Jacques Lacan, Seminar XIII, The Object of Psychoanalysis:

“It is the same thing. However large the sphere may be I can enlarge this hole infinitely so that it is going to be reduced to a simple point at the opposite pole. This means that on a surface determined by this edge that we call the edge of a disc, that this surface is in reality a sphere, all these holes that we may make are infinitely reducible to a point and, what is more, they are all concentric, I mean that, even this one that I am making outside the first cut, in appearance, can, by a regular translation, be brought to the position of this one here.”

This web page aims to provide a centralized consideration/exposition of the role of the inversion circle in art, architecture, and psychoanalysis. Below, find examples, discussions, and critical source material to aid the study of the inversion circle’s active presence in art, history, logic, and psychoanalysis. CLICK HERE TO LEARN WHAT A NEW LACANIAN TOPOLOGY MIGHT LOOK LIKE.

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WIKIPEDIA: Inversive geometry is the study of inversion, a transformation of the Euclidean plane that maps circles or lines to other circles or lines and that preserves the angles between crossing curves. Many difficult problems in geometry become much more tractable when an inversion is applied. Inversion seems to have been discovered by a number of people contemporaneously.

For a good review of the rules of inversive geometry, visit J. Wilson’s “Excursion through Inversion” page authored by Lexi Stear, Mimi Tsui, and Kendyl Wade. This visually adept summation will get you through the basics — the idea that the space inside a circle can be equivalent to the space outside the same circle.

Jacques Lacan employs the idea of the inversion circle directly, in playing-out the diverse functions of the Borromeo knot system of Real, Symbolic, and Imaginary. In particular, Lacan presents indirect evidence¹ that he knew how the inversion circle worked and why it was critical to, in this case, play up the role of the Imaginary as mediator of the Real and Symbolic domains. We owe the discovery of this connection to Iraj Ghoochani.

The inversion circle is a portable “device” with myriad physical manifestations, all which use the idea of extimité to topologize the spaces and times we live in. We can understand its properties cumulatively, by studying the circumstances of these different applications to build a polythetic² picture of its functionalities and rules of order. The inversion circle is not simply a trick used in ethnological and artistic ways, it is the means by which these ethnologies and artworks can claim to be timelessly effective. The inversion circle is the fifth causality and ground of Aristotle’s classic four.³

working through examples

In the following visuals, the inversion circle is playing a critical role. Can you (1) spot an inversion circle(s)? Can you (2) discover the critical role of relating an inside to an outside and vice versa? Then, can you (3) speculate on how the artist(s) in question seemed to have learned about inversive geometry to use it so effectively?

The inversion circle project uses examples to allow psychoanalysis to extend into ethnology, art, architecture, etc. as a way of comprehending Lacan’s most provocative original concepts: extimité, the torus, symmetrical difference, metaphor, repetition, inversion, chirality, the rim, etc. The project is tentatively built around three lists: models, examples, and ideas.

Table of Contents

I. Models (this page). Examples are paradigm-exemplars that expand to diverse other examples; most have literal representations of inversion circles or their functionalities.

A. Resources. YouTube videos give critical background tutorials on (1) projective geometry, (2) inversive geometry, (3) related mathematical issues, such as Fourier theorems, Desargues’ theorems, and topological transformations.

B. Position papers. Ranging from short one-page arguments to developed theses, these extend and consolidate the central position of the inversion circle in Lacanian theory.

II. More examples. Examples include films, music, architecture, literature, folklore, ritual practices, and jokes.

III. Ideas (annex). Lacan’s extimité was not hermetically separate from other topological concepts. At the center was the torus, and the torus’s affiliation with George Spencer-Brown’s non-numerical calculus axioms: “a cross and cross again are equivalent to no cross” (consecution)  and “a call and call again” are equivalent to one call (idempotency). Recourse to Spencer-Brown’s notation system allows for critical comparisons of examples from diverse media.

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1. The Injunction of Popilius

Lacan cites the Injunction of Popilius in Seminars XII (Crucial Problems …) and XIII (The Objet …), and “L’étourdit.”  This ancient example estabishes that inversion circles were known and applied, topologically and correctly, since antiquity. Both Popilius, consul of Rome, and Antiochus, the Hellenistic King of the Seleucids understood precisely the meaning and significance of the circle Popilius drew in the sand, which deflected the attack on Alexandria. This proof of “what did they know and when did they know it” takes the inversion circle out of the hands of mathematicians and puts it on the agenda of ethnographers. Lacan knew the meaning of this transfer and assumed inversive geometry to be native to the “subject of science” the was the subject of his psychoanalysis. In this sense, inversive geometry is this science of the subject.

Polybius, The Histories, Fragments of Book XXIX, published in Vol. VI of the Loeb Classical Library edition, 1922-1927:

“At the time when Antiochus approached Ptolemy and meant to occupy Pelusium, Caius Popilius Laenas, the Roman commander, on Antiochus greeting him from a distance and then holding out his hand, handed to the king, as he had it by him, the copy of the senatus-consultum, and told him to read it first, not thinking it proper, as it seems to me, to make the conventional sign of friendship before he knew if the intentions of him who was greeting him were friendly or hostile. But when the king, after reading it, said he would like to communicate with his friends about this intelligence, Popilius acted in a manner which was thought to be offensive and exceedingly arrogant. He was carrying a stick cut from a vine, and with this he drew a circle round Antiochus and told him he must remain inside this circle until he gave his decision about the contents of the letter. The king was astonished at this authoritative proceeding, but, after a few moments’ hesitation, said he would do all that the Romans demanded. Upon this Popilius and his suite all grasped him by the hand and greeted him warmly. The letter ordered him to put an end at once to the war with Ptolemy. So, as a fixed number of days were allowed to him, he led his army back to Syria, deeply hurt and complaining indeed, but yielding to circumstances for the present. Popilius after arranging matters in Alexandria and exhorting the two kings there to act in common, ordering them also to send Polyaratus to Rome, sailed for Cyprus, wishing to lose no time in expelling the Syrian troops that were in the island. When they arrived, finding that Ptolemy’s generals had been defeated and that the affairs of Cyprus were generally in a topsy-turvy state, they soon made the Syrian army retire from the country, and waited until the troops took ship for Syria. In this way the Romans saved the kingdom of Ptolemy, which had almost been crushed out of existence: Fortune having so directed the matter of Perseus and Macedonia that when the position of Alexandria and the whole of Egypt was almost desperate, all was again set right simply owing to the fact that the fate of Perseus had been decided. For had this not been so, and had not Antiochus been certain of it, he would never, I think, have obeyed the Roman behests.”

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2. The Day the Earth Stood Still (1951)

When a spaceship lands on the Capitol Mall in Washington DC, an implicit structure forms around the landing strip. Surrounded by military tanks and armed soldiers, the alien

visitor, Klaatu, and his robot Gort nonetheless invert the spatial order to condense and re-radiate the authority and advanced technical powers of the galactic federation they represent. Effectively, the boundary of the known universe translates to the local margin on the Capitol Mall, with the result of amplification of the larger circumference as it is concentrated in the point source of Gort’s defensive ray. Key to the inversion logic of this transformed site perimeter, the ray destroys only those objects that are potentially harmful. Guns disappear from soldiers’ hands. Tanks melt away around their crews. The evil in the object attracts the force that neutralizes it. This science fiction demonstrates all of the properties of the inversion circle (the bond between periphery and center, the conversion of lines and circles in certain conditions of tangency, the extimité of inside and outside, and the rule of ⚪︎⚪︎ = ⚪︎⃝). The amazing thing, however, is the way it performs all of these topological tricks within a smooth narrative flow, with a cinematic establishing scene, premise, obstacle plot, two functional plot points, and a romantic union of the Œdipal couple. In other words, no one needs to breathe a word about topology to enjoy this fantasy at the level of consumer cinema. No one needs to think about the fact that Gort’s precision ray is a “one dimensional subspace within the Real projective plane.” Klaatu’s escape from locked rooms does not have to be topologized. Inversive geometry is what makes the film work at the level of fantasy. Like the Injunction of Popilius, we can enjoy the story without doing the maths. This, if anything, demonstrates that topology is a “native formation” of the Imaginary, something that has been present ever since humans turned from bi-univocal concordance to the signifier of the Other.

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3. Diego Velázquez, Las Meninas

This “self-portrait,” painted in 1656 in one of the rooms of the Alcázar Palace in Madrid, is perhaps the most controversial in the history of art, thanks to the small object at the rear of the studio, which appears to be a mirror reflecting the royal couple, Felipe IV and his wife Mariana. How, given the distance of the mirror from the point where the couple must have stood, when the painter captured them as if taking a snapshot with a camera, could the king and queen’s images be the same size as if they were standing, not in the frame of the doorway but in front of a mirror, which had frozen their image and then been moved to the end of the room? Critics, art historians, and philosophers as famous as Joel Snyder, Leo Steinberg, and Michel Foucault have argued about geometry, architecture, and optics for over forty years, but a

Spanish engineer proved that it would have been possible for images painted on the canvas shown with its back to the viewer to have “reached” the mirror intact, at the unexpected scale. The result is astonishment, for the mirror image underscores the other puzzle that troubles some of the viewers of this work as they stand before it in the Prado, namely how Velázquez himself had stood in the very spot in order to paint Las Meninas. The accomplishment of the inversion circle is to allow content from one side to be projected correctly to the other side, while observing strict rules with relation to touching the periphery or center. Velázquez seems to have taken the frame of the mirror, split it into two halves, separated them the distance of the depth of the room, and projected the alternative spaces that must occupy the immediate space in front of a canvas: an artist, a subject, and (later) a spectator, in the varied forms of (1) a precise projection, (2) a dream, and (3) a logic puzzle. Hint: to understand the role of inversion circles in Las Meninas, consider how the frame of the painting and the frame of the mirror are constructed by vectors circling in opposite directions to each other, but that the space of observation is not from one side or the other, but between: the “Janusian observer.”

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4. Picasso: Les Demoiselles d’Avignon

The “text” of Picasso’s famous-scandalous 1907 portrait of five figures reputedly in a brothel has not properly theorized the role of Melancholy, seated in the traditional pose (elbow on knee) beside a bowl of fruit. Two “custodians” open curtains, one in the traditional “Parrhasian” position at the front of the scene, another who parts the sky in the same manner as the angel of the Apocalypse in Giotto’s Arena Chapel west wall mural. As with Velázquez’s Las Meninas, we have to ask about the space between the hole in front and the hole in back. Are they separate (⚪︎⚪︎) as a Euclidean space would arrange something near and far, or are they concentric (◎), as we would discover if we drew them on the surface of a 2-d torus? The stakes are high, since, following the rule of consecutive order (e. g. the signifying chain), ⚪︎⚪︎⚪︎⚪︎⚪︎⚪︎… reduces to ⚪︎. But, ◎, the mark of a space that “re-enters its own form,” presence equals absence. These two rules can be resolved only if we twist space into a Möbius band, which can be accomplished if we cut a 3-d torus “katagraphically,” rotating the knife 180º (1π) as we complete the full circuit (2π), creating two Möbius-shaped surfaces with a void between, a relation that Euler would model as “symmetrical difference” (union without intersection, ◎=  .). This equation is as permanently mysterious as it is historically established: Lawrence Sterne’s 1759 picaresque novel, The Life and Opinions of Tristram Shandy, Gentleman, presented, among other clever devices, the paradox of a blank page.

We might consider how the inversion circle as the line we can draw between nothing and less than nothing. It is the void, critical to architecture and thinking alike, since it allows us to suspend time and space in order to “do what must be done” without the condemnation of contradiction.

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5. Magritte: Memoirs of a Saint

Magritte, ever tuned into the playfulness of the inversion circle, presents us with a stripped down version that is Picasso’s Demoiselles without the figures. Its melancholy lies in its pure “Parrhasian” quality. This is the story of a contest between two painters in ancient Greece. Zeuxis, the presumed favorite, painted a trompe-l’œil mural of a bowl of fruit, so realistic that a bird flying by mistook it for the real thing and dive-bombed into the wall, breaking its neck. Magritte shows the sky as internal to the red curtain, a vague beige horizontal and black vertical plane meet to form a horizon, and an off-stage sun casts a shadow from a ~45º angle, comparable to the shadow cast by the spectator-God in Antonello’s St. Jerome. The partridge does not appear in Magritte’s cylinder, but his sky does.

This detail deserves to be connected to Vladimir Nabokov’s aviary lines:

“I was the shadow of the waxwing slain
By the false azure in the windowpane;
I was the smudge of ashen fluff — and I
Lived on, flew on, in the reflected sky.
And from the inside, too, I’d duplicate
Myself, my lamp, an apple on a plate:
Uncurtaining the night, I’d let dark glass
Hang all the furniture above the grass,
And how delightful when a fall of snow
Covered my glimpse of lawn and reached up so
As to make chair and bed exactly stand
Upon that snow, out in that crystal land!”

Within the Crystal Land, exactitude is accomplished by the curtain that allows the furniture to stand above the grass, in perfect projective position, anamorphically. Magritte’s pithy emblem of this exactitude restates the case of Melancholy in the same way the Albrecht Dürer, by misspelling m-e-l-e-n-c-o-l-i-a allowed the answer to appear as an anagram: limen coela — gate of the heavens. Vico had pointed out that cœlum was both a wedge/cut and heaven, a contronym suggesting a theory of the katagraph, the two curtains required for Parrhasius to win the contest by painting a curtain rather than a projected object beyond the picture plane.

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6. Magritte: Not to Be Reproduced

If position changes without a corresponding “rotation” of orientation (think of the reflection of the subject standing before the mirror as having left the spectral encounter to go around the world in an instant and see himself from the back), we could say that the 2π journey around the world has come disconnected from the orientation, which is 0π or, adding in the mirror’s π (180º flip) and the ’round-the-world 2π journey, a paradoxically one π short of a (spectral) image. An enterprising wag might say that it’s “a slice without the π. It is as if orientation had forgotten to turn but arrived in the same position anyway. The novel placed on the sill beneath the mirror, which escaped the curators of the Magritte show at MoMA (“Magritte: The Mystery of the Ordinary, 1926–1938”), 2013–2014, was placed tangent to the inversion circle and thus, in the mirror, is tangent to the same frame. The novel’s cover is reversed by the mirror, but not the chiastic plot of Poe’s 1838 novel, The Narrative of Arthur Gordon Pym of Nantucket. Chiasmus is a device that changes orientation without position, a paraphrase of Euler’s “union without intersection.” It was used to great effect in the story of the invention of artificial memory by Simonides. The X-shaped (hence chi-astic) design preserves position while changing orientation (in-out, up-down, divergent-convergent, etc.). Like Joni Mitchel sings, “I see life from both sides now,” but this is not the logic of the coin with two sides but, rather, the coin with only one side to begin with. You can have change of orientation without positional change or positional change without orientation change in the logic of the inversion circle. This principle connects many examples, since chiasmus has been a literary technique popular since the days of itinerate bards who added, to their standard Homeric lines, gossip about local scandals. This proves the inversion circle able to adapt to cultural examples where its logic is in force even if the set design is a bit strange. As with the example of Mr. Know-All (below), the temptation to regard the subject as standing before a mirror is replaced by the infinitely more productive thesis, that he is standing before an inversion circle, in the role of a “Janusian observer.”

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7. Breughel: Tower of Babel

The Biblical account of the Tower of Babel borrows from a tradition before it, truncates the “full story,” and leaves things a bit up in the air (hah hah). The clouds shown in many paintings (to the left is Brueghel’s) is a buffer to vision based on the status of the viewer. To see past the blur is equivalent to being given a pass to the temple on top of the labyrinth. The lower part represents the trials-and-errors of life on earth; the upper part, the lapidary perfection of architecture as a temple or tempietto. The clouds, one may easily guess, are an inversion circle to explain somehow that the labyrinth and temple are really two versions of the same thing. Orientation is key. From below, the top appears to be either destroyed or inaccessible. From the top, there is no orientation, since the view is panorama-plus: panoptical. With orientation canceled (because it is perfect in all directions) the top is demonically everywhere. Like the spotlight that roams around the set of Casablanca, an ominous power lurks, making the innocent feel guilty. The inversion circle between the labyrinth and temple have a puzzle-answer structure, as if the temple were held, anamorphically, in the winding folds. The logic of the labyrinth is like the wisdom of lalangue: the “out of the mouths of babes” effect that, in the shrine of the Roman god Vagitanus, the oracle could hear prophecy in the newborn’s cry. (The site of the present-day Vatican was intentionally built over Vagitanus’s temple.) We should say that Babel is also “out of the mounts of babes”: an invocatory force within non-orientation that produces a lapidary pivot point, from which all locations are brought within reach.

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8. Antonello da Messina: St. Jerome in His Study

This painting, created in 1474, presents a full-frontal view of the inversion circle, formed as a proscenium arch to remind us of the fundamental architectural value of the theater as an alternation division between the living and dead, motion and rest, freedom and paralysis. It is a boundary doubled, ⚪︎⃝, so that in the interior of the buffer zone, between the inside and outside frame, there are ciphers to be de-ciphered in order to reveal the password required for solving the puzzle we share with Jerome. His puzzle was that of consolidating the polyglot sources of holy writings into a single Vulgate Bible. This required going beyond the conventional assignment of signifiers to signifieds to retrieve the “Real” of religious script. Penny Howell Jolly provides an excellent guide to the iconography of this painting; but we should go beyond, to a topological understanding. Putting ourselves in the position of Jerome (and Antonelli) repeats the conundrum of Las Meninas. We have proof that we have all been standing in the same spot at different times, and like the little girl who wonders, “Daddy was born in Manchester, Mummy in Bristol, and I in London: strange that the three of us should have met!” Žižek uses this story to explain the inverse logic of why the letter always reaches its destination without involving topology but, instead, the idea of instrumental convergence. Jolly avises us to do the same with the partridge, the bird that, according to Medieval bestiaries, was so sex-minded that it could be impregnated by the wind. This is all we need to see how this limen coela allows Jerome to feel the divine afflatus of the entity who gazes at him from the auditorium set up by this small painting. Because we find the partridge, peacock and golden bowl in paintings of the Annunciation, we know we are on the right track. The confirmation of finding a potted carnation plant at the edge of the shadow cast into Jerome’s study by the inversion circle should be sufficient to expand the topology of this work to include Vico’s vera narratio, the lalangue of the unconscious that speaks the truth because, like the speech of the first humans, it was unable to do otherwise.

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9. Euripides: Alcestis

Veils, curtains, shrouds, and clothes in general play an important role in inversive geometry. The all delay the moment when “the letter reaches its destination, only backwards.” As in Magritte’s Not to be Reproduced, we suspect that a circuit has been completed without the requisite (Euclidean) complete turn of orientation. Our mirror image has forgotten to look at us. Instead of the mirror-as-cut, we have a speculum of inversion. This turns out to be the rather old story of the wife who sacrifices her life for her husband, a husband who not only forgets the favor but denies it (think of St. Peter’s three-fold denial of Christ), but is then forgiven when a gift is wrapped in the guise of a new bride.  Mozart borrowed the story to use in Le nozze di Figaro, with slight alterations to casting. We learn from Le nozze, however, that the 2π. of death can have a 2π return, as long as the resulting 4π simultaneously negates position and orientation — i. e. a full double circuit, such as the one that can cut a bagel into two Möbius bands. In the language of the theater, location is configured as the distance between the honored guest, Hercules, and the funeral going on in a distant part of Admetus’s palace. This separation is radically represented of the sacrifice Alcestis has agreed to make, given that no one else will take her husband’s place in Hades. Scandalized when he discovers the truth, Hercules performs the hero’s best trick, the “Orphic twist,” arguing to the gods of the underworld that Alcestis should not be received as dead but returned as alive. This inversion of orientation allows the hero to have a joke at Admetus’s expense, making the best of pronouns describing a “new wife” he has found for the widower. Re-re-orientation is a return of the Real, as a true après coup. Mozart at this moment adds the lyrics, corriam’ tutti (everyone run!). Late-comers will not be admitted to this miracle, whose 4π circuit has returned it to the origin point in no time at all, just like all après coups.

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10. Freud: Emma’s Fear of Dress Shops

The concept of orthography (the right-angled graphic relation of independence of the x-y axes) is critical in describing the unconscious Real’s relation to the conscious-Symbolic. This is reflected in the vertical operations of Lacan’s metaphor formula (the suppression of a signifier, S’ by M, the complementary “expression” of a signifying chain, S’ by x), in comparison to the horizontal movement of the formula from left to right: first a sudden synchronic replacement that obscures some signifier, S’, and has us thinking in an alternative medium. This is presented to the right of the multiplication sign, the •, which seems to allow us to “cancel out” the two S’s, leaving behind M/x, the metaphor with its diachronic adventures and an ‘x’ that mysteriously magnetizes some elements to allow their collection and compression in s”. The metaphor may be restated as relation of the unary (trait?) to this atomic signified. In Freud’s account of Emma, a young girl who is molested by a shop-keeper before puberty, with only her clothes protecting her from direct touch of her genitals. She does not register this trauma (it becomes an M/S’) until after the •, when she enters a dress-shop where two male clerks are in the middle of a joke, and Emma assumes that their laughter is be about her out-of-fashion dress. Although she finds one of the clerks attractive, she exits the shop in a panic and finds it impossible to go into a dress shop alone after that: M(1/s”). The role of the shop as a container and Emma’s paralysis in relation to it, on account of clothes, explains the dynamics of unconscious suppression of trauma, and the return of the Real in inverted form. Not only can the original story be seen in light of the inversion circle, the two shops are essentially the same shop (condensation), from the perspective of a “Janusian observer.” Orientation neutralized, position merged (paralysis), unconscious topologically explained: ⚪︎ ⚪︎ → ⚪︎⃝.

Sigmund Freud, (1950) Project for a Scientific Psychology  [1895]. The Standard Edition of the Complete Psychological Works of Sigmund Freud 1:28–391.

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11. Castor and Pollux; The Wizard of Oz (1939); Restorative Coma (paralysis)

The story of two twins, one immortal (Pollux) the other mortal (Castor), is the same as Dorothy in The Wizard of Oz (1939), although Dorothy must double, from a Kansas farm girl to a potential witch of a magical kingdom. The plot parallels include:

(1) Death of the mortal twin: Castor in a skirmish with Idas and Lynceus, also twins; Dorothy, comatose following a concussion from a flying window frame during a tornado.

(2) An appeal by the immortal twin to the gods of the underworld, to allow the deceased to return. Pollux won this appeal with the limitation of allowing Castor to live for half of the year while Pollux took his place in Hades. The twins became markers of the inversion circle, converted into cosmic seasonal cycle. Dorothy was rescued by the farmhands who had been transformed into magical beings (Tin Man, Straw Man, Cowardly Lion); after her coma passed, Dorothy was revived and the farmhands joined the family around her bed.

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12. Zeuxis and Parrhasius; Somerset Maugham’s “Mr. Know-All”

Lacan cites the story of Zeuxis and Parrhasios in Seminar XI, The Four Fundamental Concepts of Psychoanalysis: Zeuxis and Parrhasius were renowned painters, and they decided to compete to determine who could create the most realistic painting. Zeuxis painted a cluster of grapes so lifelike that birds flew down to peck at them. Confident in his victory, Zeuxis asked Parrhasius to unveil his painting. However, Zeuxis soon realized that the “curtain” he was asking to be drawn aside was actually Parrhasius’s painting itself. Admitting defeat, Zeuxis acknowledged that while he had deceived birds, Parrhasius had deceived a fellow artist, making him the true master of illusion. The issue of the “master of illusion” is also at the center of “Mr. Know-All,” by Somerset Maugham, a story about Mr. Kelada, a brash and talkative man of Middle Eastern descent, who dominates a shipboard conversation during a post-World War II (in the movie version) voyage. He is disliked by the narrator and others due to his overbearing nature. A pivotal moment occurs when Mr. Kelada bets on the authenticity of a woman’s pearls, claiming them to be real. However, when he notices the distress of Mrs. Ramsay, the pearl’s owner, he pretends to lose the bet, saying the pearls are fake. Later, the narrator learns that the pearls were indeed genuine, and Kelada sacrificed his pride to protect Mrs. Ramsay’s secret, revealing his unexpected kindness and depth of character. The filmed version stages the critical plot point with Kelada taking the pearls over “to the light” in front of a mirror. Mrs. Ramsay’s expression of distress is visible only from this vantage point, out of the corner of Kelada’s eye. He quickly deduces, from what he knows of the Ramsay’s two-year separation, that Mrs. Ramsay had had an affair she is desperate to conceal and reverses his judgment, losing the bet and damaging his own reputation. Although some might be tempted to think of this scene in terms of the Mirror Stage, it would be more instructive to see Kelada as being at the edge of an inversion circle. It is as if he is able to understand the Borromeo rings’ “viscosity” inside the inversion circle as, directly on the outside, as the orthography of the Symbolic and the Real. Mrs. Ramsay’s glance of panic, his unnoticed sideways witness of it, and his returned inspection of the pearls, now converted into a skeptical reversal of his prior admiration, structure a shadow of orthography that, like the shadow around Mount Analog, creates a lipogram-made-for-two.

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13. Piero’s Virgin Enthroned by Four Angels

In this painting, the symmetry of the four angels representing the four cardinal directions constitutes an “inside sphere” that we might compare to the inversion circle. Without the viewer, we might draw the ideal as ⦿, a center occupied by Mary and Jesus, a periphery held in place by angels symmetrically placed, ⊞. But, the observer does intrude, and the space is opened slightly, to witness Mary offering Jesus a flower with a name that is relevant to both a biblical tradition and a geometrical consequence of including the observer with the observed: a carnation. Can we theorize this painting in relation to, instead of an inversion circle, an inversion sphere, and a plane — the picture plane — tangent to this sphere? Is the there, then, the possibility of two carnations? Art historians have failed to understand the inversive geometry of this and other of Piero’s works. They continue to misidentify the flower as a rose, despite the repeated use of the carnation in paintings of the Annunciation and, in our sample, of St. Jerome, a specialist in the issue of Mary’s perpetual virginity.

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14. The Tiny House

ChatGPT: “The modern tiny house movement gained significant momentum in the early 2000s, particularly in the United States, as people began seeking simpler, more sustainable lifestyles. This movement was partly a response to the increasing cost of living, the environmental impact of large homes, and a desire for a more minimalist lifestyle. The economic downturn of 2008 further fueled the movement as people looked for affordable housing solutions. However, the idea of small, remote, or self-sufficient dwellings has a much older history. In the 19th century, writers and thinkers like Henry David Thoreau championed the concept of simple living in harmony with nature, as reflected in Thoreau’s experiment in self-sufficiency at Walden Pond. This idea of retreating to a small, isolated dwelling to find clarity and re-center one’s life can be seen as a precursor to the modern tiny house.”

Possibly the pandemic of 2020–23 intensified the collective desire for isolation, self-sufficiency, and ecological harmony. Typically, sites at edges (seasides, lakesides, cliff-sides, forest edges) allied this desire with the horizon which the inversion circle can claim as its own. There are two circuits in the Tiny House. One small and tangent to the inversion circle and thus is “in tune with” the periphery as an idea. The second is tangent to both the periphery and the center, converting to a line extending to infinity but which ultimately converges on an antipodal vanishing point (behind the camera in this case). The compact economy of the standard tiny house insures that the advantages of the center may be enjoyed at the periphery. Location negated, orientation is neutralized by the contrasting qualities of the preferred edges: land/water, earth/sky, forest/clearing. Architectural miniaturization comes close to making an analogy. The cabin becomes a kind of garment, a space-suit, a protection against any and all environmental challenges. The Tiny House identifies with the perspectival vanishing point, where the utility of the normal life-domain is condensed into a well-packed, ergonomic miniature. It is where hygiene meets ecology. Think of the Other, Lacan advises in Seminar XIII, 9, in terms of Richard Feynman’s version of “Dennis the Menace.”

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15. Holbein’s The Ambassadors; tempietto; camera obscura …

To understand this painting’s relation to the inversion circle you would have to know what Holbein wrote on the back, as the date of completion: April 11, 1533, 4 p.m. This over-exact designation does not relate to the vernissage, the completion of the double portrait but a use that is hinted at by the navigational instruments on the table between the two men. The vertical line between the crucifix half-hidden by the green curtain at the upper left and the point where a viewer is required to kneel in order to see the anamorphic skull titled at a 27º angle at the feet of the subjects is actually a hinge. The painting rotates to “shoot the sun” on April 11, to determine its 27º angle above the horizon of London at 4 p.m. Why this moment? The astronomer-geometer Luca Paccioli, a friend of Holbein, had determined that this would be the exact moment of the Apocalypse, determined by 3’s: 3×500, 3×11 (the age of Jesus at the time of crucifixion), and itself 2+7 = Σ9 (the sum of digits). The combination of the front and rear frame sandwich a “Janusian observer” who, inserted between the contrasting images of wealth and death, sees both circle in the same direction, although from either side, an external viewer would see them rotating in opposite directions. Compare this to Magritte’s perplexed gentleman in Not to Be Reproduced, where we become the Janusian observer and see the backs of both real and spectral images.

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16. My Man Godfrey

There are many ways to introduce the universal theme of “between the two deaths,” but the best examples are those where there is no hint of actual dying, just a period of trials endured by someone disguised as the opposite of who they were on the other side of this critical inversion circle. Unless we are watching a fantasy film, we don’t witness those who are between the two deaths literally. Instead, they are characters who have fallen on hard times and must work to achieve redemption and return. This interval could stand for the general human condition, but when it is symmetrically inserted in a conventional reality, the contrast focuses on the role of margins. In the 1936 film, My Man Godfrey, the scion of a wealthy Boston family voluntarily falls on hard times because of a failed romance, reliving the Turks’ anxieties in Freud’s account of parapraxis. The Turks he noted would rather die than lose sexual function, and Godfrey follows this advice in the attenuated circumstances of a hobo encampment at the foot of the Queens-Manhattan Bridge in New York City. He is “found” by socialites on a scavenger hunt, who are looking for “the forgotten man,” an unconscious reference to the relation of mnemonics to death that was the basis of Simonides‘ discovery of artificial memory in the 6th century BCE. His accent and manners attract the attention of the socialites’ family, the Bullocks, and Godfrey is hired to be a butler, ironically liveried in formal wear while the rest of the family go casual. Not only does Godfrey complete his labors of Hercules with flying colors, he rescues the family from financial ruin by using own money to sell the company shares short, returning the proceeds to the Bullocks to avoid bankruptcy. Godfrey’s stylishness remains the same no matter what side of the inversion circle he finds himself and, because his “circle” remains tangent to both the perimeter and the center, the hobo camp’s relation to the bridge allows it to be directly converted to a fashionable night club.

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17. the aviaries of inversive geometry

It is no accident that birds or bird-like objects keep turning up in examples of inversive geometry. There is Nabokov’s waxwing, the partridge of St. Jerome, the peacock of the runaway, the space-ship of The Day the Earth Stood Still, the neck-breaking bird that flew into Zeuxis’s mural. If it can be said that birds naturally attract the poetic imagination, it is also true that, once attracted, the birds seem to obey certain topological rules. As with Antonelli’s painting of St. Jerome, they are keys to the password between two inconvertible spaces (divine and mortal in the case of Jerome). In the disc-shaped space ship that lands on the Capitol Mall, they use both functions of dream-work, condensation and displacement. If there are skies in Picasso’s Demoiselles and Magritte’s Memoires of a Saint, there are birds in those skies.

Apparently, poets do not have to have any instruction on the subject of inversive geometry to write, as Wallace Stevens wrote Thirteen Ways of Looking at a Blackbird:

In the mythic imagination, the bird “mediates” the divine and the human. The dove is the agency of impregnation in paintings of the Annunciation. What is this aviary in relation to Lacan’s topological project? Birds biologically internalize issues of location and orientation. In fact they are the model for the way a species masters navigation by means of a “polytheistic set” of determinants, including sun angle, winds, terrain, food supplies, colors, magnetic fields (some say), … . They demonstrate a kind of synesthesia of navigational skills in nature that accompany their navigational skills for the imagination, which impressively navigate between mortality and divine immortality, inclusive parallax (the need to include the observer in the observed) and the mathematical exclusivity of the real projective plane.

18. reading as order/re-order

In James Joyce’s Finnegans Wake, as is well known, the first sentence begins with an ellipsis and the final sentence ends with an ellipsis. The two …’s are presumed to be the same, and punctuation seems to have turned the text into a self-referential circle or cycle. Another less famous novel, Julio Cortázar’s Hopscotch (La rayuela), presents its narrative in a standard order but proposes alternative asynchronous readings. Other methods are reader-originated: starting from the middle to postpone the preparatory functions of the beginning, which are consumed last; reading backwards; reading alternating paragraphs … There is even a poetry that involves redacting specific words of published poems to create a new unread poem within the original.

The essence of these literary quilting parties is the idea of the palindrome, which Lacan has addressed in Seminar XIV with his “slide-rule analogy,” where the linear interval of 1 is added to the golden ratio interval, a, which is then slid into the 1 to produce remainder of a². This is slid into the line of the a to produce a³, and so on and so on so that the powers of a point to a “vanishing point” that is the imaginary golden mean, both internal and external to the 1.

There is, here, the logic of the palindrome, the folded reading, or the “stepped reading” implied by the Fibonacci series 1 1 2 3 5 8 13 21 …., where each subsequent number is the sum of the two previous numbers. Lacan cites the self-intersecting quality of this in his reference to the equation x = 1 + 1/x, which requires the “answer” to be plugged back into the “question.” Instead of a hypertext or metalanguage, the caption intended to explain the form

is the form itself. What has changed is that the frame implied by this mode of understanding is replaced by an interactive reading that incorporates the viewer in the viewed — what Spencer-Brown said would actually be what phenomenology proposes but fails to produce.

Reading order is about the consecutive. Understanding is about seeing the consecutive in a “vertical” or “orthogonal” way. These may be understood as the principles of position — moving from place to place — and orientation — maintaining a constant directional standard. While it would be difficult to give a brief explanation, reading order is not just related to architecture on account of the verticality of this “orthography” (correct writing/reading), it is imperative to regard the architectural ideal, the tempietto, in its full functionality.

Jim Egan, a photographer who has undertaken an intensive study of the stone tower in Touro Park, Newport, Rhode Island, argues that this structure was designed by Queen Elizabeth I’s advisor, John Dee. This three-storied structure include the functions of an observatory, planetarium, a territorial marker, a camera obscura, a horologium, and a ritual center. It is a “theory of reading” not just literal texts (the royal charter, celestial patterns) but a reading that engages the landscape and sky in a comprehensive self-referential way. Dee, according to Egan’s research of his mathematical theories, was aware of the numerical and structural consequences of his design.

Can an architectural tower be a “theory of reading”? Is this just a manner of speaking? The question should be referred back to the issue of dream-work, and the relation of position and orientation, a mandate that embeds itself in all cultures, of all periods, as a latent force of metaphor: the suppression of one order, in order to reveal a puzzle-form, in another order. In this way, order gives over to disorder, form to the formless. Topology and inversive geometry address these issues as central and decisive. It is essential that, for a new Lacanian topology to emerge, we consider these possibilities.

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Notes

¹ In Lacan’s Seminar XXII: RSI, Lacan says that the Brunnian variation of the Borromeo knot was inspired by James Joyce’s interest in the Christian symbol, the circle and cross. Joyce, however, found this to be a graphic condensation of his general practice of converting linear narratives into circuits/cycles. See David Auerbach, “The Linear and the Circular in Ulysses and Finnegans Wake.” Joyce’s interest in inversion came directly from Giambattista Vico (The New Science, 1725/1744). One chapter of Ulysses for example demonstrates that Vico’s cycles are independent of scale and can be found, even in contemporary thinking, in the co-presence of expressive, representational, and conceptual functions. Whether direct or indirect, through Joyce, the inversion circle matches the standard set by Girard Desargues, the creator of projective geometry, that every line is an infinite circle with a vanishing point at infinity — i. e. an inversion circle.* A polythetic set combines two arrays that have only overlapping and/or re-ordered elements in common. Lacan anticipated polythetic correspondence in his thesis of signifiers that “slide past each other,” where quilting (points de capiton) are required to stabilized meaning. A master signifier presumes and stabilizes the polythetic set by being an open structure that resists semantic determination and is a pattern of relations rather than a relation itself.

² The inversion circle may also be the fifth discourse (comparable to the discourse of capitalism) that is simultaneously a revised template for Lacan’s classic four. Lacan himself implies as much when he tags the circular space around the quadration of the Real and the Symbolic with SEMBLENCE, JOUISSANCE, SURPLUS JOUISSANCE, and TRUTH. This is the “orthogonality” of the inversion circle, a space of pure rotation (non-orientation) that not only allows for displacement but requires it.

³ Semblance is not animal camouflage but the more complex self-mis-identification that is the alienating presence of hygiene in at the radical center of subjectivity, the kernel of objectivity within the subject and subjectivity. From the Mirror Stage on, the neurotic subject is misidentified and misidentifying. Within this context, discourse converts the standard Shannon-Weaver model of communication into a case of symmetrical difference, where the surplus of the Agent-speaker is possessed by the (defective) Other, and the surplus of the (defective) Other is possessed by the Agent-speaker.

We enjoy this semblance in Groucho Marx’s joke at the expense of Emanuel Ravelli (Animal Crackers, 1930):

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