But There Are Other Geometries

Looking up at the stars, I know quite well
That, for all they care, I can go to hell,
But on earth indifference is the least
We have to dread from man or beast.
 
How should we like it were stars to burn
With a passion for us we could not return?
If equal affection cannot be,
Let the more loving one be me.
 
Admirer as I think I am
Of stars that do not give a damn,
I cannot, now I see them, say
I missed one terribly all day.
 
Were all stars to disappear or die,
I should learn to look at an empty sky
And feel its total dark sublime,
Though this might take me a little time.
 
—W.H. Auden, “The More Loving One”

Auden taught himself to be the more loving one, though I cannot say I sense in his other work that he also taught himself to be happy. I have always loved Auden. I loved her because she understood how I loved Auden, his antiquarian referentiality, his knowing melancholy, his formal mastery of the line. But in the end, I was the more loving one; I, to her, and when it ended between us, I choked up every time I tried to read Auden under a newly black sky.

I could barely read anything at all, but I could look. I spent the nights up, looking. This is how I discovered the cube-sculpture poems of Rimma Gerlovina. She began making the cubes in the Soviet Union in 1974 and continued after moving to New York City in 1980. They are colorful, some made of plain cardstock, some covered with floral motifs. Most of them open like boxes. Almost all of them contain words, mostly in Russian, typewritten on little labels affixed either to the outside or interior. On her website, Gerlovina translates these labels herself and includes a picture of each cube. One early piece, titled After Chekov, has a label on the exterior whose Russian words are rendered as “Man should be beautiful in every way—in his.” In its interior are four small cubes whose labels read “soul,” “face,” “cloth,” and “thoughts.” Another piece is a solid red cube that reads “Who is there?” on the outside; a tiny green cube sits inside, saying emphatically with its small form and in attached letters “It’s me.”

If it seems strange to mourn a love up late looking at cubes, ours was a strange love, one in which we became each other’s formal constructs, made structures for knowing together. She changed how I saw the cube, and the rhomboid chunks of stone in the foreground of Bruegel’s Great Tower of Babel; all the Bruegels were ours, the little landscapes in their frames like capsule universes for which we had a secret language together inscribed on the back of the earth’s palimpsest. We built our own Babel of signifiers, spoke our own tongue. If I had handed her a tetrahedron, she would have known to give me a cube, and I in turn an octahedron back—we implicitly knew each other’s order, the next thought, the next Platonic solid, the next word. We burned in tandem, equal hemispheres—until we didn’t, and I burned alone. So there I was looking longingly at cubes, a toppled tower, a blackened sky. What do you do when the whole rational-passionate order of your love is gone? How do you figure the world again?

Many of Gerlovina’s cubes play knowingly off of their cubeness. A label on the outside of a now-lost example reads “The cube in the space” while an interior label reads “The space in the cube.” The space of my more loving one was not a strictly empty sky. A cube, to me, always lives on the Cartesian grid, the x-y-z coordinate plane that Descartes codified in the 17th century. We called each other the Morning and the Evening Star, a binary orbit. We called each other, sent long epistolary texts, split by the Atlantic during the pandemic, five hours apart but always together. There was tenderness in our constant distance. I kept the letters that smelled like her perfume. There was an intentional note there of curdled milk, a warning. Do you know what happens to binary stars when one goes supernova and the other becomes a black hole? The first star gets thrown into the cold, empty cosmos or sucked in and crushed. And then the Cartesian grid goes too, because black holes don’t obey those geometries; it’s like space-time dips and the grid drips down with it in ways beyond Descartes, or Newton, or the three dimensions of the cubic form. So I was off the rational grid, so to speak, but even so, the x and y coordinates are still called the abscissa and the ordinate. Abscissa, from the Latin for a line having been cut off. Which she would know implicitly, and so would Descartes, and Auden, none of which makes it any better when the burning stars of your world shut off and you’re floating in the vacuum, and the red cube isn’t open yet, and the little green cube that says “it’s me” is just sitting in the dark.

***

Here's the part I wasn’t telling you before: I was in the dark alone because my friends couldn’t know her name. I had been what they call “pocketed.” They are the psychology experts quoted in Women’s Health and Cosmo when I googled what is apparently a rising trend: keeping your partner or their status a secret from your friends. She had her reasons, good ones: trauma and circumstance. But sometimes Descartes just bangs on about the oxygenation of blood in the heart in his Discourse on Method and forgets that reason isn’t always enough, that if the soul is really in the pineal gland, and thinking is a first principle, no amount of reason can necessarily make it all right anyway. Descartes has always felt cold to me compared to, say, Montaigne. She thought we could make it if I were more dispassionate; if I could live with the reasoning of being pocketed.

***

I was told to state my needs and I did, in bullet points, in long tiresome letters, even in the formal style of Waka poetry in the Japanese courtly mode with appropriate reference to the current sub-season using the 72-part calendar of the old capital at Nara. Yes, I know, I know. The food during the season we fell apart was broad beans and cutlassfish. Such was the nature of our love that I knew she would know I had taken care to get the sub-seasonal cuisine right. Or that once she would have, and delighted in it, our mutual precisions of calendars, our mutual tongue of dead courtiers and small ceremonial deer. Anyway, it didn’t matter because now my desire was too much, my passion too great, because the more loving one wants more than indifference sometimes, because I couldn’t be Auden like that. I had always told myself I could reason my way to a good life. And yet here, profoundly, was the intimate failure of reason in the face of want.

***

When I finally told a friend, the poet Alina Stefanescu, what had been happening, she in turn told me about the history and theory of pockets, because she knew it would make me feel better. She sent me links to historical pockets in museum collections. Pockets were originally more like cubes, which is to say discrete three-dimensional forms attached separately to clothing by a band at the belt. They evolved later into the two-dimensional attachment that forms the pocket on the garment. Alina understood intuitively how it felt to be flattened, like a cube that is six squares laid out. That’s how you can make a cube with paper, how Gerlovina made hers, by the reverse process, folding up the six squares into the third dimension. That’s what it feels like to finally tell someone about your love’s demise afterward, too, the snapping up the z-axis into the dimension you’re really meant to exist in, the very thing prohibited from your life in loving.

***

There isn’t a Gerlovina cube like this, but you can imagine it: the outside says “The Closet,” or maybe also “The Pocket.” The inside says “It’s me” again. That’s partly what pocketing feels like when you’re queer, the lattice of convenient lies, the mesh that falls down over you, hiding part of yourself all folded up. In the speculative fiction trilogy The Three-Body Problem, the Chinese science fiction writer Liu Cixin imagines pocket universes, sanctuaries from a dark forest of galactic strife ring-fenced with black holes and roving civilizational destruction. Liu also imagines what happens when one intergalactic civilization destroys another. They’re not gone exactly, just flattened into one dimension, a single point. Flattening is what you do to your worst possible anticipated enemies to seize their scarce resources; it’s not how you treat a beloved. If the sky is empty in the universe of The Three-Body Problem, it’s because everyone with a star to speak of flattened everybody else—put them in a pocket, that is a closet, that is a grid with quadrilateral lines of a relationship that exists only to and for you, from which the known world is excluded.

***

There isn’t a cube for this, but there’s a cube for Descartes, one on which Gerlovina writes “You think” on the outside and “but I am!” on another smaller cube inside. This is a reference to Descartes's most famous line, perhaps the most famous line of Western philosophy: Cogito ergo sum, I think therefore I am. Gerlovina says “You think,” and “but I am,” though, not a faultless statement of first principle from which to build a reasoned trajectory but a rejoinder, a comeback, a defiance. Maybe you aren’t, actually, maybe the pretense of dispassionate reason implicit in that claim actually hides Descartes’ rather passionate justification of the existence of God and is just relying on you not to disassemble it. Projecting an acceptance of dispassion maybe never worked for anyone, or maybe is just a thing analytic philosophers do to pretend the condition of personal objectivity exists to make their work meaningful. I tried to love Wittgenstein, I really did. But I only ever really loved Gödel, whose proof of an incompleteness theorem renders the very method of mathematical proof and formal logic itself broken. This feels less self-satisfied, like Auden saying cruelly of his own face that it was “like a wedding cake left out in the rain.”

***

There is another line to plot between Descartes and cubes, cubes and love and fate. It runs through Mallarmé. The critic and academic Andrea Gadberry does it better than I can here in her Cartesian Poetics; she reads Mallarmé’s famous final symbolist poem “Un coup de dés jamais n'abolira le hasard” (“A Throw of the Dice will Never Abolish Chance”)—set in concrete typographic audacity unusually across the page—alongside Descartes’s theory of fate and free will. There is a version of this poem by the artist Marcel Broodthaers that is rendered with the text lines as just grey rectangles, all stark modernity as concept in 1969. This is a popular year to set modernity in, when we aren’t shoving it at the 1914 edition of the poem or Modernism on the French left bank or Descartes himself. Anyway, the modernity problem is another capricious turn of definitional reason that annoys me, and I’m more interested in the throw, and the dice, those two delectable little cubes with dots that are called “pips” like an orchard fruit. I have two sitting on my desk right now, retired casino dice from Caesar’s Palace in Vegas. I am not the gambling type, but maybe I should be, since now I love throwing them, and how their sharp industrial-precision-crafted edges land with a satisfying thunk. They are red and transparently tempting like Jell-O.

***

Alea iacta est is another famous Latin phrase that sort of betrays itself. The die has been cast is what Caesar uttered as he crossed the Rubicon to seize Rome. Caesar was a soldier and had a soldier’s unimaginative Latin, which is why we traditionally teach him to schoolboys. The Empire unsuccessfully tried to ban dice games and their catastrophic losses many times and always failed, mostly because the soldiers loved them. Cicero is more refined the Caesar, if grammatically excruciating, in his history of dice in De Divinatione, or On Divination. He notes that a throw of the dice might abolish chance, if you’re seeking the advice of dice oracles across the classical Greek world and Asia Minor, who used astragali, uneven knucklebone dice from sheep, to generate prophetic verses. The poems of the dice oracles mostly survive cut into monumental stone columns in places where packs of wild dogs still like to chase archaeologists. Thus, the corpus is not well known by the public, but deserves to be. Consider the following example, translated by Fritz Graf:

The sun has gone down, and terrible night has come.
Everything has become dark: interrupt the matter, about which you ask me…

ήέλιός τε δέδυκεν, ἐφέστηκεν δ'ολόη νύξ ·
πάντα ἀμαυροῦται · παῦσαι, περὶ ὧν μ' ἐπερωτᾷς

This doesn’t feel like sublimity is coming for me anytime soon. This feels like another Gerlovina cube, a green one that says “GRAVE” in all caps like a mossy seat for a stone, and inside a black one that says “I’m already dead.” This feels like the experimental dystopian project from the late ’60’s architecture group Superstudio, which imagines a postapocalyptic cube in the midst of Central Park:

In the most charred, devastated and molten area of that grey space that once was New York, and, more precisely, where Central Park once was, at about 81st, there stands the city. When the others realized that the explosion had irrevocably contaminated all the inhabitants of New York, and that their bodies were rotting without recourse, it was decided to build the city. It is a cube, with a length, width and height of 180 ft, covered in quartz tiles measuring 10 x 10 inches, in each of which there is a lens 9 inches in diameter. This covering condenses light onto the photo sensitive layer behind, which transforms it into energy necessary for the functioning of the city.  

Will my rotten body be next to hers? That would be awkward, at this point, especially since it was me, my flesh, that wanted too much, that wanted her more than she wanted me. Too much and never enough, my body. Maybe if I were the mechanical body of Descartes’ anatomy, I could get over it, but I’m not, I’m all guts and sinew and want. I watched all three Cube horror franchise films in the writing of this essay. They were sort of like existentialist plays, if the dialogue was somehow charmingly bad, and about stereotypical protagonists played by Canadian budget actors trapped in the same evil cube puzzle-prison set over and over. And, if such plays included more limbs being spattered apart by military-industrial grade lasers. This woman is not beautiful in her thought, I imagine Chekov saying of me, plucking out a paper cube gesturally. But I wasn’t ever beautiful that way, that was never the point, I was the more loving one to her, and remain so to many things, texts, and people in the world, because I cannot be dispassionately ordered into something more lovely. Perhaps I am then either uniquely suited or uniquely unsuited to the reading of philosophy.

***

The Cartesian grid exemplifies elegant ordering. But there are other geometries; Riemann geometries are those of spaces stretched over something else, for instance, a grid over a sphere. Stretch me a manifold of a world where nothing is neat or reasoned or reasonable and maybe I make sense; maybe I don’t love too much anymore. Maybe someone will have me. I order a lot of mixed modern divination dice, which inexplicably ship from France—Un coup de dés jamais n'abolira le hasard, indeed. Her French was exquisite, better than mine ever will be. Maybe she would have been a better reader of both Descartes and Mallarmé here. I loved once the simultaneous precision and almost lyrical sorcery with which she read, her voice reedy and quicksilver. There is no way she beats me at the dice oracles, though, at their gnawed little flesh-stripped bones of fate configured into modern plastic conveniences. The bodily turn was always mine; I belong in the charred flesh cube of another well-designed modernity after all, clean lines that hide a mass of decay and despair you must unfold to find.

When the divination dice arrive in the mail, I cast them out from the package directly onto the table. The following appears on each of the visible faces:

The number 13
A queen that is also a drummer clutching a plant (?)
A tree
Six pips
A cartoon turtle
A hook (or a scythe?)
A hand writing with a pen
A magnifying glass
A smiley face with $ eyes
Either a shrine or a porch swing
A person in a hardhat building a brick wall
A question mark

First, before I consult the prophecy, let me tell you that a cube in four dimensions is called a tesseract. Think of it like a cube is folded from squares, but as being sort of folded cubes, extruding beyond the capability of human sight or the three-dimensional geometry of our interior vision. Gerlovina has a sculpture-poem of one representation of the form of a tesseract. Pyramidal insets lead to a cut-out smaller cube set within a solid larger one. On each side is a text triangle that reads “exit into another dimension.” This probably feels like the shift to Bach’s fifteenth Goldberg Variation, like the mode just inclines and the leaves all fall off the trees in the wind, and you’re not eating broad beans and cutlassfish, but it’s Rittō (立冬), just the very start of winter. We both loved the Goldberg Variations, loved them like we once loved each other, as part of loving each other, all these shared infatuations—the insistently cartilaginous harpsichord of Jean Rondeau, and Glenn Gould’s version with its intimacy, famous for the pianist’s humming. Now I’m listening to Víkingur Ólafsson’s new recording of the Goldbergs, and I can feel the edge of the cold and it hurts less, this version, the one that was never in our binary orbit, for which no cost of pocketing was extracted, to whom I was never a secreted thing. I love the fifteenth variation’s start of darkness, I, alone now, unfolding into the black vacuum of sublime minor keyed space, point and counterpoint unfurled together.

Will I ever love again? I read the dice: In thirteen months, or six, or six minutes, or the thirteenth month of a non-existent year. When the drummer-queen of succulents declares it, when I stop missing the empty sky and learn to love turtles and trees and the terrestrial and hew more to the x-axis, covered in an early rime of frost. When the ominous fishhook that looks like an inverted question mark, which is either pre-determination or free will, or whatever some philosophy that permits interchange of admirers allows. When the brick that is the stretched rectangular form of the cube settles into its grout. When I see this new lover for the first time, and we both see Mount Fuji or just the beach, maybe the sand between our toes will feel like equal affection. Maybe she’ll love me as I am; as a tesseract, as an extra-dimensional thing, as someone who cannot be flattened, for all my variational too-muchness. Maybe we’ll meet at the reason-passion axis. Maybe I won’t be the more loving one anymore, maybe, if we come up snake-eyes or box-cars, we can make our methodology together there; in the new darkness of space before matter enters the universe in an explosive boom—before there were stars to give a damn about, before you could cast onto the plane of the table the six faces of fate. What if this new love could be an uncertainty, a blackout sky, an incompleteness theorem, an unknown season, an incalculable gambler’s probability?

References and Commentary

The Great Tower of Babel, painting by Pieter Bruegel the Elder (ca. 1563), is one of two surviving paintings Bruegel did of the Tower, but only this one shows the rhomboid blocks and the workers who move them under the stern supervision of royalty. This, in my opinion, makes it the better of the two.

Descartes’ Discourse on Method was originally published as Discours de la Méthode Pour bien conduire sa raison, et chercher la vérité dans les sciences in French in 1637 in Leiden. The Latin edition was published in 1656 in Amsterdam and was probably more widely read than the French one, since it was the academic standard language of the time.

Cixin Liu wrote The Three Body Problem as part of the Remembrance of Earth’s Past trilogy. The trilogy is called 三体 (Three Body) colloquially in Chinese as a whole. Excerpts were first serialized in Science Fiction World in 2006, with the first novel released in 2008. Tor published an English translation by Ken Liu in 2014. It is part of a boom of new translated speculative fiction from China in the Anglosphere.

Alina Stefanescu is a poet and critic living in Birmingham, Alabama. Along with the critic and essayist Ryan Ruby, she is one of two friends who carried me through the difficult end of this relationship, and the subsequent writing of this essay. Only one of the two actually seems to believe in divination, dice, and astrology.

Gödel’s Incompleteness Theorem was actually published as two interconnected theories, in 1931. The way in which it uses the unique properties of prime numbers is particularly elegant.

"Un coup de dés jamais n'abolira le hasard" ("A Throw of the Dice will Never Abolish Chance"), the poem by Mallarmé, was originally handwritten in a quite unconventional manuscript, then published in print in 1897. The autograph unicum of the manuscript was sold at Sotheby’s, but a picture of the title page remains on Wikipedia for the public.

Andrea Gadberry’s Cartesian Poetics: The Art of Thinking was published by the University of Chicago Press in 2020. It was a joy to think alongside in writing this piece, and it is an exercise both in scholarly rigor and lyrical thinking.

Marcel Broodthaers’ version of "Un coup de dés jamais n'abolira le hasard," based on the first printed book edition of 1914, rather than the magazine run of 1897, was printed starting in 1963 in three editions on three different types of paper. A 1969 edition is currently on display at the Metropolitan Museum of Art in New York City. Perhaps it will make you, too, feel modern.

Cicero’s De Divinatione was written in 44 BC. Practically speaking, most people now read it in the Loeb Classical Library volumes devoted to all of Cicero’s works, specifically volume 154, with a dual English translation from 1923 by W.A. Falconer. These are also available online through Perseus for free in multiple translations and the Latin original. Cicero is merely a cognomen that is thought to originally reference a chickpea; his full name is Marcus Tullius Cicero.

The text of this particular dice oracle is transcribed and translated, as well as given context in both archaeological and theological/methodological detail in a scholarly paper by classicist Fritz Graf—dating these fragments, or attributing authorship, is difficult by nature. The paper—“Rolling The Dice”—can be found in the collected volume Mantikê: Studies in Ancient Divination published by Brill in 2005, and edited by Sarah Iles Johnston and Peter T. Struck. Epigraphy is a highly specialised subject within classics, and I am in Graf’s debt for his accessible and comprehensive work on this subset of texts. Authorship as we conceive of it in contemporary terms would not be recognizable to the semi-random compositions of dice oracles and their layered historical interpretation, transmission, and inscription.

The postapocalyptic New York cube was described in “Twelve Ideal Cities - Twelve Cautionary Tales for Christmas, Utopian project, 1971” by the architects of the Superstudio group (Alessandro Magris, Alessandro Poli, Piero Frassinelli, Cristiano Toraldo de Francia, Roberto Magris, and Adolfo Natalini). It is unclear why this was published at Christmas in particular, but it’s a fun holiday tradition.

The three films of the Cube horror franchise that I watched during the writing of this piece are Cube (1997), Cube 2: Hypercube (2002), and Cube Zero (2004). They are most definitely period pieces and should be enjoyed as such.

The Nara calendar was codified before the Heian-period move to Kyoto, in the old Japanese imperial capital of Nara, during the historical period of the same name, from roughly 710-784 AD. Japanese writing systems have changed since then—I use modern simplified Kanji here.

The Goldberg Variations (BWV 988) were published in 1741. Glenn Gould’s two recordings of the full set of thirty date to respectively 1956 and 1981 were published by Columbia Music. Víkingur Ólafsson’s recording came out October 6, 2023 with Deutsche Grammophon. He plays Carnegie Hall’s Stern Auditorium in February. I have a ticket.