’s essentially the opposite of spherical geometry. In hyperbolic space, parallel lines diverge, and instead of closing in on itself, a hyperbolic plane gets progressively larger as you move away from the origin. This can be hard to visualize, but as it turns out, examples do exist in the real world. You can crochet a hyperbolic surface, but if crafts aren’t your bag, nature has also provided an example: the humble coral lettuce, whose leaves occupy more and more surface area as they grow away from the plant’s stem.

The hyperbolic board that mannymakes uses in his game has octagonal tiles, which extend for a radius of four tiles from the board’s center. This sounds like it’d be way too small for a chess setup, but it’s not: this board already has 161 tiles, and adding a further layer would add another 448 of them. (A plain old 8×8 chessboard, by contrast, has 64 squares.)

So what’s it like playing chess in this strange environment? Well, it’s … different. As its inventor explains in a fascinating video on the project, translating the rules of “normal” chess to a hyperbolic chessboard with octagonal squares is not always straightforward, and the game remains very much a work in progress. The whole thing is made more complicated by the fact that, instead of just black and white, the hyperbolic board requires three colors to avoid having adjacent tiles of the same color.

As with many such games, the experience of playing is more interesting than it is, y’know, fun. But if you want to be cast astray into the cyclopean depths of non-Euclidean space, where the geometry is all wrong, and Brown Jenkin lurks beyond the limits of sanity, his tiny bearded face cackling as he rubs his hands together in anticipation of your next move—well, hyperbolic chess awaits! You can saddle up a fellow masochist or even play the computer, although on the latter point, mannymakes cautions that the site’s AI hasn’t been updated for the new variant, so it might be just as confused as you.