Lacan and Topology

by Bruno de Florence on October 8, 2011

The use of Topology by Lacan has baffled many of his readers, most especially those with no background in Maths. Lacan himself did not help as what he presented was his own version of Topology, just as he did with Logic. It is only in Seminar 12 (1964­1965) that he extensively presents and explains his approach to Topology, while up until then, only the word itself had been mentioned here and there. Throughout that seminar, the topological figures used to represent the non­linearity which characterises Freud’s description of the psychical apparatus are the Moebius strip, the Klein bottle and the Cross­cap. Those figures have now become emblematic inside the Lacanian psychoanalytical community. Almost 10 years later, in Seminar 20 (1972­1973), Lacan presented a figure which has now become the icon of Lacanian psychoanalysis, the Borromean knot.

Nowadays, good quality graphics and videos of those figures can very easily be found on the Internet, notably on Youtube, thus assisting first time readers in coming to grip with Lacan’s elaborations, most especially those who have no direct experience of psychoanalysis.

Although they may not seem to be any connection between Topology and psychoanalysis, it is worth pointing out that Topology can be used in other non­mathematical areas as a pedagogical tool, as well as as a creative tool. As a composer and a musicologist, I was able to use it as a framework for composition, and as a musical analysis framework for some of Bach’s pieces, showing that Canon 2 from Bach The Musical Offering could be accounted for as a simultaneous double walk along a moebius strip.

Lacan considered the use of Maths in general as important, since a mathematical equation (such as E=mc2) is a letter superimposed on the real, that act producing knowledge about the real, even though the real cannot entirely be “knowledgised”. It also minimises the ambiguity inherent to metonymy, while at the same time allowing for further interpretation and elaboration. This knowledge remains imaginary, an imago (the latin word for image), something which stands for something else. It is all in the eye of the beholder.

Can Topology be considered as a metaphor in its use in psychoanalysis? Lacan at first says no, then in his typical fashion of saying the same thing and its opposite, from one year to the next, or within the same seminar, says yes: “the efforts I am making to bring you a topology are to account (my Italics) for a form to allows us to conceive of these anomalies which are ours, concerning those problems of inside and outside” (Seminar 13, 8 June 1966). So Topology is and is not a metaphor! This statement is more logically consistent than appears at first, if you place it on a moebius strip, that is inside a non­ Euclidian logic, where the principle of non­contradiction is not part of the initial axioms. Charles Peirce had elaborated such a logic, which he called Logic of the vague. Nowadays, it would be called Fuzzy Logic.

Lacan then asks a rhetorical question: does an analyst need to learn Topology in order to practice analysis? That is not the question, is his reply, for a topological  object is what he will be dealing in his daily practice, and if his topology is all wrong, it will be at the expense of his patient.

Is the use of Topology  by Lacan justified? My answer is a definite  Yes, as per the Freudian  text. First, Freud conceives  the Ego as a surface, which therefore  participates  of both the outside and the inside of the psychical system. Second, in his 1916 Introductoty Lectures on Psychoanalysis, Freud says than an idea can have the propero/ (my Italics) of being conscious or unconscious, while saying in The Interpretation of Dreams that the conscious and the unconscious are locations (my Italics) between which ideas circulate. And precisely, the mathematical of Topology is “the  mathematical study of the properties that are preserved through deformations, twistings, and stretchings  of objects”. For instance, a stretched  circle is topologically equivalent  to an ellipse. Another exemple  is if you remove a point from a circle, you obtain a line. More generally, Topology  is the study of the properties of an object when a transformation  has been applied to it. Do they remain the same, that is are they invariant, or not? Thus, if an idea goes from one location (the conscious)  to another (the unconscious), Topology  allows us to ask what happens to its properties.

Here is a video example of how the Moebius strip can account for a surface being both an inside and an outside. Its main characteristic is that, in Mathematics language, it is non-orientable, that it cannot be decided if it is left or right orientated.

You should really go through the experience of making a Moebius strip and slide your finger along it to see what happens.  You can follow the practi ea! steps given  in this video:

Note that a Moebius strip can have any number of twists, not just I, providing the number of twists is odd.

Let’s  see now how we can use Topology  to account for some of the Freudian and Lacanian categories. A comment made by Chris Oakley during a session of the Lacanian  Forum Reading Group on Seminar  10 had led me to surmise  that the foetus, upon birth, goes from one topology to another, and that this could for account for the fact that “the act of birth, as the individual’s first experience of anxiety, has given the affect of anxiety certain characteristic forms of expression” (Freud,  Inhibitions, Symptoms and Anxieo/ 1926).

Initially, we can say that the foetus is inside a sphere. Nutrients and oxygen are brought directly into his blood stream, he does not have to breath and does not have to ingest and masticate food. Upon birth, that is at the moment when he has completely  exited the mother’s womb, the autonomous respiratory  system kicks in and he starts breathing. For the very first time, he is traversed  by something  outside of himself. We can therefore say that his sphere has had a hole punched into it, thus turning it into a tore.


Two other phenomenas  contributes  to the creation of this hole: the experience of hunger (unpleasure), and the delay between the apparition and the disappearance of that sensation, that is the satisfaction brought about by the ingestion  of food. Whereas inside the womb, this process was a feedback loop with no delay in its regulation controlled  by biology.

We can also say that that particular experience constitutes  the first encounter  with the real of the ex-uteros, the real of the pneuma. In his etiology of anxiety, ‘The Pathology of the Nightmare’ 1910, Ernest Jones (Seminar X : 12th December 1962) had remarked that the physical symptoms of anxiety are directly related to breathing, such as the sensation of a weight on the chest.

At a later stage of his existence, the baby experiences a second hole. He comes to realise that he is not everything for his mother, that she has other centres of interest. His imaginary supreme importance is punctured. Inside or on the surface of that hole, he will place something that his mother may want, that he possesses but others don’t, and which will make him again the centre of her attention. Lacan called that something the phallus (not to be confused with the genital male organ), notated minus Phi. What that object actually is does not matter. What is important is that a place holder for it has been created, allowing the baby to engage in a dialectic of objects and love. If a particular object does not produce the expected effect, another one will be tried out. Hence, L’amour fait son objet de ce qui manque dans le réel (Love makes its object from what is missing in the real, Écrits, La psychanalyse et son enseignement, 1957, my translation).

The notion of the phallus has a particular place in Lacan’s teaching. It is a signifier with no signified, and it only refers to itself. It acts as a clutch to allow for other signifiers to come into existence, since for Lacan, a signifier is characterised by its differential aspect. It corresponds in Freud’s notion of Einzige Zuck (unique trait or feature). All other signifiers become a variation or metonymy of this initial trait or mark. It also corresponds to Gottlob Frege’s logical proof for the concept of zero.


Here, the relationship to Topology consists in how a hole is defined: a topological hole is a structure which prevents an object from being continuously shrunk to a point. Therefore this is not the hole in your pocket or socks. This distinction is important.

The figure of the Borromean knot was used by Lacan to illustrate the characteristics of a subjective position. It is the result of a coinçage (blocking, freezing) obtained when 3 loops are linked with another in such a way that if one of the loops is cut, the other 2 are free. Each loop represents a category: the real (all that exists, but has not yet been signified), the imaginary (produced knowledge), and the symbolic (the treasure chest of signifiers).

borromean knot

In the picture below, His Majesty the Baby sits atop the conjunction of all 3 loops. That conjunction is what hold us “together” as subjects. You can then guess what happens when the structure comes loose.


However, there is more to the Borromean knot. I do not know whether Lacan was aware of it or not. Knot theory is another branch of Mathematics. Of particular interest, is the Brunnian knot class, of which the Borromean knot is part, being the simplest knot. A brunnian knot is made of several non­ knots, that is closed loops without any knot.

Instead of loops, we can have strings forming knots:

The knots of Celtic art are another example of Brunnian knots. Lacan was particularly  interested in the Borromean  knot as it required 3loops, which fitted his 3 categories  of real, symbolic and imaginary, as well as Freud’s 3 categories  of conscious, unconscious and pre-conscious, and later his 3 categories  of ego, id and superego.

A Brunnian knot can be made of any numbers  of loops.

brunnian knots

In a Borromean  mesh,  four  Borromean  rings are linked into a mesh, so that three sets of rings are mutually interlocked.  No two sets of rings are linked, so if any one set is removed, the rest falls apart. If you use Wolfram’s MatbP.roatjr,a software, you will find on his site numerous applets  to generate such rings.

borromean mesh

Further, the circles need not be Euclidian circles, but for instance, Moebius strips. In this instance, what happens if you cut one of the loops is visually more obvious. To my mind, using Moebius strips is a more effective method.

moebius knot

Lacan worked his way from separate loops to the structure formed by their binding. It may be more visually “intuitive” to consider the binding structure first, then its unknotting. In the video below, the non­orientability of the brunnian formation  is visually obvious, as it cannot be decided if it is the bars which keep the string together, or if it is the string which keeps the bars together.

It is interesting  to note that Molecular  biology makes use of the concept of Brunnian knots to account for the way biological  phenomena  are linked together to produce an actual  biological structure.  From this, I have surmised  the notion of Coherence Field, whereby in order to exist, a phenomenon  must conform  to the laws or exigencies  specific  to the field in which it manifests itself. For instance, Einstein’s space-continuum coherence  field is always curved.  The path taken by any object must proceed along that curvature. Therefore, the shortest  path between any 2 points is not a straight line, as per Euclidian geometry (and immediate  intuition),  but a curved line.  Much can become clearer if we consider jouissance as a coherence  field.

Further information:

Posts for the “Topology and the clinic” category : here

Posts for the “Dreams” category: Available here

Posts for the “Lacan Jacques”  category: Available here Posts for the “Freud Sigmund”  category: Available here Ecrits : 1966: Jacgues Lacan  or here

Autres Ecrits: 2001: Jacgues Lacan or here

Working Group to start in the Autumn of 2014

Interrogating Freud & Lacan- Your Invitation  to Participate in a Working Group gives details. This examines  the use of formalisations (mathematical, topological, etc) within psychoanalysis. Further information: Reason for ‘Interrogating Freud &Lacan’ & how you may join in or he re. Please download and circulate the poster from he  re. Note: To participate please pre-register  by 31st August 2014.