# Topology in non-mathematical literature

A great piece of knowledge that I heard from a talk of Robert Ghrist, is that one of the earliest instances of non-trivial manifolds (i.e. of dimension higher than 2) appears in Dante's Paradise, where he essentially describes the cosmos as the union of two balls, one containing Earth, and the other containing the paradise and the celestial spheres, "glued along their boundary", hence essentially describing a 3-sphere.

I would be extremely interested to know if anyone else knows of similar instances where "complicated" topological spaces in fiction and classic, non-mathematical literature.

To be clear, I am trying to stay out of the usual torus/moebius strip stuff.

EDIT: I have been asked to give evidence of my statement about the 3-sphere in Dante. Before that, answering to those who say that this is being generous toward Dante, of course he did not describe the three sphere intentionally. His poetic, especially in the paradise, is allegoric, he explain theological concepts poetically through half-impossible images (the figure of god, made of three interconnected rings that mirror each other, is hardly a tentative to tell us about the Borromean links) and yet in the seemingly impossible description of the cosmos, he ends up describing a 3-sphere.

This being said. In Canto 28, Dante gets to the Empyrean, and looking down he sees the Earth surrounded by the celestial spheres. He then turns around and looks up, and he sees God as a point of light, surrounded by angelic spheres. Then

La donna mia, che mi vedea in cura forte sospeso, disse: «Da quel punto depende il cielo e tutta la natura»

My lady, who saw my perplexity — I was in such suspense — said: “On that Point depend the heavens and the whole of nature.”

The word "depende" translates from latin as "to hang from". So one interpretation is that the whole cosmos hangs from the point of light that is god. It then becomes extremely natural, with the modern mathematical language, to interprete this description as god being the "north pole" of the cosmos, earth being the "south pole", the angelic spheres being the "north hemisphere parallels, the celestial spheres being "south hemisphere parallels", and the primum nobile being the equatorial sphere in between. More terminology used by Dante goes in that direction, e.g. the fact that Dante insists that the universe is made of these two parts, the "original" (paradise and angelic spheres)nand the "copia" (earth and celestial spheres). See also this: https://mathinees-lacaniennes.net/images/stories/articles/dante.pdf, where these concepts are explained a bit better (i am no theologist, nor historian of literature).

• I think this is generous to Dante, but being similarly generous I suspect many ancient religious scholarly writings can be interpreted as making reference to the formation of reality as some sort of manifold-boundary-glueing. I’m not sure MO is an appropriate place for a question like this, though. Jan 17 at 22:35
• I don't suppose you're interested in the Heinlein science fiction short story, "And He Built a Crooked House", invollving a tesseract – archive.org/details/Astounding_v26n06_1941-02_dtsg0318-LennyS/… Jan 18 at 1:45
• @AlecRhea Really? Many ancient writings can be interpreted as referring to manifold-boundary-gluing? Can you name one? I can't. Jan 18 at 11:23
• In fact, I'm surprised to hear about this interpretation of Dante. The "standard" interpretation is that the geometry of Dante is simply a series of concentric spheres. Various artists through the ages, including Michelangelo, have drawn pictures of Dante's fictional world, and I don't recall any of them as suggesting an "exotic" topology. Can @CuriousUser give any more details to support this interpretation? Jan 18 at 11:38
• @AlecRhea Well, if your point is that there is zero evidence of exotic topologies in Dante, then I agree. Jan 18 at 13:07

Maybe something like Borges "The Library of Babel". In this story he wrote "The Library is a sphere whose exact center is any hexagon and whose circumference is unattainable". You can think of it as a manifold of dimension 3. I think I remember it can be visualized as a Klein bottle.

Also "Pascal's sphere" (La esfera de Pascal) is interesting. He wrote "sphere with center everywhere and circumference nowhere". I read it long time ago, so I don't remember the details, but there is this type of content here.

By the way, in general Borges books have, implicitly, lot of mathematical ideas.

• Borges' library is infinite in all directions, filled with books of finites size printed in finite alphabet. He concluded that the books are not unique, but also assumed that this implied that the library is periodic in all directions. The latter error aside, if we accept that the library is periodic in 3 directions, we get an explicit universal cover of a 3-torus. Jan 18 at 17:20
• Now I'm wondering if there are examples in Lewis Carroll's Alice books. He has every other form of mathematical oddity in there somewhere... Jan 18 at 17:24
• You can get something like a "sphere with centre everywhere and circumference nowhere" in the framework of non-standard analysis by taking a sphere with radius the reciprocal of an infinitesimal. If you move any finite amount, you have only moved an infinitesimal proportion of the radius away from the centre. Jan 18 at 17:36
• @Kevin Unfortunately I haven't found a good translation of Alice to Spanish. Although I understand and write English I don't have enough level for read the word games of Carroll. The Spanish translations I have read are so bad... Jan 18 at 17:41
• @Kevin I wonder if Dodgson condensation can be found somewhere in Alice :-) Jan 18 at 18:20

The classical 1884 fiction Flatland contains a dialog between the square and the sphere, where the former tries to convince the latter that if Spaceland exists, then there must also be a space of four dimensions and a being living there made of multiple spheres in one.

In the short story And he built a crooked house by Robert A. Heinlein, an architect builds a 3-dimensional house made of 8 cubic rooms arranged according to the template of the faces of a tesseract. During the visit of the house, an earthquake actually folds the rooms and the protagonists end up trapped in what is now the surface of an hypercube.

• The Heinlein work is a short story, not novel. Jan 19 at 11:07
• You mean the boundary of a hypercube, not the "surface". Jan 19 at 15:39
• @kapil I guess hypersurface is the correct word. Jan 19 at 19:55
• I had mentioned the Heinlein story in one of the first comments on the question. Jan 19 at 21:00

Jacques Lacan is a french psychoanalyst who made use of topological concepts to discuss his theories. The Borromean knot played a special role in his 70's work, illustrating the relationship between the Real, the Imaginary and the Symbolic. His use of topology is controversial, Lacan is criticized for example in the book of Sokal and Bricmont, Fashionable nonsense.

• Similarly, Alain Badiou writes about topology a fair bit, and similarly, your mileage may vary on how much insight you find there... Jan 19 at 0:12

This one is a bit of a stretch, but I've always associated the Egyptian deity Apep with the long line.

A depiction from the tomb of Ramesses I:

Keeping in mind that hieroglyphics are a pictorial language, these artistic depictions of their cosmological origin/sustenance stories amount to one of the best windows we have into their collective cultural interpretations of these stories; this is reinforced, to my mind, by the fact that we find them alongside actual 'text' on tombs as above, with the text seemingly as 'background material'.

As the story goes, Apep arose as the animated umbilical cord of Neith after giving birth to Ra, the sun god. While Ra went on to embody all that Egyptians considered 'good' (order, light, truth, etc.), Apep became an embodiment of 'evil' as Egyptians saw it (chaos, darkness, falsehoods, etc.)

The day/night cycle in Egyptian culture was embodied by the struggle between these deities, with Ra riding across the cosmos daily to maintain order against the background chaos of existence, embodied nightly by Apep. Each night, Ra (assisted by Set) would slay Apep, only to have him rise again the next night -- these 'slayings' are typically depicted as a spear towards the snake's body/head, presumably chopping it into pieces:

Despite this nightly dismemberment, there was always yet more of Apep ripe for the slaying the following evening. This seems like an informal consideration of $$\omega_1\times[0,1),$$ where Ra slays and dismembers some number of the $$[0,1)$$'s off the end each night and finds more waiting for him the next day. Conflating divinity with the 'highest possible infinite' a la Cantor, we could also interpret their discussions as being about $$O_n\times[0,1),$$ a much longer line.

• How are you arriving at $\omega_1$ (uncountable) instead of just a countably infinite snake?
– AnoE
Jan 19 at 10:42
• @AnoE Personally, I feel like $\omega_1$ is still too small to discuss what they were alluding to; a more appropriate interpretation would be the latter one above, where Ra is allowed to slay any set-sized ordinal amount each night and still arrives to a proper-class length snake the next day. These are gods, after all; why should they be constrained to the first few piddling infinities? Jan 19 at 14:17
• The sense I get is that Apep was ‘counterintuitively long’, even to other gods like Ra and Set. Jan 19 at 14:28
• Honestly this reads much more to me as a snake of length $\omega$. Jan 21 at 21:31
• @NoahSchweber (This is all mostly in jest, in response to Timothy calling me out above. I am by no means an egyptologist lol) Mar 25 at 23:05

Not sure if the single point topological space qualify (but why not?). In Cosmicomics, Italo Calvino has a hilarious story on life on such a space, All At One Point, before the Big Bang started the Universe.

Some puzzling literature not only for topologists but also for differential geometers: J.R.R. Tolkien's The Silmarillion is the book that tells the story of Arda, the world in which The Lord of the Rings is set. According to the book, Arda was created flat, but then turned into a round world by an act of god, with a cataclismic event that happened during the Second Age, so that humans could not reach the sacred land of Valinor anymore. (However, lore is elves can still somehow 'sail the straight road' and get around the round-world structure to reach Valinor.)

Image from Wikimedia / Author: Ian Alexander.

• Maybe the Tolkien deity Ilúvatar used the exponential map and the straight road is just a lift to the tangent space. Jan 21 at 22:07

At the Bourbaki Seminar in November 1968 the participants were handed a (premature) announcement of Bourbaki’s death.

At the end it says

Car Dieu est le compactifié d'Alexandrov de l'univers. Groth IV 22.

“For God is the Alexandrov compactification of the universe.” Groth. IV.22

You can find the complete history here

• I believe this Kabbalist pov dates back even before Bourbaki. Jan 19 at 15:23
• This is essentially the same point as Dante's in the body of the OP, no? Jan 19 at 20:51
• Monotheism, how dull. I want to join a religion in which God is the Stone–Čech compactification of the universe! Jan 19 at 23:44

In Ted Chiang's 1990 novelette Tower of Babylon, the miner who climbs the tower finds that the shape of the world

is like a seal cylinder:

When rolled upon a tablet of soft clay, the carved cylinder left an imprint that formed a picture. Two figures might appear at opposite ends of the tablet, though they stood side by side on the surface of the cylinder. All the world was as such a cylinder.

The topology is $$\mathbb{S}^2\times\mathbb{S}^1$$.

• Thanks @FedericoPoloni for the formatting — the need for three clicks captures at least a bit of the drama.
– user44143
Jan 22 at 23:41
• I think the expression is "cylinder seal" rather than "seal cylinder"; see en.wikipedia.org/wiki/Cylinder_seal Jan 22 at 23:54
• @JohnStillwell, I almost switched it to “cylindrical seal”, which is the version that sounds right to me, but “seal cylinder” is as in the story.
– user44143
Jan 23 at 1:28

This thread wouldn't be complete without something from Greg Egan. For instance, his short story The Infinite Assassin contains a "multiverse" that behaves like a Cantor set, and Diaspora is set in a six-dimensional universe that behaves like a fiber bundle.

And in Schild's Ladder, one of the characters states this:

My earliest memories are of $$\mathbb{CP}^4$$ — that's a Kähler manifold that looks locally like a vector space with four complex directions, though the global topology's quite different.

Not sure if this is what you are looking for, but recently i came across a book that claims to explain psychology using topology. The book in question is the "Principles of Topological Psychology" by Kurt Lewin. I do not know if the book is a "serious" attempt to mix topology and psychology or just a joke as i have not read the book.

• It's actually "Topological Psychology", not "Psychological Topology". It is not intended to be a joke, but it's certainly good for a chuckle. Jan 21 at 14:06
• @RobertFurber my bad. Jan 25 at 18:56

I know you want to "stay out of the usual torus/moebius strip stuff" but I can't help mentioning the infinitely long short story "Frame Tale" by John Barth.