In the spring of 2024, Iraj Ghoochani presented to his reading group (studying Seminar XIII: The Object of Psychoanalysis) with something astounding. They had been studying, as a supplement, Lacan’s address to the Seventh Congress of the Freudian School in Paris, held in Rome in 1974, “La troisième.” On page 50 of the translation by Yolanda Szczech was, according to the translator’s notes, a variation of the Borromeo knot that extended the “toruses” of the Real and Symbolic along Cartesian coordinates. In fact, Ghoochani pointed out, this was a correctly made representation of transferring a Borromeo knot drawn inside an inversion circle to the outside — a Lacanian enactment of extimité, applied to the RSI.
The R and S circles converted — correctly, according to the rules of inversive geometry — to lines tangent to the inversion circle, lines that were at a right angle because their position internal to the inversion circle were tangent to both the circumference and center, and were oriented at π/2 to each other. They were not Cartesian but, rather, “Desarguean,” as Lacan had suggested, because in projective geometry a straight line extends to a single point of infinity and is, by that means, another circle, with a vanishing point at its single infinity: —∞—.
the joyce-vico connection
Also in 1974, Lacan used this Brunnian variation of the Borromeo knot in Seminar XXII: RSI. There, he cited James Joyce’s Finnegans Wake as his source of inspiration. This is not a surprising connection, since Joyce, in both Ulysses and Finnegans Wake, used the circle and line as interchangeable narrative instruments. In “The Linear and the Circular in Ulysses and Finnegans Wake,” David Auerbach noted that “James Joyce made the superimposition of conflicting patterns one of the central principles of both Ulysses and Finnegans Wake. … One very central opposition/superimposition is the idea of linear and circular motion. … [E]ven though Joyce gave Finnegans Wake a clear cyclical structure by having its first sentence complete its final sentence, the linearity is there everywhere as well, from Jaun’s evolution, journey, and quest narrative in the third part to the very process of children growing up and defeating/killing/usurping their father. The one-way journey from birth to death, from child to adult, is neither trumped nor negated by a cyclical view of history.”
The key however is to move beyond characterization to the root of Joyce’s conflation of lines and circles. The evidence on this is conclusive. From Portrait of an Artist as a Young Man (1916) on, Joyce realized the implications of a theory of “metaphoric evolution” discovered by Giambattista Vico (1725; 1744). Vico called this the storia ideale eterna, the “ideal eternal history,” which guided the development of any and all cultures from a first stage guided by an expressive logic, into a representational or heroic phase, ending with rationality, abstraction, and conceptualization. Ernst Cassirer, following Vico, elaborated his own version, the three “functions” of Expression, Representation, and Conceptualization.
Joyce’s grasp of Vico’s idea was radical and accurate. Vico applied his idea of three forms of thought in succession, followed by a ricorso to another cycle, then another etc. not just to history as a whole but to any “unit” of human experience. Joyce demonstrated this independence of scale in the Dalkey episode of Ulysses where, as the Canadian poet-critic A. M. Klein demonstrated, the ideal eternal history could be compressed within any single sentence beginning with an exclamation, proceding with a comparison, and ending in a concept.
The portability of the ideal eternal history cycle is itself an example of the inversion circle’s ability to convert circles into lines and lines into the effect of orthogonality, the independent but simultaneous action of chance and necessity.
inversive geometry as a basis of ethnological conflation of cycles and horizons
Inversive geometry has never been mentioned in Lacanian literature, although Lacan’s use of it and comments on this use in “La troisième” were correct, and his connection of the inversion circle to projective geometry was mathematically brilliant. Ghoochani, who is also an electrical engineer by training, had aided the group’s understanding of many mathematical particulars of Lacan’s “quantum” thinking,* but this particular discovery has profound consequences. In short, and at the very least, it is a way of re-thinking Lacan’s interest in the torus’s peculiar ability to “prove” that two separate and independent holes on a surface are equivalent to two concentric holes: ⚪︎ ⚪︎ > ◎. The British mathematician George Spencer-Brown has made the same claim when, at the end of his Laws of Form, he asks his readers to consider the calculus as if it were drawn on a closed-curved surface, the same spherical medium Lacan asserts that gives rise to the torus.
Inversive geometry has not been discussed anywhere in the Lacanian literature. Yet the “Brunnian transformation” (again, Ghoochani enlightened his group on the meaning of these variations discovered by Hermann Brunn’s discovery in 1982) can be connected to the extimité that Lacan coins but discusses only “two or three times” (according to Jacques-Alain Miller). Although Miller with perfect precision identified the function of inversion in relation to a marriage contract that stipulated that, as soon as a husband would cease to desire his wife, the marriage would end, he does not mention inversive geometry as a consequence of extimité.
Although Lacan himself does not apply the term more than “two or three times,” according to Miller, the idea of inversion is everywhere in his thinking. It is even present in his solution of the Three Prisoners’ Dilemma in his essay on Logical Time. This predates Lacan’s own estimation of his first interest in projective geometry by some twenty-four years (1937–1961).
What does Ghoochani’s discovery mean? First, it allows readers to understand his reference to the “Injunction of Popilius” in Seminar XII, Crucial Problems (p. 310): “[W]hat is outside the circle of Popilius is a circle, just like what is inside it. And the proper of a surface which is called globe is that it is from a closed cut that one separates a fragment.” Lacan mentions Popilius elsewhere but nowhere else is he as explicit about the relation of this bit of history to inversive geometry.
Extimity is, as Miller says, everywhere and nowhere in Lacan’s writing. Like this minimalist mention of inversive geometry in “La troisième,” it is foundational without being trumpeted and, thus, had gone unnoticed by the majority of Lacanians. In mathematics, inversion circles are used to solve problems that are otherwise intractable in other presentations. In psychoanalysis, the challenge is to re-think the role of the “katagraphic cut” — the historically correct name of the circle Popilius draws around Antiochus — in relation to what these two gentlemen from the second century BCE knew about the world: that it was toroid! And, we should add, they knew it without exchanging any words about the matter. With a single closed-curved line drawn in the sand by a “wand,” the inevitability of what Einstein would assert some two thousand years later would require only this graphic gesture.
What does this say about the mentality of the “ancients”? Where else is the katagraphic mark so well understood that, as soon as it is made, everyone seems to “know what must be done”? This is as important a question as the possible meaning and applications of inversive geometry to psychoanalysis.
The same effect (of silent withdrawal) happens with the only use of the word in the Bible (John 8: 1–20), when Jesus, confronted by the Sanhedrin, kneels and begins scribbling in the dust on the floor of the Temple. The magic circle here is the one drawn by the enthymeme, the rhetorical syllogism where the middle term is omitted in order that the audience supply it silently. Missing in the Sanhedrin’s case was testimony from the male adulterer, an omission that would have, in any legitimate court, justified dismissal. With faulty legal standing, the Sanhedrin had thought to trap Jesus into contradicting his principle of non-violence and forgiveness. Instead, Jesus’s response, “let he who is without sin throw the first stone,” let the pronouns draw the injunctive circle, the cause of extimity.
When psychoanalysis turns to ethnology, its most powerful Baedeker is topology, the most powerful chapter in this guide is inversive geometry. The inversion circle did not have to wait for Brunn, or Desargues, or even Pappus to be discovered. It was in the hunting thaumatropes discovered in Magdalinian caves by the French archaeologist Marc Azéma. It is the logic of the witches who understand the chirality of the inversion circle when they circle the victim in a clockwise direction to effect a blessing, counter-clockwise to effect a curse. With the example of the destruction of Jericho, and the independent geometry of Pavel Florensky,** this evidence cannot be insignificant. Without committing the grave crime of art interpretation, psychoanalysis, armed with topology and inversive geometry, becomes the clavis universalis.
the role of orthography
Orthography means the independence of two forces or variable; but it also specifies the means of their convergence, where a “truth” is revealed. It would be hard to overstate the importance of orthography, the product of the inversion circle. In Lacan’s annotation of the fundamental polygon of the torus, a centrifugal force begins at the upper right-hand corner (“Repetition”). In Slavoj Žižek’s essay on “Why Does a Letter Always Arrive at Its Destination,” we are told the joke about the little girl who wonders why, given that her mother was born in Bristol, her father in Manchester, and she in London, how they all managed to get together,” the conflict is between a Stoic and an Epicurean reading of time. The standard model of free choice is the outward-branching tree, with every decision selecting one of a set of alternatives and moving on to the next set of choices. In this model, the idea of the “choice not taken” haunts and perplexes the subject who wonders if he/she has made the right decision. The Stoics however assert the opposite. No matter what choice is made, actions will converge on a determined conclusion, assumed to be fate/fortune.
To reduce this contrast to a question of belief would be absurd. Rather, I would cite a case of “Effective Cause,” a fifth form to be added to Aristotle’s standard quartet that, Žižek suggests, uses inversions (of effect and cause, virtual and real) to show how things actually happen. As an example we need look no further than ancient device of the confidence trick. A modern version of this is vividly presented in Alfred Hitchcock’s 1958 thriller, Vertigo. A wealthy San Francisco ship magnet, Gavin Elster, enlists the help of his lover, Judy Barton, to bamboozle a retired police detective suffering from acrophobia (Scottie) to be the ideal witness at the inquest into his wife’s “suicide.” The couple plan to throw Elster’s wife Madeleine off a bell-tower of a convent church near the city, but Judy will play the part of Madeleine and convince Scottie that she is suicidal. To compound this belief, Judy must make Scottie fall in love with her to intensify his sense of guilt. Thanks to his vertigo, he will be unable to follow Judy/Madeleine up the tower stairs and forced to watch, helplessly, as the identically dressed actual Mrs. Elster falls to her death.
The con (“con-fidence trick”) depends on the effectiveness of two opposed and conflicting models of time. Scottie, the “mark” of the con, must regard his choices as freely made, the result of his desire to, first, follow “Madeleine,” who Elster convinces him is haunted by Carlotta, her great-grandmother who committed suicide 100 years prior (1857), to keep her from self-harm. His rationally made free choices, however, are guided to converge on the convent tower, where Elster and Judy know in advance he will be unable to reach the top of the stairs, only to watch helplessly as “Madeleine” falls to her death.
This convergence is both INSTRUMENTAL and EFFECTIVE. It assures the conspirators that Scottie will, of his own accord, provide the testimony that will free them from any suspicion. It’s only when Scottie, after a nervous breakdown, accidentally runs across the now-abandoned actress Judy in downtown San Francisco, that he begins the second phase of discovering the con. Initially, he thinks Judy’s resemblance to Madeliene to be a coincidence. Still haunted by Madeleine, he attempts to re-make Judy in her image, unaware of the irony that Judy actually was the Madeleine he lost. In the final scenes of the movie, his discovery of this truth is a blinding revelation that, without the instrumental effectiveness of the “non-orientation” of these models of time, would not have had the slightest effect on the audience.
The ancient confidence trick, updated by the modern 1958 Vertigo example, is as fresh as it was since the beginning of human trickery. It is not necessary to prove to any would-be skeptic, that the inversion circle has the ability to convert lines into cycles that have, as Girard Desargues put it, the ability to curve the horizon back onto itself at a point at infinity. The proof is in Shakespeare’s lines,
If this be error and upon me proved,
I never writ, nor no man ever loved.
*Inversive and projective geometry was suggested by another Persian colleague, Alireza Moharar, as early as 2002.
**This work has been explored by Anya Yermakova, “Mathematical Foundation in Pavel Florensky’s Philosophical Worldview,” MA Thesis, Oxford University, 2011, extended in her PhD dissertation, “An Embodied History of Math and Logic in Russian-speaking Eurasia,” Harvard University, 2022.