The list of paradigm-exemplars on the project’s main page provided a core that quickly expands to a broad range of cases. Each case deserves comprehensive study, appended either to this list or the POSITION PAPERS page. To examples drawn from art, architecture, film, popular culture, etc., we encourage clinical case studies where, as in the case of hygiene, boundaries, insulation, and inversions play a key role.
If it is true that inversive geometry is the logic behind Lacan’s extimité, it could be argued that, as Jacques-Alain Miller claims, it’s co-extensive with the full range of his thinking, if only because it is involved with jouissance, the demand-desire relation of the subject with the Other, and the structure of the signifier. Miller reviews Lacan’s inside-outness without once mentioning topology, but Lacan himself, especially in the middle seminars, uses many examples, from the Möbius band and cross-cap to the torus with its interior-8’s and Villarceau cuts, in combination with citations of Richard Feynman’s principle of the conservation of energy, to insure that his readers comprehend the self-sufficiency of the subject’s topology. This is not an idealism any more than it is a case of “physics envy.” Topology is not an analogy of how subjectivity works, it is an argument that subjectivity is fundamentally topological, whatever that might mean.
Literary and Filmic Topologies
Finding inversion circles in fictional writing requires a code-book with a special chapter on coincidentia oppositorum. Devices such as chiasmus, plot points, a-temporality, point-of-view, and unreliable narrators aim to make space, in the fictional imagination, for a pre-Boolean sense of determination. Without convergence, the work cannot close with any satisfaction; there must be a sense that, as in the rule of detective fiction, that the reader has been given all that is required to reach the resolution of the fictional puzzle without the author’s assistance. Even in science-fiction and fantasy, where magical agency is allowed, there must be some sense that, within any set-up, the rules of the game must accommodate both chance and necessity.
Jorge Luis Borges, “The Aleph.” Wikipedia: In Borges’ story, the Aleph is a point in space that contains all other points. Anyone who gazes into it can see everything in the universe from every angle simultaneously, without distortion, overlapping, or confusion. The story traces the theme of infinity found in several of Borges’ other works, such as “The Book of Sand”. Borges has stated that the inspiration for this story came from H.G. Wells’s short story “The Door in the Wall”.
René Daumal, Mount Analogue. René Daumal’s Mount Analogue is a surrealist novel that follows an expedition to an invisible, symbolic mountain representing the quest for spiritual enlightenment. The mountain, only reachable through a fusion of science and faith, serves as an allegory for the pursuit of the absolute truth. The unfinished novel involves an “ultimate” inversion circle between solid matter and air, using voids in an inversive way to represent the soul.
David Lynch’s Mulholland Drive. This surreal, nonlinear film that blends mystery, thriller, and dreamlike sequences to construct a variety of inversive situations, some of them literally circular. It follows an amnesiac woman and an aspiring actress unraveling Hollywood’s dark underbelly. Lynch employs an extreme version of a classic device: two women play one role and one woman plays two roles. The inversion circle plays a key role in structuring scenes where there is an overlap between these multiple actors with multiple roles, cf. the point on Mulholland Drive where, first, “Rita” is almost assassinated and, later in the film, the limousine drops off Diane/Betty to walk up a pathway to Adam Kersher’s house, which we had not realized existed in the first encounter.
Dead of Night, (1945) is a British horror anthology film directed by Alberto Cavalcanti, Charles Crichton, Basil Dearden, and Robert Hamer. The story unfolds within a narrative framework where an architect arrives at a country house and shares his unsettling dream of events yet to come. The guests each recount their own eerie tales, which include ghostly encounters, a haunted mirror, and a ventriloquist tormented by his dummy. The film is notable for its psychological tension and cyclical structure, with the architect’s dream ultimately merging into reality, leaving the viewer in a loop of suspense and dread. Not only is the architect’s dream self-intersecting, the stories inside his cycle play out various aspects of the inversion theme. This is the place to theorize inversion relations to the classic themes of fantasy: the double, travel through time, story in the story, and contamination of reality by the dream/fiction.
Portrait of Jennie (1948). Wikipedia: An impoverished Depression-Era artist meets a perky young girl in Central Park, dressed in old-fashioned clothes. The two strike up a friendship, but as months pass, Jennie magically transforms into a young woman. Her accelerated development aims to enlist the artist in a plan to intervene in a tragic event in the past. The film uses the park’s inversion status as an oculus through which Jennie, traveling through time, appears out of a different time and space.
Hitchock’s Vertigo and the thaumatropic double. This masterpiece, like Mulholland Drive, involves the double-double trick. One actor plays two roles, “Madeleine” and “Judy,” and one role is “played” by two persons, the fake Madeleine and the real wife who is murdered. In one study, the double double is compared to the prehistoric hunting charm, the thaumatrope, a disk spun so that the two images of the same animal are shown as simultaneously dead and alive. This instrumental convergence traps Scottie, the retired detective afraid of heights, so that he will provide the necessary testimony at the inquest following Madeleine’s “suicide.” Caught in a scam, he does not realize the truth until the actress Judy forgetfully wears a necklace that had been one of her props. Again, a lapidary ending! One version of this thesis will published in Cinematheme.
Vladimir Nabokov, Pale Fire. A visiting faculty from the eastern European (imaginary) kingdom of Zembla steals John Shade’s poem and flees, in the firm belief that he is the only one who understand’s Shade’s true intentions — to tell the life story of the deposed King of Zembla who, disguised as a Slavic Languages professor, has eluded his assassins by escaping to the U. S. As Shade’s university colleague and neighbor, the two have discussed the poem in detail, but Kinbote (a botkin is a fly that lays its eggs in a corpse within minutes of death) knows for certain that it is a cleverly constructed code. This is presented to a reader with the maximum expectation of skeptical rejection, but by the end of the novel there is more than a shadow of a doubt that this “tale told by an idiot, full of sound and fury” is in fact the correct account. Read this early effort to find instrumental convergence in Pale Fire.
Charlie Chaplin, The Circus, turntable scene. Chaplin was known to plan his routines in great detail, perfecting the timing, camera angles, editing, facial reactions, etc. to orchestrate a precise comic result. Evidence of this is the famous turntable scene in the 1928 film, The Circus. Chased by a policeman, Chaplin runs into the performance tent where clowns are running around a turntable. Charlie distances himself on the circular device so much that he is running directly behind the policeman, who seems to be unable to detach himself from the linear logic of the chase. The further away Chaplin gets, the closer he is to the policeman’s back. Interestingly, he checks his pocket watch, like any good Janusian observer.
Architectural Topologies
At the material level of built form, architecture’s topological aspirations would seem to be overwhelmed by literality. However, when use is taken into account, simple constructions such as doors and windows, roofs, attics, and cellars begin to divulge their mysteries.
Vico: Clearings in the Forest. The 18c. Neapolitan philosopher of culture Giambattista Vico can well claim the distinction of being the first to base an entire theory of human settlement on the (correct) idea of the inversion circle. According to Vico, generalizing from reports of explorers and missionaries, formed the idea that the first swidden clearings in forests — the first fire-created fields agricultural fields in tropical ecotomes — were geometrically determinative. Not only were the clearings the sites of incipient fixed-field farming practices, they were simultaneously ritual sites protected by altars, the sites of astronomical observations, and ceremonial sites of marriages and burials. In his famous work, The New Science (1725/1744), Vico described how clearings were, as the basis of Cyclopean culture, a circularity carried into more developed urban forms: the feature of the pomœrium, a toroid boundary maintained by ritual as a spiritual defense supplementing military enclosure. In the story of the foundation of Rome, Romulus’s symbolic plowing of the circumferential trench mimicked the sacrificial function of the center. This was preserved in the murder of Remus, the twin, for allegedly violating the furrow.
The Memory Theater of Giulio Camillo. Working in the court of Francis I until 1537, the polymath philosopher developed a memory system in the form of a theater, described in his book L’Idea del theatro. Lou Beery Wenneker has provided an able translation and explanation of Camilla’s idea of a microcosm whose proportioned divisions duplicated a universal structure based on astronomy and astrology. Most reconstruction assume that the theater was semi-circular, but it is possible that the rows revolved on wheeled tracks to enact different combinations, and that the visible half of the wheels was complemented by a dark half, to simulate the diurnal cycle and adjust for seasonal differences.
The Zairja (زايرجة) is a 12c. (at least) astrological mechanical device for generating ideas. One could claim that it is the first AI, since by reversing the calculative function and multiplying the number of causes for any one effect, the Zairja sampled relational meanings to achieve a balance point between (in Lacanian terms) the Symbolic and the Real. Like Camillo’s memory theater, the user of the Zairja became, in effect, its critical extension, just in in 2001: A Space Odyssey, the on-board computer HAL used the astronauts to execute a Second Program that its human programmers were not able to conceive. The Zairja uses instrumental convergence to optimize random associations into a “palintropic harmoniē.” The inversion circle simultaneously expands and contracts around an initial “instance,” as when the Chinese version, the I Ching, casts yarrow-sticks while a question is posed. This chance leads to an ambiguous necessity typical of all oracular pronouncements. Ambiguity IS the determinative result. The orthography of the Real and Symbolic, set within the ambit of the Imaginary, produces a prismatic time-machine that lives up to Freud’s specifications for transience.
The Thesean Labyrinth. Some Lacanian theorists have not understood the wit of the Thesean Labyrinth. Noting that it has only one passageway, they marvel that it could be used as puzzle for any entrant, let alone a prison for a menacing monster, the Minotaur. The idea of the entrapment that magicked Daphne or the paralysis of the Cyclops’ cave has escaped them. The key is the fractal (self-intersecting) pattern of the folds that deepen the main scheme of ABA to AabaBabaAaba. The pattern creates a sinusoidal doubt that springs forth with any pause, which is but cured by a thread from a sympathetic maiden. In/out, trap/escape, is the binary that, like the sine wave itself, is generated by the rotation of a vector that, to complete its circuit, involves four instructions: the i that, for π/2 is √-1, for π is –1. The circuit ends with an ambiguous ±1, the pure imaginary. The proper geometrical model of the labyrinth is a space of rotation, whose vector of rotation defines a 2π radius sphere, within which a circuit is complete at 4π, when the vector re-enters the space at the antipode.
Touro Tower of John Dee. “Clues in the historical record and a close study of the design of the Newport Tower in Touro Park, Newport, R.I. have [Jim Egan] to conclude it was built to be the city-center of the first Elizabethan colony in the New World in 1583. The colonizing effort ultimately failed, but the solidly-built structure remained. The architect of the Tower, the polymath John Dee, even provided an ‘Owner’s Manual’ containing cryptic clues about how the Tower functioned, as well as a ‘hidden blueprint’ of its dimensions.” Critical to the design were features that combined chronometry, astrolabe, camera obscura, and observatory in addition to the main function as a geographical monument. The tempietto design forces a retroactive revision of the tempietto idea and recovery of meanings that, as in the example of the Temple of Winds in the Roman forum at Athens, have been up to now enigmatic.
Sadra Tehrani: The Extimacy of Central Park. When Olmsted and Vaux conceived central park as a means of humanizing New York’s urban gridiron, they made the dramatic move, to a full inversion, putting an imagined, complete peripheral countryside into the center of the city. Did the rules of inversion persist beyond this initial gesture? Tehrani showed that Central Park, used as a location for more film scenes than any other place in the world, seemed to come with its own internal rules of visualization, modifying and in some cases completely converting standard scene-blocking techniques to take advantage of the “outside as inside” condition. Inversive geometry textbooks could be written on the evidence of such filmic adaptations. In this essay, Tehrani focuses on three films that make a case that extimité is “built in” to Central Park from the beginning.
Kingo Gondo’s House in Kurosawa’s High and Low. Akira Kurosawa’s High and Low (1963) is a tense crime drama that explores themes of morality, social disparity, and human nature. The film follows Kingo Gondo, a wealthy executive, whose son is mistakenly kidnapped along with the chauffeur’s child. Gondo faces a moral dilemma when the ransom demand threatens his financial future. As the police work to track down the kidnapper, the film shifts from Gondo’s luxurious home (the “high”) to the grimy streets of Yokohama (the “low”). The narrative delves into the contrasts between wealth and poverty, and the choices that define character and integrity. Because the kidnapper surveils Gondo’s house from below, the edge of the balcony cuts a shadow line beneath which the police, forbidden to be contacted, may circulate. Circle-ate.
The Department Store’s Lapidary Imagination. The rise of the department store in the U.S. and Europe at the end of the 19c. and early 20c. would seem to be a direct expression of the cultural shift to consumerist capitalism, but the inner voids that were an architectural signature suggest the inversion circle logic of the crystal: a faceted space to establish a 1:1 relation of viewer to viewed, canceling distance and excluding peripheral noise. This idea was articulated by Raymond Roussel’s procédé: an impossibly long line to a distant object (“cathesis”) that “by chance” reveals the object’s infinity of detail. In the department store, the procédé is a function of an interior crystal structure of the shopper’s experience, reversing the metaphoric process so that the parapraxis of movement through the store alights on an encounter with an object of agalma, which will always be under-priced.
Jeremy Bentham’s Panopticon. There can be no purer architectural expression of the relation of orientation to location than this 19c. proposal for the “perfect prison.” Bentham credited his brother for the idea but was quick to theorize its ingenuity. A ring of prison cells around a tower blinded so that it would be impossible for prisoners to know whether or not guards were actually present converted the structure into a “doubt machine.” Pascal’s wager come to mind, in that the safe money seems to be on the side of caution, given the extreme repercussions of the opposite choice. While not quite the eternity in hell of Pascal’s wager, Bentham thought that the anonymity of authority in the Panopticon would be sufficient for what Gaston Bachelard would later call la surveillance intellectual de soi. How are orientation and location related to this concept? Paralysis is clearly the idea of imprisonment. This restriction is purified by the idea that one paralyzes one’s self, out of prudence. The spatial non-orientation of the blinded tower surrounded by a periphery of cells is less about the need for the guards to see into each cell equally and more about the prisoners’ spatial equalization. There is no prisoner who is less or more subject to the hypothetical gaze of the guards, who perfectly represent the “interior Janusian observer.” The Panopticon reveals an important truth about the Janusian insider. He is not looking at, but rather being looked at. The turn that is met by equipotential faces is no longer orientable. It is non-orientation, distilled and purified. The guard(s) are defective in this model. Although standard interpretations say that it is the prisoners’ view that is the effective cause of discipline, the reverse is the case. The guards are blind. This seemingly irrational reversal is corroborated by Vladimir Nabokov’s novel, Invitation to a Beheading, where the prisoner is led to condemn himself, in the style of Kafka’s The Trial. Consider this consultation with ChatGPT on the relation of cancelled position (freedom of movement) to cancelled orientation (the internal “Janusian” observer). With cancellation (negation) as the critical operator, the role of negation in the Freudian unconscious must be the next consideration. Given that there can hardly be a better architectural depiction of the inversion circle, Bentham’s panopticon should be elevated to the role of a paradigm exemplar.
Persian Inversion Circles.
The principles of inverse transformation were derived from astrology and astronomy, so it is no wonder that cultures who invested heavily in observatories and accurate records developed traditions around the inversion circle, sometimes as a small object containing unlimited infinities or holes, voids, or defective containers that constituted miraculous passageways.
- The Cup of Jamshid Wikipedia: The cup (“Jām”) was said to be filled with an elixir of immortality and was used in scrying. As mentioned by Ali-Akbar Dehkhoda, it was believed that all seven heavens of the universe could be observed by looking into it (از هفت فلک در او مشاهده و معاینه کردی).
- The Emerald Tablet Wikipedia: “‘Tis true without lying, certain and most true. That which is below is like that which is above and that which is above is like that which is below to do the miracle of one only thing. And as all things have been and arose from one by the mediation of one: so all things have their birth from this one thing by adaptation.”
Seal of Solomon, from a 17c. grimoire - The Seal of Solomon Wikipedia: It is often depicted in the shape of either a pentagram or a hexagram. In mystic Islamic and Jewish lore, the ring is variously described as having given Solomon the power to command the supernatural, including shedim and jinn, and also the ability to speak with animals. Due to the proverbial wisdom of Solomon, it came to be seen as an amulet or talisman, or a symbol or character in medieval magic and Renaissance magic, occultism, and alchemy.
The House of Mourning [to be written by Iraj Ghoochani]
The Holy Grail [to be written by Claudio Sgarbi]
More Art Topologies
Finding inversion circles in fictional writing requires a code-book with a special chapter on coincidentia oppositorum. Devices such as chiasmus, plot points, a-temporality, point-of-view, a
The Magic Square of Albrecht Dürer’s Melencolia§1 This silent video reviews the methodology behind Albrecht Dürer’s magic square, a 4×4 “Square of Jove,” possibly intended to balance the bad effects of Saturn’s magic 3×3 square. Magic square are ancient. Chinese and Indian examples have been recovered. Thanks to Vedic mathematics, however, Dürer’s square seems to have been strategically manipulated to effect a unique result. This video presumes the viewer already knows the basics about magic squares, i. e. that the rows, columns, and diagonals all add up to the same number. For Dürer’s square, this sum is 34. A 4×4 magic square is constructed from rows of consecutive numbers, 1 through 16, broken into two halves, 1–8 and 9–16. Each is ordered as a boustrophedon, right to left, then left to right for the first half, left to right then right to left for the second. The middle columns (15, 10, 6, 3; 14, 11, 7, 2) are then reversed, but Dürer performs another flip with the two middle rows, so that the counts 5–8 and 9–12 create a zig-zag. This may be, as the video suggests, inspired by Dürer’s famous monogram, a D inscribed beneath a large A.
Running Backwards. During the American Civil War (1861–1865) a common battlefield phenomenon was the sight of soldiers in retreat running backwards. This awkward, even comic phenomenon was explained by the fear, uppermost in soldiers’ minds, of being found on the battlefield shot through the back, suggesting cowardice. Even though running backwards greatly increased the chance of being shot and killed, this was the price of honor: a literal turn to “make the film run backwards” before there was knowledge of cinema. The cinematic correlate is suggestive, nonetheless. Charlie Chaplin could act and speak backwards and was reputed to have filmed scenes doing this. Reversing them, he would appear to be the only one moving and speaking normally, while everything else around him was running in reverse. Michael J. Anderson, one of David Lynch’s favorite character actors, played a mysterious dwarf who would speak backwards, have the recording reversed, and then synch his lips to match the “corrected” version. It is tempting to add something about Lacan’s insistence that the “letter always arrives at its destination, but in reverse.”
Examples from Lacan
Lacan’s essay “La troisième” shows a literal case of circular inversion. The classic three rings of the Borromeo knot are reconfigured as a circle (the Imaginary) crossed by two lines to infinity, representing the Real and the Symbolic. However, there are other cases where the idea of inversive geometry is used without any literal depiction of circles or transformation. Contributions (citations plus brief explanations) are encouraged.
The Fundamental Polygon of the Torus. Although we can draw a variety of circles on the surface of the torus 2-d itself, the most overlooked example comes with the diagram known as the Fundamental Polygon, the standard graphic way of defining topological forms. With four vectors, usually colored red and blue, the torus’s fundamental polygon looks initially like a normal instructional folding diagram, asking us to first join the blue vectors to make a tube, then join the end vectors to create a circular tube. But, this would be a. 3-d torus, the familiar donut, bagel, or bicycle tire. The fundamental polygon of a toroid Riemann Surface pairs the blue and red vectors twice, the enact a divergence and convergence. Lacan seems to be aware of this difference although many of his followers are not. He labels his fundamental polygon from the upper right corner (“repetition,” related to a reversal of Descartes’ je pense, donc je suis). The vectors then diverge to a diagonal relation of positions outside the Symbolic (the psychotic) and inside (neurotic “acting-out”), then converge again at the lower left corner, labelled “sublation.” How do we think of these paired vectors’ diverging and converging? One suggestion is to follow Slavoj Žižek’s story of the little girl who wonders how, seeing that her mother was born in Bristol, her father in Manchester, and herself in London, how they all managed to get together. The fundamental polygon of the torus shows how the logic of convergence is simultaneous with divergence, how in the midst of our exercise of free will, we establish a deterministic fate. The line we might imagine between the two forces is, the torus tells us, an inversion circle. The fundamental polygon facilitates a convergence (of fate) that is simultaneous with divergence (of choice). Ethnologically, this is an essential feature of of the con (“confidence trick”), where a mark (the victim) believes him/herself to be acting voluntarily but, thanks to the structure of the scam, every choice actually brings the mark one step closer to the con’s pre-determined conclusion. A paradigm case is the feature of Hitchcock’s 1958 film, Vertigo.
The Three Prisoners’ Dilemma. Lacan first heard this puzzle over a dinner table in 1937, but the story stuck with him and became the basis for his essay “Logical Time and the Assertion of Anticipated Certainty.” Derek Hook has made the critical point, that the inflexion of the puzzle occurs between an intersubjective and transubjective mapping of the space where three prisoners with colored dots pinned on their backs, realize that the responses of the other two will create a symmetrical difference that will split the “moments” of their promised release. The warden has, however, set up the contest as having only one winner whereas it is necessary that all three prisoners will discover the solution simultaneously. What exactly is the role of this “foregone conclusion”?
Without the simultaneous action of the prisoners (they all rush for the door at the same time, seeing that they all had failed to move, also at the same time) the puzzle would not be solved. Yet, without the structure of the contest, where only one would be victorious, the prisoners would not have been individually motivated. The three prisoners form a circle that marks their imprisonment; the offer of freedom constitutes an identification of the “outside” of this circle with the discovery of the “interior” logic. The periphery corresponds to the prisoners’ silhouette, where the two dots of the others are visible to each. When Lacan associated the Brunnian variation of the Borromeo RSI as an inversion, he saw this as a means of connecting the R-line to metaphor and S-line to metonymy. The vertical-Real is the relation of semblance to truth and jouissance to surplus jouissance in his revised version of the discourse matheme template. If the prisoners are, as it seems they must be, hysterics, their semblance is literal: the barred S represents the critical difference between each prisoner’s face and invisible (to him) back. The dot is that part of the subject of which he is unaware but is visible to others — namely the unconscious, in a metaphoric relation of semblance to truth. As time metonymically shifts from the first pause to the second, the Other of jouissance is realized to be barred, S(Ⱥ). Its surplus is the the knowledge (S2) that is felt as a surplus, lying beyond the “rules of the game.” The two pauses, designated as jouissance and surplus jouissance, precede the Truth, the objet a as a gateway, an escape, the gap in the circle that is simultaneously the violation of the rules and and the new game that emerges through the playing.
In chess, it is said that there are two sets of rules. The first is the set of rules determining the moves of each piece, the order and timing of the players’ actions. The second is the set of rules that are created as the game evolves, rules that are discovered silently and shared without explicit agreement but nonetheless followed equally.
