H. P. Lovecraft and non-Euclidean geometry
After creating the first versions of HyperRogue, I have learnt that people often associate non-Euclidean geometry with the works of H. P. Lovecraft (which I have not read by then). Later, I have also learnt that people say things like “non-Euclidean geometry is just the geometry on a sphere, Lovecraft was afraid of spheres”, suggesting that H. P. Lovecraft did not understand the meaning of the technical term he was using.
It appears that most people saying this got their knowledge from a popular video by Overly Sarcastic Productions. OSP seem to hate Lovecraft for his racism, and claim that whatever he did was stupid. However, it appears that Lovecraft actually understood non-Euclidean geometry better than OSP.
Misconceptions
The video says “since we live on a globe, all of our geometry is non-euclidean”. However, this is wrong for two major reasons.
The first reason is that all this curvature of Earth does not matter for the day-to-day life. While it does matter when you are creating maps or planning flights — it is the reason why all maps have to distort something, and why routes taken by planes appear curved — people believed for a long time that Earth is flat (and some still do): even for large structures such as the Great Wall of China, the non-Euclidean geometry of Earth’s surface causes the side walls to be only 675 nanometers longer.
More importantly, originally non-Euclidean geometry is about what would happen if straight lines did not behave as Euclid thought (for example, a triangle with three straight edges could have angles not adding up to 180°… and we do not really know whether they do add up to 180° in our real world, maybe they don’t, but our instruments are not precise enough). It is more accurate to say that a sphere is a model of non-Euclidean geometry. This is a model where straight non-Euclidean lines are modeled by curved Euclidean lines (great circles). Models are useful to explain a mathematical structure or prove that it makes sense, but they should not be taken literally (like you should not say that movies with airplanes are bad because your model airplane does not fly; or that the set of real numbers is countable because there is a countable model of ZFC): for a being living inside a non-Euclidean space, this triangle would have all its edges straight. Specialists and communicators often gloss over this difference (and consider “non-Euclidean” and “curved” to be essentially the same thing —because they behave the same when seen from the inside), but this seems to be confusing to non-specialists.
The issue is that we are not two-dimensional beings living on a sphere, but we are actually three-dimensional beings. If you fly in a plane and light a flashlight, the light rays won’t circle around the Earth and light the back of your head. If we were actually living in two-dimensional spherical geometry, this is what would actually happen. The video above, our our introductory video Portals to Non-Euclidean Geometries, or the Hyperbolica video, shows how weird would perspective in three-dimensional spherical space be. Curved surfaces seen from the three-dimensional Euclidean space they are embedded in are much simpler than curved three-dimensional manifolds that you witness from the inside. (Our spacetime is curved, according to the General Relativity Theory, but the effects of this are even smaller than the effects of the curvature of Earth.) Also Lovecraft probably meant the negative curvature, which is rarer in our lives than the positive curvature of spheres.
Non-Euclidean geometry in Lovecraft has also been discussed in this blogpost from 2014. The author includes some nice pictures and some historical information, but seems to make a similar mistake — he thinks that non-Euclidean geometry just means that the surfaces are curved, which does not describe why R’Lyeh feels so alien: “I see the buildings and structures actually changing shape simply by viewing them from different points of view”. But, if the space itself is non-Euclidean, the light rays will follow the curvature, and buildings will change their apparent shapes from different points of view!
Recently, 3D visualization of non-Euclidean geometry are appearing. This includes non-Euclidean VR (playable online), 3D modes in HyperRogue, and upcoming games such as Hyperblock or Hyperbolica. These visualization allow us to see how an Euclidean explorer would feel when thrown into such a curved manifold. In the light of this knowledge, let’s see whether Lovecraft’s writings can be interpreted as accurate descriptions of such explorers (whether he actually understood non-Euclidean geometry himself, he consulted a mathematician, or this was just a coincidence).
Since the works of Lovecraft are currently in the Public Domain, we can download his complete works for free (or donating to the Equal Justice Inititative, as a counterbalance to Lovecraft’s racism), and learn that Lovecraft has actually used the term “non-Euclidean” just twice, in The Call of Cthulhu and in The Dream in the Witch House. In At The Mountains of Madness (1931), he mentions “There were geometrical forms for which an Euclid could scarcely find a name”, although this does not really refer to non-Euclidean geometry, but rather seems to simply say that the shapes were weird
(Euclid worked mostly with second order curves, there are lots of fractal shapes in the nature, and even not counting fractals, mathematicians today know many other shapes). In The Dreams in the Witch House (1932), the main character is studying “non-Euclidean calculus” (a rather weird term, it is geometry, not calculus), Riemannian geometry, quantum mechanics, and general relativity theory (“Planck, Heisenberg, Einstein, and de Sitter”) and hypothesises that our world is embedded in a higher dimensional space, and he could move through these extra dimensions by drawing weird angles. This seems to be more fantasy (if you draw angles in our three dimensions, you won’t exit them; also a manifold need not be really embedded in a higher dimensional space), and no details related to non-Euclidean geometry are given.
The Call of Cthulhu
The Call of Cthulhu (1926) is the most interesting; still, the references to non-Euclidean geometry are rare enough that we can basically list and comment on all of them in this post.
He talked of his dreams in a strangely poetic fashion; making me see with terrible vividness the damp Cyclopean city of slimy green stone — whose *geometry*, he oddly said, was *all wrong* […] — this does not say much, but we will learn more about what Wilcox dreamed about later: Without knowing what futurism is like, Johansen achieved something very close to it when he spoke of the city; for instead of describing any definite structure or building, he dwells only on broad impressions of vast angles and stone surfaces — surfaces too great to belong to any thing right or proper for this earth, and impious with horrible images and hieroglyphs. I mention his talk about angles because it suggests something Wilcox had told me of his awful dreams. He had said that the geometry of the dream-place he saw was abnormal, non-Euclidean, and loathsomely redolent of spheres and dimensions apart from ours. Now an unlettered seaman felt the same thing whilst gazing at the terrible reality.
So Lovecraft says that there was something weird about angles in R’Lyeh. This is correct — non-Euclidean geometry is indeed about angles (in impossible spaces, recently often incorrectly called non-Euclidean by gamers, there is nothing weird about the angles, you see squares and rectangles with four right angles everywhere). Lovecraft could be describing hyperbolic geometry here; in hyperbolic geometry, you can fit much more in a ball of small radius than in Euclidean (see e.g. a ball of radius of 20m which has the area of Earth), so “surfaces too great to belong to any thing right or proper for this earth” makes sense. As shown in the video above, hyperbolic planes look circular, so they indeed could remind of spheres.
The very sun of heaven seemed distorted when viewed through the polarising miasma welling out from this sea-soaked perversion, and twisted menace and suspense lurked leeringly in those crazily elusive angles of carven rock where a second glance shewed concavity after the first shewed convexity.
Yeah, geometry in R’Lyeh is all wrong — not only the architecture. The space is curved, so the sun will look very weird. Since parallel lines diverge in hyperbolic space, faraway objects (such as the Sun) will look smaller than they are. They also will behave weirdly, as can be seen in first-person perspective modes in HyperRogue, where it appears that Sun and stars are quite close to the HyperRogue world (as they illuminate only a very small fragment of the world) — in fact, far away objects in hyperbolic geometry actually behave like this, because the non-Euclidean parallax effects are different. Probably the curvature does not change sharply from zero (Earth) to very high (R’Lyeh), but rather the space becomes more and more curved gradually. While the sailors were in the transitional area, the changing curvature could cause the sun to appear distorted.
The quote about concave and convex angles is very interesting. When I have read this for the first time after exploring only the hyperbolic plane, and mostly in the Poincaré model, I thought that Lovecraft was wrong — an angle is either concave or convex, no matter how you look at it. However, see the video above — it is the {4,3,5} hyperbolic honeycomb but with some of the cubes filled. The video starts quite normal, a world created of cubes, similar to Minecraft. However, soon, the cubes start looking weird —when I have perceived this effect for the first time, it was striking that sometimes we do see one face of the face and don’t see the other face we would expect; sometimes that other face is seen at a more acute angle than expected, sometimes we see two faces which appear to be at a right angle (and thus belong to the same cube), but their colors are different, which means that they are actually different cubes.
These effect happen for two reasons: first, the dihedral angles of our cubes are 72°, not 90°; second, the perceived projection of hyperbolic space is the Beltrami-Klein model, in which the straight lines are straight but the angles are not mapped faithfully, so the same angles may look concave or convex depending on where you are looking at them from. (This is the opposite of e.g. the Mercator projection or the Poincaré model, which both map angles faithfully while distorting straight lines.) This effect is hard to see in the honeycomb images from Wikipedia, or non-Euclidean VR; our visual system is so used to the Euclidean geometry that it does not interpret the image correctly, what you see as an angle of a cube is likely the angle of two adjacent cubes (144°). Here is the online version where you can explore this yourself — press “full screen” button, rotate camera with the mouse, go forward with End — although it is very slow, the native HyperRogue executable, upcoming HyperRogue VR, or the upcoming hyperbolic voxel game HyperBlock, should show the effect better.
It was, Johansen said, like a great barn-door; and they all felt that it was a door because of the ornate lintel, threshold, and jambs around it, though they
could not decide whether it lay flat like a trap-door or slantwise like an outside cellar-door. As Wilcox would have said, the geometry of the place was all wrong. One could not be sure that the sea and the ground were horizontal, hence the relative position of everything else seemed phantasmally variable.
It is not clear how the gravity would work in hyperbolic space, but it would definitely feel unnatural to us. In our world, every object falls down, and the “down” directions taken by all falling objects are parallel. Parallel lines in non-Euclidean geometries work differently, so it is not clear how this would work. Various gravity lands in HyperRogue explore various kinds of gravity. Johansen and his team experienced this weirdness with their own senses.
Parker slipped as the other three were plunging frenziedly over endless vistas of green-crusted rock to the boat, and Johansen swears he was swallowed up by an angle of masonry which shouldn’t have been there; an angle which was acute, but behaved as if it were obtuse.
It is very easy to get lost in hyperbolic space, because there is much more space than in Euclidean. In Euclidean geometry, if you go 10 meters to the North then 10 meters to the East, you end up in the same location as when you go East first and then North; in hyperbolic geometry, the two locations might be almost 40 meters from each other. Holonomy effects also bring some confusion. (The easiness of getting lost, as well as the mind-bending size of the hyperbolic structures, are crucial for the design of various quests and game mechanics in HyperRogue.) It appears that, while the sailors were escaping, they encountered some kind of structure (for example, a right-angled pentagon) which they needed to go around; Parker went in the wrong direction, and he got lost (since it was a right-angled pentagon, not a square as he assumed).
Conclusion
This seems to be all non-Euclidean effects mentioned in The Call of Cthulhu; as we can see from this analysis, Lovecraft’s description gives a rather impressive and accurate description of how an Euclidean would feel in a non-Euclidean world, which is quite amazing given that he did not have access to modern visualizations of non-Euclidean spaces. Interestingly, the narrator in The Call of Cthulhu is named Francis Wayland Thurston, and William Thurston is one of the most important mathematicians working on non-Euclidean geometry in XX century — some of the so called Thurston geometries are even weirder than spherical or hyperbolic geometries, with structures changing their apparent shapes a lot more. If you know any other interesting places where Lovecraft described weird geometrical effects, please say so in the comments!
