Sometimes nothing less than a manifesto is required to combine aspirations with assertions. The relation of inversion geometry to Lacanian psychoanalysis is just such an occasion to list the points that critical for making Lacan’s topology able to make the critical leap from mathematics to ethnology.

1. Real>Structure>Topology> … INVERSIVE GEOMETRY! Although projective geometry (Pappus, Desargues) is the foundation of Lacan’s topology, there is beneath topology a “foundation of the foundation”: ithe regulative role played by the “inversion circle”  — fundamentally, Lacan’s idea of extimité. What is mathematically fundamental is also, according to Lacan, fundamental for the theory and practice  of psychoanalysis.
2. Inversion/extimacy is the result of the subject’s severance of the space of viewing from the space of the viewed. Because the severed space is boundless but finite, it can be mapped to a sphere, where the geometrical consequences can be localized. Any viewpoint generates an antipodal point at infinity, whose position has a “spooky correspondence” to the point of view. At infinity, the world looks back at the subject, but this returned gaze must be localized at the sites of all visible objects and persons in the visible half of the spatial sphere. As a result objects and subjects gaze back at the viewer, who must suppress this reverse gaze and insulate the viewing position with an “inside frame.” The space between the two frames corresponds to an unconscious of the visible world, a-temporal and without hierarchical differentiation. Should the insulation of this double frame wear then, “short circuits” break through in the form of uncanny appearances. In the visible field the localized gaze returned from the antipode creates “perfect shadows” behind all objects, which create a continuous circulatory plenum (all shadows are connected). Watch the silent video, “Mapping the Topology of Inversion to a Sphere.”
3. Inversion has many names. Thanks to the many terms referring to inversion, it is clear that Lacan’s interest begins much earlier than the first appearance of the word extimité, in Seminar VII: The Ethics of Psychoanalysis. Although the “inversion circle” is identical to extimité, Lacan’s interest in inversion only rarely appears under that specific heading. Lacan employs many terms to talk about inversion: crisscross, the interior-8, the Villarceau cut, the Injunction of Popilius, symmetrical difference, the void, diagrammatic overlays, Brunnian transformations, Euler circles in the relation of “union without intersection.”Because inversion is fundamentally the factor of non-orientation, the interior-8 and Möbius band portray inversion in its simplest forms. The torus, represented by the “fundamental polygon,” inverts repetition into sublation, and even the subject-signifier relation involves inversion.
4. Inversion geometry’s alternative terms are widely and evenly distributed across Lacan’s writings. To relate these terms to the central cause of inversion, is necessary to learn how to (re-)read Lacan using what Freud called Gleichswebende Aufmerksamkeit — evenly distributed attention. Consider: (1) This is the way the Analyst listens to the equally-evenly distributed blah blah blah of the Analysand, who is encouraged to randomize thoughts through free association. And (2) as a methodology, Gleichswebende Aufmerksamkeit leads to a method of “polythetics.” Like any point on a sphere, where any element is a center surrounded on all sides, inversion in whatever form it appears lies in the middle of other Lacanian concepts.
5. Polythetic ordering is employed by the Unconscious and, thus, is precisely that which the Analyst must await in the blah blah blah of the Analsyand. Key to this awaiting is lalangue, which we take to be the sum total of “left-overs” after phonemic units are subtracted from speech. This includes sighs and gestures along with the nonsense slips of the tongue, spoonerisms, or other language dysfunctions. As Lacan argued, lalangue, in avoiding the truth, tells the “truth of truth.”
6. Inversion is evident in the dream’s combination of metaphor (condensation) and metonymy (displacement), as a means of allowing the Imaginary to create bridges, tunnels, and defiles in the Symbolic of dream content. The same RSI relations pervade ethnological formations in the topology of the “portable ratio,” where any two rings are held together by any third ring and any ring functions as the “inversion ring” (both top and bottom) — the “effective cause” of the rings and, by extension, the RSI.¹ Through the concept of inversion, the logic of the RSI extends not just to all incidents of Lacan’s thinking, it extends to all “stages” of cultural developments.
7. The RSI cannot be used as a marker of Lacan’s historical sequential development (Imaginary, Symbolic, Real). Rather, it proves that his theory is “polythetic” (complete at every historical moment) and “hologramic” (the whole is deducible from the part; synecdoche). Everyone one encounters the 2:1 ratio in Lacan’s thinking, there is the RSI. This is true for the Fibonacci numbers, where each number is the sum of the previous two; or Pascal’s famous triangle, which he used to calculate odds for gambling.
8. Fractalization of Lacan’s presentations (mi-dire) is the crystalline structure of his thinking, which uses the function of subtraction/sublation in relation to completion. Completion of a circuit requires the temporary removal of a “one” that becomes virtual and “anamorphic” in relation to the salient remainders.²
9. There is a temporality in Lacan’s thinking, but not a literal historic sequence of three stages, each of which is dominated by the Imaginary, Symbolic, and Real, respectively. Rather, time is present as an ideal narrative logic in the same way it is present in the logic of the Three Prisoners’ Dilemma. This implicit temporality is the same sliding sequence that defines the Fibonacci numbers in Lacan’s “slide-rule analogy.”
10. The time- and scale-independence of this fractal structure is simultaneously time- and scale-intensification. There is no Lacanian theoretic without relation to this co-productivity of intensification and indifference, which can be developed equally productively from the perspective of its a-temporality (“hologram”) or a-spatiality (“polythesis”).

immediate implications of the nine points

Anamorphosis. The “non-spaces” and “non-times” of topology intensify with Lacan’s foreys into anamorphosis. Mladen Dolar ambitiously proposes a more extensive role for anamorphosis, putting it on the level of the proposed for extimité by Jacques-Alain Miller,³ but inexplicably he misses the essential anamorphic logic of Holbein’s 1533 painting, The Ambassadors, that would support this project. This is explicated by John North (The Ambassadors’ Secret).⁴ The painting’s anamorphic skull forces the viewer into a fixed position (“paralysis” of the Real) in order to invert the painting, using the left edge as a hinge to reveal contents of the obverse, making it work like an astrolabe shooting the position of the sun over the horizon of London on Good Friday (April 11, 1533). In combination with the triplicities of the date and location (3×500, 33, 27º, etc.) this 27º angle specifies that the completion of the painting (vernissage) converged with the date of the Apocalypse predicted by Luca Paccioli and others.

The astronomical/astrological/numeric involvements of Holbein’s anamorphic paradigm is not interpretive, supplementary, or coincidental. It was key to the 16c. idea of anamorphosis, as fundamentally structured by (1) paralysis of the point of view, leading to (2) an osculatory and clairvoyant/divinatory parallax-within-parallax, the discovery of the “truth of truth” within Euclidean appearances. Anamorphosis is thus the dialectical companion of perspectival observation, a point missed completely by contemporary theory’s most authoritative texts on this subject, Jonathan Crary’s Techniques of the Observer (1992) and Martin Jay’s Downcast Eyes  (1993).

It is no coincidence that both of these authors advocate a radical historicism of visuality and optics based on the development of instrumentation and techniques. Joan Copjec, in “The Strut of Vision,” first published in Qui Parle 9, 2 (Spring/Summer, 1996), 1–30, perceptively noted that these historicists had only thinly disguised their positivism. Crary had used optical studies of the nineteenth century to create a forced choice, of deleting the body entirely or having to accept a “bodiless body,” lacking both generic (i.e. human) and individual bodily specificity.

Extimity. Just as the role of anamorphosis is limited when theorists rely on a restricted historical and literalist definition, extimity is a general concept that is not limited by the single rare literal term. Readers who do now learn to see the inversion principle of extimity in other terms thus fail to map Lacan’s interest. They unnecessarily confine it to specific historical periods, ignoring the actual development of the idea through a continually evolving set of references. Terms change but the idea remains fundamentally the same. For example, Lacan refers directly to a famous example of extimity in the “Injunction of Popilius,” but he does not use the word extimité. Does this mean that this relevant example is not a case of extimity? Clearly not, but to date no commentaries have connected this defining case with other examples of inversion.

A new program. Anamorphosis and extimity, which have been independently cited as central and significant for all of psychoanalysis, are equally strategic for the extension of the restrictive mathematical definition of Lacanian topology to the expansive praxis of that same topology to inversive geometry by means of examples from culture: rituals, customs, folktales of the first human societies; the literature and art of ancient classical periods, eastern and western; and the modern enterprises of literature, the arts, popular culture, architecture, and theater and film.

RSI. In the place of the Real, Symbolic, and Imaginary, inversive geometry finds the second parallax within the unexpected terms, Paralysis, Hygiene, and Marriage, a triad that can be found in the earliest (“Cyclopean”) cultures as well as the most modern. This re-interpretation carries anamorphosis and extimity into the full range of ethnology by refashioning practices of the three terms into the space-time of inversion via the mandates of the inversion circle’s principle of equality of inside and outside. See Iraj Ghoochani’s brief explanation of inversion circularity.

Polythesis as Praxis. Applying Maril Praz’s principle, that nothing appears or disappears in history, only the proportion of relations changes, the contemporary practice of locating the first and final uses of specific terms (e. g. Samo Tomšič’s delimitation of Lacan’s extimité, or similar projects to locate anamorphosis or the RSI) gives way to a “polythetic” and “fractal” paradigm, where all of Lacan’s concepts are structurally implicit in any one case. This dispels the allure of historicism for once and for all, following Copjec’s advice. Polythesis is set theory under the expanded mandates of pre-Boolean logic, where the subtraction and re-insertion of elements is not only tolerated by required, in the spirit of the 16c. Italian mathematician Gerolamo Cardano’s solution of the extraction and re-insertion of the imaginary numbers to solve quadratic equations. Psychoanalysis mandates the same extraction and re-insertion in the

orthogonality of the Real and Symbolic, managed by the (implicitly inversive) circular operations of the Imaginary. In the spirit of a manifesto, this mathematical innovation translates directly into a psychoanalytic innovation first introduced by Lacan when, in Seminar IX, Identification, Aristotle’s square of logical oppositions (universal-particular, positive-negative) undergoes an excission of what Lacan celebrates as a unary: the universal negative.

Re-insertion of the extracted unary negative is Lacan’s principle of polythesis, applied to create the Gleichswebende Aufmerksamkeit (evenly-distributed attention), the essential component of the foundational principle of free association. When Freud advocated the equal-and-opposite practice of the Analyst’s attentiveness to the Analysand’s blah blah blah, Lacan fashioned the mi-dire principle of his presentations and writings into a research program. This allows us to read the AEIO square as an inversion circle, with an “osculatory” role played by the universal negative. In the Brunnian/inverted version of the Borromean RSI, this emphasizes the “dial” role of the Imaginary and its cultural correlate, Fantasy. Just as all cultures allow for the extraction and re-insertion of imaginary formations (“Once upon a time …”), psychoanalysis permits — or rather mandates — the “treasury of signifiers” as (1) a silent language of cultural production, collectively or individually realized, while (2) the sequence of extraction and re-insertion introduces a structural concentricity. See the silent video “Lacan’s Inversive Geometry” for an expansion of these ideas.

In narrative theory, concentricity relates directly to the signifying chain and the metonymy of cause/effect conversions. For psychoanalysis, concentricity generates a program of the different modalities of the (literary) fantastic, correlated to Lacan’s four/five discourses. Concentricity, in turn, is structured by Lacan’s metaphor matheme, a sublation (M) orthogonally/mathematically opposite repetition (S’…S’), signifying the suspended ‘x’ that “returns” in the form of s”, signified of the unary 1, the “one of 1.” By this extraction and restoration, metaphor becomes foundational and inclusive, a full and structural presence of the RSI within human enuciation.

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The discovery that Lacan was using inversive geometry to expand the “Brunnian braid” of the RSI Borromeo rings (“La troisième“) was made by Iraj Ghoochani. The case for correcting Grigg’s translation of “Other Side” (Seminar XVII, The Other Side of Psychoanalysis), to preserve the relation of l’envers to inversion geometry and projective geometry is owed to Quinn Foersch. Don Kunze discovered Grigg’s excision of Lacan’s graphic of the “slide-rule analogy,” presented in Seminar XIV, The Logic of Phantasy. This graphic, present in the STAFERLA edition (p. 70), supports the idea that the recursive relation of (unary) 1 and a is the key to the theme of inversion in XVII. Foersch and Kunze also contend that recursiveness, implicit in the Borromeo knot’s 2:1 structure, where any two rings are held together thanks to an absent third, justifies regarding the RSI as the hallmark of Lacan’s interest in inversion.

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¹ “Effective cause” is the hypothetical fifth form of causality Lacan proposed to correct Aristotle’s misconception of passivity in his fourth, Material Cause. Slavoj Žižek developed this in his podcast on “The Reality of the Virtual,” 2004.

² The  mathematical concept developed by John Conway, a “sliding number relation” akin to the summations involved in the Fibonacci series and Pascal’s triangle. This portable form of self-reference creates an internal loop (lac) that supplements at the same time it is extracted, in the same way an anamorphic construct (the skull in Holbein’s Ambassadors) supplements the meaning of the main image at the same time it antagonizes the perspectival order.

³ Mladen Dolar, “Anamorphosis,” S: Journal of the Circle for Lacanian Ideology Critique 8 (2015): 125-140.

⁴ John North, The Ambassadors’ Secret: Holbein and the World of the Renaissance (New York and London: Hambledon, 2003).