Non-Euclidean geometry and games

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:

  • Look for parallel lines. In Euclidean geometry, they are in a constant distances from each other. In spherical geometry, they converge, and in hyperbolic geometry, they diverge.
  • Look at the angles of a triangle. In Euclidean geometry, they sum up to 180 degrees. In spherical geometry, they sum up to more (for example, take the North Pole, and two vertices on the equator as the vertices). In hyperbolic geometry, they sum up to less.
  • An easy way to tell whether a game uses truly non-Euclidean geometry is to look for rectangles. In non-Euclidean geometry there are no rectangles, anything that looks a bit like a rectangle actually has its angles smaller than 90 degrees, or its edges are curved. So, if you see rectangles, the game is (probably) not non-Euclidean.
  • In Euclidean geometry, a circle of radius r has perimeter 2πr. In spherical geometry, it is 2πsin(r) (which is bounded), and in hyperbolic geometry, it is 2πsinh(r) (which grows exponentially). In a three-dimensional hyperbolic world with “absolute unit” of 1m, a ball with radius 100m will have greater volume than the observable Universe!
  • In truly non-Euclidean 3D games and simulations the parallax works different. In Euclidean space, things that are far away from you (stars, distant mountains) are seen in roughly the same place as you move. This changes in non-Euclidean geometries: in hyperbolic space, everything moves, while other non-Euclidean geometries are even weirder.

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.

Manifolds

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 and interactive demos using non-Euclidean geometry

  • our HyperRogue — a roguelike game taking place in the hyperbolic plane (i.e., two-dimensional hyperbolic geometry). This uses a hyperbolic plane (without any topological surgery or boundary), so its world is larger than No Man’s Sky, MineCraft, or anything Euclidean.
  • Bringris — our non-Euclidean falling block game (similar to Tetris), made with the HyperRogue engine.
  • MagicTile — like Rubik’s Cube, but in non-Euclidean 2D manifolds.
  • Hyperbolic Maze — a maze in a hyperbolic 2D manifold.
  • Hypernom — this uses three-dimensional spherical geometry.
  • Uniform Polychora — more three-dimensional spherical geometry.
  • Non-Euclidean VR (H3) — this is three-dimensional hyperbolic geometry. See also H2xR (hyperbolic in some dimensions and Euclidean in other dimensions) and a new version.
  • our Virtual Crocheting — a demo in three-dimensional spherical geometry.
  • Curved Spaces — fly through three-dimensional non-Euclidean manifolds.
  • Hyperbolic Games — simple games in 2D hyperbolic manifolds.
  • HyperSweeper — Minesweeper in hyperbolic plane.
  • Sokyokuban — Sokoban-like in the hyperbolic plane, playable in a browser. Holonomy makes it interesting. (See also this for another puzzle based on holonomy.)

Games in development

Recently there are several cool non-Euclidean game projects in development!

  • Hypermine — this is a Minecraft-like in three-dimensional hyperbolic space. The screenshots in the Gallery are quite impressive, and the development is progressing quite well! (update: unfortunately the development is going slow recently :( )
  • HyperBlock — another Minecraft-like. This uses H2xR geometry, i.e., a hyperbolic plane with the ‘z’ coordinate working in Euclidean way.
  • Hyperbolica — a non-Euclidean game in development. The trailer shows hyperbolic geometry and a bit of spherical geometry. Contrary to HyperRogue, Hypermine and Hyperbolica which are focused on gameplay in an infinite world, it appears to be more of a story-based game, with walking, puzzle, shooting elements, and more mainstream graphics. (The sun in hyperbolic space does not work like it is shown in the trailer — it should become visibly brighter as we move towards it — but hopefully it will be changed :)
  • Non-Euclidean billiards in VR — the idea of mapping a real right-angled square table to a hyperbolic right-angled pentagon, or spherical right-angled triangle, is very cool!
  • Last but not least, HyperRogue is also in development —its non-Euclidean engine and unique world is a great testing grounds for various experiments with game genres or other weird geometries, and the results of these experiments are added to the game. By changing the options, you can get something completely different than the original roguelike in the hyperbolic plane.You can experiment with spherical geometry, various manifolds without boundary, 3D geometries including non-isotropic ones; roguelites, racing, puzzles, and so on.
  • Spaceflux —the existing videos show “fractal geometry”, but the plans in the kickstarter page mention hyperbolic geometry and even non-isotropic geometry (Solv).

Examples of notable games played on manifolds

  • Asteroids (1979) — when you go through the east edge of the world, you appear on the west edge; similarly for north or west. This is a two-dimensional flat manifold without boundary (called a flat torus).
  • Pac-Man (1980) — like Asteroids. In most versions you can only go through the E-W edge but not through the N-S edge, making it a cylinder (a manifold with boundary).
  • Civilization (1991) — as mentioned above, the surface of a sphere is non-Euclidean. This is why it is impossible to make a flat map of Earth which does not distort anything. Unfortunately, most games taking place on a spherical planet do not take this non-Euclidean geometry into account; they take a flat map and pretend that this map has no distortions. Civilization is played on a cylinder (you cannot go through a pole, while in the real world, the shortest flight from Europe to Hawaii would go through the North pole). Some other games are played on flat tori, which is in some sense even more different from a sphere.
  • Portal (2007) — once you place some portals, the world becomes a manifold with boundary.
  • Manifold Garden (2019) — it uses the term “manifold” correctly. I have not played it yet, it seems to be mostly a three-dimensional flat torus (i.e., a three-dimensional flat manifold without boundary), but it has some portals too.
  • Fragments of Euclid, Paradox Vector — these games are on Escheresque Euclidean manifolds. Escheresque like in Escher’s Relativity or Another World: the directions are not consistent. Fragments of Euclid is a puzzle game while Paradox Vector is an FPS.
  • Maquette (to be released in 2020) appears to be a game with portals, where one end of the portal can be larger than the other end, and consequently objects can become larger or smaller after going through the portal. Mirror stage (2009) is a similar idea in 2D; see also Sierpiński’s Tomb. It is also possible to have portals where one end is a square and the other end is a rectangle, causing the objects to be stretched by portals (see also my old demo based on a similar idea). This is no longer a Euclidean manifold, but rather an affine one (we could call it a “similar manifold” if only scaling is allowed, but that term does not seem to be used). Affine/similar geometry is different than Euclidean geometry (3rd axiom becomes meaningless) but it is still not called non-Euclidean, since parallel lines are not affected.

Other notable games which are geometrically weird

  • Antichamber — this game is probably responsible for popularizing the mathematically incorrect usage of the term “non-Euclidean”. This is mostly a Euclidean manifold (with boundary), but also exhibits some effects that would not happen in a manifold (e.g. you end up in a different place when you go some steps and back). I believe almost all the weird things in Antichamber could be (and probably have been) implemented with the teleportation trick desribed above.
  • Four-dimensional games. Some people may think of these games as non-Euclidean, because four spatial dimensions would not fit in our three-dimensional world. However, a world which works just like our old three-dimensional Euclidean space, except that it has more dimensions, is still definitely Euclidean (according to definition). It is of course possible to have a four-dimensional non-Euclidean space, but at the time of writing, it appears that no game tried to implement this.
  • Perspective tricks, such as Fez, Echodrome, Monument Valley, Naya’s Quest, or Perspective. Superliminal has some perspective and “affine manifold”aspects. These games are weird and cool, but should not be called non-Euclidean either. I would call some of them Escheresque.

Videos claiming to be non-Euclidean (correctly or not)

  • Not Knot — a classic video featuring non-Euclidean 3D geometry.
  • Non-euclidean virtual reality — this is non-Euclidean in the mathematical sense.
  • our Temple of Cthulhu in 3D — the “squares” are actually curved. At a first glance it appears that this world consists of a sequence of smaller and smaller balls. In fact, these “balls” are horospheres (a shape from hyperbolic geometry that does not really have a Euclidean analog; interestingly, while the 3D world here is non-Euclidean, the geometry on the horosphere is Euclidean), and they are all infinite. (more similar videos)
  • our SolvRogue —while in two dimensions we have only spherical, Euclidean, and hyperbolic geometry, there are even weirder non-Euclidean geometries in three dimensions. Go here for more.
  • Non-Euclidean Worlds engine — this video starts with Circle Limit by M. C. Escher, which is indeed based on non-Euclidean (hyperbolic) geometry. However, the most of the video presents a plain old affine manifold with boundary.
  • “No! Euclid!” GPU Ray Tracer gets an upgrade! — this is quite interesting, because this is indeed a curved space, not based on surgery.

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

Mathematics, game development, art, roguelikes, hyperbolic geometry. Sometimes all at once.

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