Non-Euclidian Horror: The Writhing Spheres

I've always wanted to write an article on why "non-Euclidian" should be a synonym for "horrifying".

If you're a horror academic, this post will probably be trivial nonsense at best, and outright misleading at worst. I strongly suspect all this material has been covered before. So it goes; this is ostensibly a D&D blog.

Shamefully, I haven't read Jonathan Newell's book, "A Century of Weird Fiction, 1832-1937: Disgust, Metaphysics and the Aesthetics of Cosmic Horror", despite hiring him to draw the maps for Magical Industrial Revolution. It's probably dreadfully clever and full of useful facts, but it's outside my normal areas of study, all the local library lending programs are shut down due to the plague, and I've run out of bookshelf space three times since March. To atone for my sins, you should buy a copy or three.

As penance for not doing my assigned reading, I've decided to publish this post.
The Flammarion Engraving

Possibly Gilman ought not to have studied so hard. Non-Euclidean calculus and quantum physics are enough to stretch any brain; and when one mixes them with folklore, and tries to trace a strange background of multi-dimensional reality behind the ghoulish hints of the Gothic tales and the wild whispers of the chimney-corner, one can hardly expect to be wholly free from mental tension.

-The Dreams in the Witch House

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.

-The Call of Cthulhu

Even the pictures illustrate only one or two phases of its infinite bizarrerie, endless variety, preternatural massiveness, and utterly alien exoticism. There were geometrical forms for which an Euclid could scarcely find a name—cones of all degrees of irregularity and truncation; terraces of every sort of provocative disproportion; shafts with odd bulbous enlargements; broken columns in curious groups; and five-pointed or five-ridged arrangements of mad grotesqueness. As we drew nearer we could see beneath certain transparent parts of the ice-sheet, and detect some of the tubular stone bridges that connected the crazily sprinkled structures at various heights. Of orderly streets there seemed to be none, the only broad open swath being a mile to the left, where the ancient river had doubtless flowed through the town into the mountains.

-At the Mountains of Madness

That looking-glass had indeed possessed a malign, abnormal suction; and the struggling speaker in my dream made clear the extent to which it violated all the known precedents of human experience and all the age-old laws of our three sane dimensions. It was more than a mirror—it was a gate; a trap; a link with spatial recesses not meant for the denizens of our visible universe, and realizable only in terms of the most intricate non-Euclidean mathematics. And in some outrageous fashion Robert Grandison had passed out of our ken into the glass and was there immured, waiting for release.

-The Trap

Part 1: Geometry Class

Primary and secondary mathematical education has fundamentally changed in the past few decades. Proofs are out; they might be discussed by a teacher, but students are not expected to work through a proof on their own. Practical applications are in. Classical texts and ancient authorities are no longer cited. While some schools still require students to purchase geometry sets, they tend to be used for art class or prodding classmates instead of geometry.

But for hundreds of years, Euclid was central to mathematical education. Students learned his axioms by heart. Euclid took the visible world and transformed it into elegant mathematics. A pastoral natural scene, under Darwin, becomes red in tooth and claw. Because he assumed the points, lines, and circles of his system were the points, lines, and circles of the real world, he horizon, the columns of a building, and the shape of a cone of sand become living mathematics under Euclid.

All principles of Euclidean geometry - or, for thousands of years, simply "geometry" - derive from five postulates:
1. A straight line can be drawn between any two points.

2. Any straight line segment can be extended into an infinite straight line.

3. A circle can be drawn given a straight line segment as the radius and one end point as the centre.

4. Any right angle is equal to any other right angle.
The first four postulates are elegant and brief. They feel intuitively true. The last one even feels tautological; of course one 90 degree angle  is equal to any other 90 degree angle.

But the fifth postulate is troublesome. It bothers students when they learn it, even if they can't say why.
5. If two straight lines are drawn which intersect a third straight line in such a way that the sum of the interior angles on one side is less than two right angles, then the two lines, if extended infinitely, must inevitably intersect each other.
Draw a horizontal line on a flat sheet of paper. Drop two sticks on it. If the sticks are exactly perpendicular (vertical), then they will never cross, even if they're infinitely long. If one stick is angled to the right and the other to the left, then they'll cross once. If they're both angled to the left or to the right, they'll still cross, unless they've fallen at exactly the same angle.

The first four postulates are axioms, as solid (within their system) as bedrock. They're the the bottom of the stack and cannot be reduced further. The fifth remains a slightly wobbly postulate and requires more assumptions. This bothered mathematicians. Anyone who could untangle those intersecting lines would "purify" Euclid and earn eternal fame. Proofs of increasing complexity were published over the centuries, but every time some dreadful flaw emerged. The search consumed the lives of many great mathematicians. In 1820, Farkas (Wolfgang) Bolyai wrote to his son.
You must not attempt this approach to parallels. I know this way to its very end. I have traversed this bottomless night, which extinguished all light and joy of my life. For God's sake! I entreat you leave parallels alone, abhor them like indecent talk, they may deprive you from your time, health, tranquility, and the happiness of your life. That bottomless darkness may devour a thousand tall towers of Newton and it will never brighten up in the earth... I thought I would sacrifice myself for the sake of the truth. I was ready to become a martyr who would remove the flaw from geometry and return it purified to mankind. I accomplished monstrous, enormous labors; my creations are far better than those of others and yet I have not achieved complete satisfaction. For here it is true that si paullum a summo discessit, vergit ad imum. I turned back when I saw that no man can reach the bottom of this night. I turned back, unconsoled, pitying myself and all mankind... I have travelled past all reefs of this infernal Dead Sea and have always come back with broken mast and torn sail. The ruin of my disposition and my fall date back to this time. I thoughtlessly risked my life and happiness - aut Casear aut nihil.
But despite this warning worthy of any Gothic Horror novel, the son, Janon (Johann) Bolyai dared to continue his father's work. In 1823 he wrote back.
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.
If that's not Lovecraftian prose, I don't know what is! The dreadful warning, the obstinate investigation, the transcendent discovery that smashes reality; this is a horror plot writ large.

In one of those odd coincidences of history, a half dozen or more mathematicians discovered non-Euclidean geometry at the same time. Euclid, the bastion of stability for generations of schoolchildren, was toppled from his pillar, and a realm of curving chaos enthroned in his place.

Part 2: A Matter of Perspective

The fifth postulate, as described above, matches our expectations of reality. Railway lines, sticks, towers; all seem to converge, meet once, and then diverge forever. But it is not necessarily so. Two equivalent postulates exist.

Draw a horizontal line on a flat sheet of paper. Drop two sticks on it, and imagine one stick is angled to the left, the other to the right. If they are extended infinitely and meet only once, then we have Euclid's fifth postulate, and all is well.

But imagine the sticks extending to infinity and never meeting, just continuing forever. That doesn't make sense if our piece of paper is flat, but what if it's curved? What if it, and the horizontal line, and the sticks, are on the surface of a sphere? Then the lines could extend infinitely and form a loop without ever meeting. This is elliptical geometry.

Or imagine the sticks cross once, then cross again  (at least once). That's odd. As they extend towards infinity, they loop back and forth like a pair of skaters or stitches in cloth. If our piece of paper is flat, that makes no sense, but if it's a sort of saddle-shape, the seemingly straight sticks can bend back towards each other. This is hyperbolic geometry.

Amazingly, for both elliptical and hyperbolic geometry, Euclid's first four axioms remain true. Strange bendy triangles and pointed circles arise, but with consistent and unvarying results. This raises a distressing possibility. Since all three variants are equally "true", which one is the "real" one? While the universe appears to follow Euclidean geometry, it might be because we can only see lines of limited size. If the universe is not flat but curved, it could easily appear locally flat. Compared to the universe, a railroad track or Grecian column is very small indeed.

Any scientific discovery takes time to enter popular consciousness. Lovecraft was born in 1890; just late enough to be educated on textbooks drawn from Euclid by schoolteachers steeped in Euclid, but well within the time when non-Euclidean geometry was percolating into popular culture. In every field of study, a world of fixed absolutes and thousand-year laws was collapsing into incomprehensible chaos, and geometry - the last bastion of law - was not immune.

I think it is impossible for a horror writer to fake disgust or terror. To make a scene authentic, they must be repulsed by what repulses their characters, horrified by the same scenes, staggered by the same conclusions. Horror must be drawn from life

H. P. Lovecraft was fascinated and horrified by the difference between perception and reality. What we see, and what is true. Do the sticks meet only once, or do they merely appear to meet? What is real, and can the human mind survive true comprehension of reality?
One can only hope that Lovecraft's racial views, his "fascinated disgust", will one day require such additional explanatory footnotes.

Part 3: Phase-Contrast Microscopy

When Meillassoux writes about the idea of the world-in-itself he invokes the idea of a “great outdoors” or “absolute outside” – a world that exists “whether we are thinking of it or not” and which “thought could explore with the legitimate feeling of being on foreign territory – of being entirely elsewhere”. It is precisely such an “outside” that preoccupies Schopenhauer when he writes of the will-in-itself, and while Schopenhauer inherits much from Kant, including an insistence that the world of our senses is one of mere phenomena or “representation,” his “strange immanentism of noumena”, as Thacker puts it, links the will-to-live to the phenomenal world, since the latter is but the manifestation in space and time of the indifferent and inaccessible former.

-The Daemonology of Unplumbed Space: Weird Fiction, Disgust, and the Aesthetics of the Unthinkable, Jonathan Newell
This is good academic writing. As far as I know it's solid gold. I can't tell Schopenhauer from Schubert, and Jonathan Newell strikes me as the sort of person who knows what they're talking about. There are several plausible-sounding quotes, and scattering of dashes. I'm convinced, and I'm not even on his dissertation committee.
The horror of the story lies not merely in the contemplation of an alien world, but, crucially, in the realization that the world has always been suffused with alienage. “From Beyond” reveals that reality has been already, always contaminated. [...] We could also read “From Beyond” as a sort of microscope-story, a science fiction tale about seeing things which could not normally be perceived but which science can now reveal, and which are omnipresent. The polypous beings vaguely resemble blown-up bacteria, made visible by the Tillinghast resonator just as bacteria are by a microscope.

This is a horrifying realization, and in the abstract, it's horrifying to anyone. The idea that slimy things with legs do crawl within the slimy air, all around you, invisible and lurking, is horrifying. It needs no additional context. But there's an aspect that Newell didn't emphasize, a twist on mere microscopy.

In 1932, Frits Zernike invented phase contrast microscopy. He won a Nobel prize for it in 1953, but the Nobel committee is always at least two decades behind the times. Phase contrast microcopy would have saturated the scientific and near-scientific papers Lovecraft read. It was, and still is, astonishing. In 1934, H.P. Lovecraft published “From Beyond”.

The speed of light in a vaccum is constant, but the speed of light in different materials can vary. Imagine two photons in lockstep (in phase). Their peaks and troughs match. They are traveling in the same direction at the same speed. One passes through water, the other through air. The photon passing through water moves slower compared to the photon passing through air. Its amplitude and wavelength stay the same. If they both then enter air, the photon that passed through water will be shifted behind. It's out of phase, like Left Shark.

To the human eye, both photons appear to be identical. They've got the same wavelength (colour) and amplitude (brightness*). Light passing through a glass of water doesn't appear significantly different than light passing through the air next to it, right? If it does, get your tapwater tested. Light, as we see it, is wavelength modulated. Longer wavelength fall towards the red end of the spectrum, shorter wavelengths towards the purple. And that's it. Our feeble eyes can't see polarization or phase shifts. Phase contrast microscopy breaks down the barrier of mere human perception.

*yes, I know, brightness=/=amplitude but this is a D&D blog, give me a break.

Phase contrast microscopy uses very clever optics to turn phase shift - the delay in light - into a wavelength or amplitude difference. Suddenly, things that were invisible become visible. Everyone who uses phase contrast microscopy says something like "it opens up a whole new world", and it really does. Glassy organisms become sharply defined. An invisible world pops into full visibility.

I don't really "get" the optics behind phase contrast microscopy, and, bluntly, neither did Lovecraft. The principle is enough, and he didn't have full colour youtube videos (or a phase contrast microscope to play around with). A whole new world, parallel to our own, but hidden by our paltry and limited perception. This "newly visible world that lies unseen all around us" actually exists. It's real! It's under the microscope! Ick!

To the modern reader, phase contrast microscopy is fun, even adorable, but like anything else it can be turned into horror if taken to an extreme conclusion. Lovecraft saw the horror in reports of phase contrast microscopy, and turned it into a story. We can see, in living colour, what he could only imagine.

Conclusion: There Are No Straight Lines

They say of the Acropolis where the Parthenon is that there are no straight lines.
-Stephen Fry, QI Series E Episode 1

From our infancy, the idea of certain contrasts becomes fixed in our minds: water appears to us an element that moves; earth, a motionless and inert mass. These impressions are the result of daily experience; they are connected with everything that is transmitted to us by the senses. When the shock of an earthquake is felt, when the earth which we had deemed so stable is shaken on its old foundations, one instant suffices to destroy long-fixed illusions. It is like awakening from a dream; but a painful awakening. We feel that we have been deceived by the apparent stability of nature; we become observant of the least noise; we mistrust for the first time the soil we have so long trod with confidence.  
-Personal Narrative of Travels to the Equinoctial Regions of America, During the Years 1799-1804, Alexander Humboldt

Non-euclidian geometry turns straight lines and solid ground into curving and suppurating folds of incomprehensible space. Phase contrast microscopy turns a glass of clear water into a writhing column of vibrant life. H.P. Lovecraft was horrified by these things, and he tried to convey this horror to his readers. Without context, without knowing why an author was horrified, stories lose a degree of vibrancy.


  1. Just to say: keep doing what you're doing. Inspirational.

  2. Great post!

    I was just DMing a sequence of sessions in a non-Euclidian multi-plane. A place echoing the past of world. Pockets dimensions connected by cylindrical rivers. Each one of them containing fragments of the past. Probably partially inspired by your Book of Speculative Ages. Also inspired by the myths of the dungeons the PCs have found: snakemen (Tomb of the Serpent Kings), apemen (The Portal Under the Stars), weird elves and goblins (from Dolmenwood)... And a tale named The Third Bank of the River (http://www.sphstigers.org/ourpages/users/jasher/SSBootCampWebsite/The%20Third%20Bank%20of%20the%20River.pdf)

    It was interesting, but I have felt that many times the players did not feel comfortable for experimenting with the world. They were afraid of taking actions, because they did not understand the physics. They were afraid of making "wrong" choices. Therefore, I was afraid that they were finding the weird physics too arbitrary, and they would end up losing some interest in the game. Abstractness is daunting.

    1. A very good point. RPGs and theater-of-the-mind stuff relies heavily on intuition. The size of a room. How a door works. How people behave. A GM can say "a city" and a city instantly appears in the minds of the players.
      But if the GM says "a riemannian manifold", what tends to appear is "error 404, image not found". Intuition breaks down.

  3. Great post, and I'm very flattered by the references! This totally gets at something core to Lovecraft, I think. He's horrified by the idea that the stable-seeming, comforting, everyday world we all know, the one revealed by our senses and the one that culture gave meaning and order, is a lie, and that the truth is that the universe is this alien, bizarre, non-intuitive thing. Particularly adored those quotes from the Bolyais!

    It's unsettling and compelling to him not just because of the sickening gap between perception and reality, but because that gap dethrones human beings from the exalted vantage we like to place ourselves on; we're not as special as we've told ourselves. And then with microscopy and evolution, we discover that we're not "pure, unpolluted" beings, we're crawling with non-human life - and ultimately we pulled ourselves from the primordial muck. It speaks to this powerful tension in Lovecraft's mind and his writing, a tension between the realization that the universe is this queasy, strange, indifferent, place, that human beings are porous, fleshy accidents of evolution, and his desperate desire to foreswear all this and cling to the comforts of tradition, to these arch-conservative ideas. That tension made his politics so incredibly poisonous and also gave his fiction its virulent power.

    1. Thanks!
      As a side note, I suspect the contempt heaped upon any math curriculum change (i.e. the New Math) comes from the same roots. Math is supposed to be math; how can the math I learned be different than the math you learned? This new and confusing system is beyond human comprehension; best beat a retreat to safer traditional ground.

    2. I think it’s interesting how a lot of contemporary, consciously post-Lovecraft cosmic horror is by and/or about people for whom the regular everyday world isn’t particularly comforting, who live in cultures that are in some way alienating or hostile to them.

  4. I'm not quite sure whether the main point of this article is the science-horror part, or the excuse to talk about scientific topics part, or equally important; if the middle, then this comment may be a bit pedantic, so apologies on that.

    Anyway, I want to push back a bit on the science-horror thing. I understand the appeal of it, and Lovecraft has of course influenced me as he has many of us, but even the obvious stuff like his racism aside, increasingly I just find science-horror unpalatable. In 2020, do we really want to be talking about "forbidden knowledge", reveling in ignorance, and discussing the dangers and horrors of exploring and understanding the universe? I'm not saying in an absolutist sense that we shouldn't have concerns, like for instance with weapons or weaponizing technologies to manipulate or control people, but even then, I think it would be more productive to do it in a tempered and considered way.

    It is just wild to me that someone could be smart enough, educated enough, and open-minded enough to challenge their preconceived understandings of the world, but so myopic that they can't get over the abrasiveness inherent in that. Again I don't want to overstate it or be absolutist about this, I'm not saying existential horror can't be a thing, but I really think with Lovecraft it's less so existential horror per se- that's just a pretext for challenging ones personal beliefs and biases, it's a much more personal and petty response, I think. When I read those Bolyai quotes you shared- I think one could just as well interpret those as saying "if this is true, we have a lot of work ahead of us and I don't want to do it", than existential horror per se. It sucks when you realize something you've spent a lot of time on is wrong, or doesn't work the way you thought it did in some critical way, and you're going to have to start over or scrap large portions of it, or on an even more basic and primal level than that, you're just going to have to change your thinking. As a former scientist and current software engineer, I understand those pains. But letting it scare me or trying to deny or ignore it has never worked in my favor.

    You can call the things revealed by phase microscopy creepy crawly monsters, or you could call them fairies. You can be appalled by the implications of a curved universe, or revel in the magic of the possibilities in that.

    I really think nowadays, I mean there certainly are people who read Lovecraft and experience the existential horror of it, but I know for myself, and probably for many others, the stories are meant to be horror but what I read in it is not horror but wonder, at the possibilities of what is or what could be, of thinking about the world in a totally different way.

    1. This comment has been removed by the author.

    2. It's interesting - horror and wonder really slip into one another at times; they're not opposites. Lovecraft was deeply unsettled by what science told us about the univese, but he was also an ardent materialist and science enthusiast. Lovecraft talks a lot about forbidden knowledge, but on some level he finds that insight really powerfully attractive.

      I feel this really palpably at the end of "The Shadow Over Innsmouth," which on one level can be read as a pretty odious story where a lot of Lovecraft's uglier views are on full display - but then that ending, at the same time, is ecstatic, blissful, fascinated.

      "Stupendous and unheard-of splendors await me below, and I shall seek them soon. Iä-R'lyeh! Cthulhu fhtagn! Iä! Iä! No, I shall not shoot myself—I cannot be made to shoot myself! I shall plan my cousin's escape from that Canton mad-house, and together we shall go to marvel-shadowed Innsmouth. We shall swim out to that brooding reef in the sea and dive down through black abysses to Cyclopean and many-columned Y'ha-nthlei, and in that lair of the Deep Ones we shall dwell amidst wonder and glory for ever."

      I do think this year has demonstrated the necessity of confronting ways that the universe can be terrifyingly hostile to human life - and, to your point, the awful cost of denying or ignoring that horror. Covid-19 isn't quite the colour out of space, but 2020 has felt deeply Lovecraftian to me, reminding us that nature is not here "for-us," and that even with all our science and knowledge, we have not sealed ourselves off from it.

    3. One of the nice things about blogging is that articles don't need a main point! I'm free to post as many half-baked ideas as I'd like. :D

      As Jonathan said, horror and wonder are cousins. I am fascinated by the things I find in a drop of pond water, but part of me goes "ick". Those beautiful glassy forms, those elegant chains of proteins, the perfection of microcystals... but also, the writhing tentacles, the ravenous blind hunger, and, of course, the fact that they simply are - they exist without needing interpretation or context.

      I am also deeply cynical about hopeful stories. Humans love stories, and stories can be a mental trap. More notes on the subject here: https://coinsandscrolls.blogspot.com/2017/06/stories-hard-science-fiction-and-moloch.html

    4. I think I agree with the idea you two suggest that wonder and horror can be complementary.

      That being said, I only partially agree with the covid example. If anything, covid, to me, speaks less to the horrors of the world per se, than it does the horrors of willful human ignorance and the things we're willing to allow in lieu of addressing problems head on. But that's a whole separate conversation and in any case also not necessarily mutually exclusive with the idea of horrors of the world. It sounds like we're probably agreeing more than disagreeing on that anyway.

      And I agree that overly optimistic / hopeful stories have their own problems (I'll have to read that article to get your thoughts on that).

    5. Horror and wonder are two sides of the same coin.

    6. I'd definitely agree that part of the horror of Covid is our own refusal to confront it, or respect it - clinging to anthropocentrism, perhaps.

    7. I've always said sense childhood is that one of the great things about tigers is that on one hand they are like big and cuddly kitties, on the other they can rip your face off.

  5. "One can only hope that Lovecraft's racial views, his "fascinated disgust", will one day require such additional explanatory footnotes."

    I will say that, while I am admittedly sheltered from the worst realities of racism by both living in a fairly multicultural and tolerant part of the world and being part of the majority there, it wasn't until years after reading Lovecraft's "The Shadow Over Innsmouth" that I realized it was meant to be a derogatory allegory for interracial marriage. It's one of the worst of his stories in regards to his racism (though one of his best as a story itself), but it all just flew over my head because I've never viewed interracial relationships as something I even have a stance on, because no one here really has a problem with them. Which I guess is a form of privilege, but the kind that's better to extend to everyone than to eliminate.

    1. I suggest reading Michel Houellebecq's essay "H. P. Lovecraft: Against the World, Against Life" about how Lovecraft's racism is the basis of the horror in its writings. Excellent food for thought

  6. That was a very good piece, and thought-provoking. Thanks! If you’ll pardon an unsolicited-but-friendly copy-edit:

    >Lower wavelength fall towards the red end of the spectrum, higher wavelengths towards the purple.

    In fact, greater (longer) wavelengths are at the red end of the spectrum, shorter wavelengths at the violet or purple end. Other way round for frequencies: low frequencies red, high frequencies violet or purple. The latter is probably what you meant: we generally use ‘lower’ and ‘higher’ for frequency, versus ‘longer‘ and ‘shorter’ for wavelength.

    Dominic Brown

    1. Oh the shame! I was thinking "longer/shorter" in my head, but wrote "lower/higher".

    2. What I figured. Great article—keep up the good work.