Lacan’s Torus is the 3D Spatial Approximation of the Combinatorial Unworld — Işık Barış Fidaner

To interpret what Lacan’s torus really means, let us examine how he figuratively attempted to derive meaning out of it. In Seminar 9, Lacan overlaps his torus with the image of two intersecting sets A and B (Gallagher, page 179):

torus3

On this image Lacan marks the “symmetric difference” of the two sets, which consists of A–B and B–A. He leaves two remaining areas unmarked: (1) The intersection A∩B that occurs between the sets, and (2) the complement of their union (A∪B)’ that occurs outside both sets. Thus he suggests that these two unmarked areas carry meanings that are different from the marked areas.

The marked area of symmetric difference approximately overlaps with the volume of the torus and the two unmarked areas approximately overlap with (1) the middle hole of the torus, and (2) the empty space around the torus. Please keep this image of Lacan’s torus in mind while we generalize the combination of A and B.

This is a relatively simple image because the symmetric difference of A and B is a uniformly significant area if we treat A and B symmetrically. This area marks the possibilities where A excludes B or B excludes A. Let {A, B} indicate the set that combines the two symbols A and B. Now we can distinguish the meaning of marked areas from the meaning of unmarked areas.

The marked symmetric difference area indicates the possibilities where the combination {A, B} is divided: It indicates either just {A} or just {B}. In contrast, the two unmarked areas indicate the possibilities where the combination remains undivided. It’s either (1) the full combination {A, B} which includes both of the sets, or (2) their non-combination {} which includes neither of the sets. Let’s use this simpler notation instead of unions, differences and intersections.

What would happen if there were three sets A, B, C? The unmarked areas would include (1) their full combination {A, B, C}, and (2) their non-combination {}. There would now be two kinds of marked areas: (1) the areas that belong to a single set: {A}, {B} or {C}, and (2) the areas that belong to two sets: {A, B}, {A, C} or {B, C} [1]. Both kinds of marked areas are shown in the image below:

torus4

In this case the one-set areas (marked 1) and the two-set areas (marked 2) would both divide the combination, but they would not divide the combination equally. So which of these (one-sets or two-sets) divide the combination {A, B, C} more? I did the math of this in my PhD thesis [2]. Here’s a figure from the paper:

torus1

It turns out that one-sets divide the combination {A, B, C} more than the two-sets because 1/3 is closer to 1/e ≈ 1/2.72 which achieves the greatest possible division.

If we generalize the combination to any number (n) of sets {A, B, C, … } there will be several different possibilities of divisions 1/n, 2/n, 3/n, etc. which will all belong to the divisive areas that Lacan marked. The division amounts of these marked areas have curves that reach their maximum at 1/e (the maximum they reach is also 1/e). These curves very roughly resemble a cross-section from the volume of Lacan’s torus. Here’s a figure from my PhD paper:

torus2

The two unmarked areas will always include the two non-divisive combinations: (1) n/n where everything is combined (this part overlaps with the middle hole of the torus), and (2) 0/n where nothing is combined (this part overlaps with the space around the torus).

Finally, let us interpret what the full combination n/n means. What occurs at the full combination of all the symbols is a paradoxical singular entity that affects all the symbols. It is at once the most special occurrence and the most commonplace occurrence. At the center of all the symbols, what is most particular overlaps with what is most universal, and there is a universal singularity. In fact, what all symbols have in common is the mysterious object that causes our desire for these symbols and the symbolic in general. That’s why the hole in the middle of the torus signifies the objet petit a for Lacan. The hole in the middle of the torus decenters the sphere [3]. By using the spatial image of the torus, Lacan approximated how the combinatorial unworld decenters the spatial world [4].

Işık Barış Fidaner is a computer scientist with a PhD from Boğaziçi University, İstanbul. Admin of Yersiz Şeyler, Editor of Žižekian Analysis, Curator of Görce Writings. Twitter: @BarisFidaner

Notes:

[1] Translated into the more complex notation: {A, B, C} indicates A∩B∩C. {} indicates (A∪B∪C)’. {A} indicates A–(B∪C). {B} indicates B–(A∪C). {C} indicates C–(A∪B). {A, B} indicates (A∩B)-C. {A, C} indicates (A∩C)-B. {B, C} indicates (B∩C)-A

[2] See “Summary Statistics for Partitionings and Feature Allocations” by Işık Barış Fidaner, Ali Taylan Cemgil, presented at NIPS (or NeurIPS) 2013. See also its errata on this page.

[3] See “Echology, Echosystems, Echocide”

[4] See “Spatial and Combinatorial”, “Making the combinatorial unworld of the unconscious permeable”

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