Non-Euclidean geometry and games

Zeno Rogue on 2019-05-01

The term “non-Euclidean” is often used by gamers (game developers, journalists, etc.) to mean any kind of game where the space does not work exactly as in our world. While such games typically tend to be amazing and very fun, this is not what “non-Euclidean” traditionally means for mathematicians, for whom it has a more precise meaning, which is not “anything that is not a perfectly normal space”. This article provides a summary of what “non-Euclidean” means, and the various weird geometries used in games.

A hexagon in the hyperbolic plane can have six right angles.

Non-Euclidean geometry

The discovery of non-Euclidean geometry is one of the most celebrated, surprising, and crazy moments in the history of mathematics. It is something that many great thinkers for more than 2000 years believed not to exist (not only in the real world, but also in fantasy worlds). So many popular expositions of mathematics discussing non-Euclidean geometry have been created that the term has rightfully entered the general public conscience, as something extremely alien, important, crazy, and difficult to understand. In general, something extremely cool!

Recently, the term “non-Euclidean geometry” has been appropriated by some game developers for any kind of game space which works in a different way than ours. This is unfortunate, as players are attracted to such games, thinking “hey, at last I will have a chance to understand that weird and important thing what all these mathematicians were crazy about!”, which is nowhere near the truth —while these games are usually very cool, they are usually based on relatively straightforward concepts that have nothing to do with the original thing.

Euclid has shown how everything in geometry (Pythagorean Theorem, etc.) could be derived from a small set of very simple postulates… but there was one thing he was not happy about: his fifth postulate, which was not actually that simple: If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.. Euclid believed that his fifth postulate could be proven from the other ones, and he failed, and so did many mathematicians through the ages. The mystery has been solved in 19th century.

I am resolved to publish a work on parallels as soon as I can put it in order, complete it, and the opportunity arises. I have not yet made the discovery but the path which I have followed is almost certain to lead me to my goal, provided this goal is possible. I do not yet have it but I have found things so magnificent that I was astounded. It would be an eternal pity if these things were lost as you, my dear father, are bound to admit when you seen them. All I can say now is that I have created a new and different world out of nothing. All that I have sent you thus far is like a house of cards compared with a tower. — János Bolyai

Bolyai, Lobachevsky, and Gauss have created a new world, where all Euclid’s postulates hold except the fifth, thus showing that the fifth postulate could not be proven from the other ones. Since Euclid believed that such a thing could not exist, it has been called by Gauss non-Euclidean geometry.

Today, we call this hyperbolic geometry, while (two-dimensional) non-Euclidean geometry could be hyperbolic or spherical. A sphere is curved in the third dimension; we say it has constant positive curvature. (The surface of Earth is a good approximation, though the curvature is not exactly constant: it is slightly more flat on the poles.) Euclidean geometry has curvature 0, while the hyperbolic geometry has constant negative curvature.

The meridians on Earth are straight (and they eventually meet in the poles), while the parallels (except the equator) are not straight. The picture above shows an analogous situation in the hyperbolic plane. Red lines (“meridians”) are straight and they diverge, the central green line (“equator”) is straight, but the other green lines (“parallels”) are not. The red lines are all straight, and the red segments between two green lines are all of the same length; the picture may suggest that this is not the case, but this is an artifact of the projection used (it is impossible to render non-Euclidean geometry on a flat picture without distortion).

You can easily tell whether you are in an non-Euclidean world in the following ways:

Play our HyperRogue to explore a non-Euclidean world and get some intuitions about how non-Euclidean geometry works. The main gameplay is designed for the hyperbolic plane, but you can also experiment with other 2D and 3D geometries.


Games claiming to be non-Euclidean usually have worlds obtained by performing some kind of “surgery”: we cut some fragments (chambers) out of a Euclidean space, and then glue them together in some non-standard way. In 3D games, the place where we performed surgery typically looks like a portal, but the game may also make the surgery appear seamless. Mathematically, this is called a Euclidean (or flat) manifold (with boundary); Euclidean/flat because it is made of fragments of Euclidean space, and “with boundary” because there are typically some walls which you could not go through, and some points inside such walls could not even be modeled consistently (walls of the portals). It is also possible to have manifolds without boundary; typically these look like periodic spaces.

Such games are probably called non-Euclidean because their geometry is impossible to interpret consistently as a part of a world similar to ours. In a Euclidean world, when you go 10m, turn 90 degrees right, go 10m, turn 90 degrees right, go 10m, turn 90 degrees right, go 10m, and turn 90 degrees right, you return back to your starting point and orientation. In a manifold (and also in non-Euclidean geometry as described above) it is possible to end up in a different point. (A great example of this is the VR project Tea for God, where the VR world you are exploring is huge, while in the real world you are just walking back and forth around a small room.) It is also possible to make a loop which brings you back to your starting point inside the manifold, but would be different in Euclidean world. However, this is not what non-Euclidean geometry means to a mathematician. Surgery changes the topology of the space, but it does not change its geometry.

In a manifold you can sometimes find triangles whose angles sum up to something else than 180 degrees, or parallel lines which stop being close when one of them goes through a portal. However, in a truly non-Euclidean world, these phenomena happen even for very small triangles, and for every pair of lines. Effects like this animation could not be achieved using portals — in non-Euclidean geometry it is possible to see the whole right-angled pentagon at once, while with portals, one of the five right angles will always be hidden behind a portal.

An easy (but limited) way to implement a manifold in a game is to make invisible teleportation devices, which seamlessly teleport the player to another location which looks exactly the same. That technique works in basically any game engine (even in Minecraft). I have seen many comments under videos using this technique saying “this is not non-Euclidean, you are just using teleports!” These comments are right that this is not non-Euclidean in the mathematical sense, but using teleports has nothing to do with that. In general, I find that sentiment weird. It is the effect that matters, not how it is implemented. Any video game is an illusion, after all.

Of course we could also do this starting with non-Euclidean space, obtaining a non-Euclidean manifold. Hyperbolic manifolds are typically bounded, thus they lose their exponential growth (and, depending on the game design, this exponential growth may be a huge technical problem); however, parallel lines and triangles still work differently.

When the distance is not the Euclidean metric

I have seen some people argue that any games played on square grids are non-Euclidean. This is because, in such a game, the number of steps you need to take to reach point (x,y) from the point (0,0) is given by the formula |x|+|y| (so called taxicab metric) or max(|x|, |y|) (so called Chebyshev metric), or some other formula where the set of points in d steps is an octagon, while the Pythagorean theorem says that the distance between these two points is actually the square root of x²+y² (so called Euclidean metric). Similarly, one could say that HyperRogue is not hyperbolic, since it is a grid-based game.

In fact, we do not really need a grid for this problem: if you play a top-down game with continuous space using the keyboard, you can usually move in eight directions, so the distance will still be given by one of the formulas above. So this would make lots of games non-Euclidean.

This seems to be again a confusion arising from having several things named after Euclid. “Non-Euclidean” means that Euclid’s parallel axiom is not satisfied, not that the metric is different than the Euclidean metric. Grid-based games are not normally perceived by people as anything weird, and this is expected, as many important properties of these spaces are similar to that of continuous spaces. Parallel lines in a square grid work like in Euclidean geometry, while Great Walls in HyperRogue work like straight lines in hyperbolic geometry. A square grid grows quadratically, just like the Euclidean plane, while the HyperRogue world grows exponentially. And so on. A rather impressive phenomenon arises when you are simulating how effects spread on a square grid — for example, you are simulating heat transfer (in time 0 one point of the grid is very hot, and you let the heat spread to other points), or random walk (in time 0 there are many particles in one point of the grid, and then each of them moves randomly). Even though it might appear at the first glance that the waves should spread in square or octagonal shapes (because of the structure grid), they are in fact perfectly circular! This happens on any sufficiently symmetric grid on the Euclidean plane, but will be different in other grids!

Artists associated with non-Euclidean geometry

M. C. Escher has created many great artworks based on impossible geometries, which has in turn inspired many amazing games. If you read that Escher used non-Euclidean geometry, this is true, he did use non-Euclidean geometry in his Circle Limit series. However, if a game reminds you of e.g. Ascending and Descdending, Waterfall, Relativity, Depth, or Another World II, well, these artworks do not have much to do with non-Euclidean geometry. Commonly used terms for such spaces include impossible space/geometry or Escheresque.

Another artist commonly associated with non-Euclidean geometry is H. P. Lovecraft: surfaces too great to belong to any thing right or proper for this earth […] the geometry of the dream-place he saw was abnormal, non-Euclidean, and loathsomely redolent of spheres and dimensions apart from ours […] One could not be sure that the sea and the ground were horizontal, hence the relative position of everything else seemed phantasmally variable. […] an angle which was acute, but behaved as if it were obtuse. (H. P. Lovecraft, Call of Cthulhu) These descriptions are very vague, but they describe some of the feelings a layman has where exploring a non-Euclidean simulation quite well, even amazingly well given the fact that Lovecraft had no access to such simulations: he does mention that there is something very weird about angles in R’Lyeh, and you get this feeling in a non-Euclidean simulation, while in games using “non-Euclidean” in a non-mathematical meaning, the angles look mostly normal; they remind the player more of Escher’s impossible architectures than R’Lyeh. This article explores this in more detail.

Games using non-Euclidean geometry

Most of these are free (and also the source code is available).

Interactive demos using non-Euclidean geometry

These let you explore non-Euclidean spaces but have no actual gameplay.

Non-Euclidean games in development

Recently there are several cool non-Euclidean game projects in development! Hypermine and HyperBlock are very promising, but unfortunately the development is going slow recently :(

In the following sections, we list games that are incorrectly called non-Euclidean. The common question is: what is the correct term, then? Well, the geometry in the game may be weird for many possible reasons, and thus, it depends on the game in question. The categories below are the suggested terms.

Games in wrapped spaces

Games in wrapped spaces are sometimes called non-Euclidean, even though it is one of the oldest tricks in game design!

Games with portals

In these games, we have “portals”, which take you into another part of the world. The portals tend to be explicit: you know where the portals are and as long as you do not go into a portal, the space works in a totally normal way.

Using portals to simulate non-Euclidean geometry

We can take six square rooms, and connect them with portals as if they were the faces of a cube. This way, we get an approximation of spherical geometry (as a cube approximates a sphere). You get some effects typical for non-Euclidean geometry, such as holonomy.

Games in impossible spaces

In the previous category we had games with explicit portals. A more interesting effect is obtained when the portals are not visible, producing an illusion of an “impossible space”.

Recursive games

Some games use constructions which contain a smaller copy of themselves, or more precisely, contain themselves. This is no longer a Riemannian manifold, since we cannot uniquely define the distance between points in the space — if you go outwards, you become smaller and smaller, and thus things become larger and larger in comparison. Mathematically it is called an “affine manifold” (if any affine transformations are allowed) or “similarity manifold” (if things can only become smaller or larger). Affine/similarity geometry is different than Euclidean geometry (3rd axiom becomes meaningless) but it is still not called non-Euclidean, since parallel lines are not affected.

Relativistic games

Relativistic games are based on Einstein’s special relativity theory, which is in turn based on Minkowski geometry, which is different from Euclidean geometry, and is related to hyperbolic geometry. Possible velocities of objects could be said to follow the rules of hyperbolic geometry — which is why you cannot go faster than light. These effects happen in our world but we normally do not move fast enough to see them, unless we simulate a universe where we can easily reach relativistic speeds, as in the following games.

Other notable games which are geometrically weird

Related Videos

No time to play games? Videos are great too!

Thanks to Henry Segerman for suggesting improvements, and to all the developers who try to create these mindbending geometric experiences!