Flattening-Out

Flattening-out (aplatissement) names the operation Lacan performs on oriented Borromean knots when he projects their three-dimensional topology onto a two-dimensional plane in order to render visible the structural identity — the "sameness" — of the knot across its different spatial orientations. The procedure is not merely pedagogical: because the Borromean knot is a three-dimensional object whose distinctive property (the removal of any one ring frees all others) cannot be read off from an arbitrary planar view, a controlled flattening is required to make that uniqueness demonstrable. The concept thus marks the point at which the topology of the Real, Symbolic, and Imaginary must pass through an Imaginary medium — the two-dimensional diagram, the visual schema, the model — in order to become legible at all.

The deeper theoretical stakes, as Lacan frames them in Seminar XXII, are epistemological and ultimately ontological. The necessity of flattening-out exposes what Lacan calls a fundamental limitation of conceptual thought in grasping the Real. 'Consistency' — the Imaginary register's topological property — is what the flattened diagram preserves and communicates; but consistency is similarity, not identity. In being made demonstrable, the knot is simultaneously distorted: the very operation that lets thought 'highlight the sameness' of the knot is the one that cannot render the full dimensionality of the Real. This structural irony — that showing requires a reduction that betrays what is shown — is what underwrites Lacan's formula that 'there is no sexual relationship': the impossibility of writing the sexual relationship without remainder is homologous to the impossibility of flattening the Borromean knot without loss.

The concept belongs to jacques-lacan-seminar-22, a seminar in the late Borromean period (1974–75) where Lacan's primary formal instrument is the oriented Borromean knot rather than algebra or the graph of desire. Flattening-out sits at the intersection of several of the corpus's canonical concepts. With respect to the Borromean Knot, it names the specific technical problem that arises when the knot's three-dimensional structure — whose three rings correspond to R, S, I — is rendered on a surface: the knot's uniqueness can only be highlighted by a procedure that is itself an Imaginary simplification. With respect to the Real, flattening-out dramatises the Real's resistance to symbolisation and to imaginary capture: it is precisely because the Real ek-sists outside the plane of consistency that no flat diagram fully grasps it. With respect to the Imaginary, the concept specifies what the Imaginary's property of consistency does when pressed into service as a means of demonstration — it can show similarity but not identity, presence but not ek-sistence.

Flattening-out also resonates with Showing (Monstration) and the Matheme: where the matheme aims at integral transmissibility by escaping the imaginary register, flattening-out is the moment at which topology is forced back into imaginary representation in order to be transmitted at all — an admission that even Lacanian formalization cannot entirely escape the imaginary support it needs for demonstration. This connects further to the Concept in the Hegelian sense: just as the Hegelian Concept cannot grasp the Real without contradiction, flattening-out shows that conceptual (or diagrammatic) representation of the Borromean structure must pass through a reduction that is simultaneously necessary and deficient. Finally, the concept feeds directly into Sexuation and the formula of the non-existent sexual relationship, which Lacan treats as the properly human consequence of this same structural impossibility.

Seminar XXII · R.S.I.Jacques Lacan · 1974 (p.124)

why must it pass by way of what I am calling, and what I already called, my flattening-out of the knot?… the fact that it has to pass by way of the flattening-out to highlight the sameness of the knot

The phrase "has to pass by way of" encodes a structural necessity — flattening-out is not optional but is the unavoidable condition for making the knot's identity demonstrable — while "highlight the sameness" immediately reveals the limitation: what is shown is sameness (consistency, an Imaginary property), not identity or ek-sistence, so the very operation of demonstration introduces the gap between the Real and its conceptual capture.