1

Index

Base Definitions

Characteristic

is defined beginning with Euler's theorem as vertexes (sommets) - edges (aretes) + faces = X.

This X is the characteristic of a surface.

Obviously, to be able to calculate it, we must give to the surfaces a form provided with

vertexes, edges, and faces. This presentation is called "rigidified in planes" ("en rigides par

plaques").

Classification of surfaces

There exist several fashions to classify topological surfaces, according to their genre or

characteristic.

There are two large families of surfaces: non-orientable surfaces and orientable surfaces.

Two different surfaces can have the same characteristic.

Here is the table for non-orientable surfaces:

Characteristic:

1

projective plane

0

Klein bottle

or 2-plane projective

-1

3-projective

-2

4-projective

and for orientable surfaces:

2

sphere or zero

torus

0

torus

-2

two-holed torus

or 2-torus

-4

3-torus

Between these two families, there is a sheath relation (relation de doublure), which

means that orientable surface can envelop the corresponding non-orientable surface.

Dual cuts or section couples

These are two cuts made on a surface, which have only one point in common.

Thus, on the torus there is a section couple.

Dimensions

Each of the sizes necessary for the evaluation of figures and solids. We often define

time with the term fourth dimension.

2

Genre

The maximum number of closed, disjoined lines that can be made on a surface without

fragmenting it (or cut).

This number permits a classification of surfaces.

Immersion and plunging (plongement)

Our space is three-dimensional. One can speak of an immersion of a surface as soon as

one makes an abstraction of this space and makes such impossible phenomena as resectioning

(recoupement) or the triple point intervene in our space . . .

On the other hand, a surface is plunged when it does not make an abstraction of its space,

of the environment (milieu) where it is. A sheet of paper constitutes an environment, just as

does our everyday three-dimensional space.

Intrinsic and Extrinsic

A property is intrinsic to a surface when it is maintained whatever its space of plunging.

A property is extrinsic when it depends on the plunging space of a surface.

For mathematicians, the twist is an extrinsic property. (All that matters is knowing

whether the number is even or odd.)

Moebian, Moebian space

A hasty denomination of the space proper to the projective plane. Its immersion known

by the term "cross-cap," constructed on a Moebius strip with one half-twist, allows us to

understand this usage.

Thus, this adjective is often used as a synonym for non-orientable or unilateral.

Orientable and Non-orientable

As soon as we leave surfaces in two dimensions, the concept of unilateral no longer

functions.

Orientation then comes into play.

To define it, we must return to the law discovered by Moebius (and which specifically

permitted him to discover non-orientable surfaces).

We begin with a tetrahedron (a polyhedron composed of four triangles).

We define a direction of reading the vertexes of the triangles composing the polyhedron.

When we turn the orientation of our reading in the same direction for all of the

polyhedron's triangles, the edges are crossed in an opposed direction depending on whether

consider them part of one face or the adjacent face.

This quality is an invariant of a surface.

(We should add that all polyhedrons are decomposable into triangles.)

Let us also note that this direction is not the same for the observer placed at the exterior

of the polyhedron.

3

On the other hand, a surface is called non-orientable when, in this decomposition into

triangles of a polyhedron, two edges are found to be not oriented in the same direction (cf. the

heptahedron of Reinhart).

Envelopment of two thicknesses (Revetement á deux feuillets)

A topological manipulation that consists in giving to a surface the form of an

envelopment of two thicknesss from another surface. When the surface is in this position, on can,

without violating the law of continuous transformation, make these two thicknesses stick

together and transform the first surface, doubled, into the second.

This procedure serves when one can produce a turning-inside-out of an orientable

surface. This envelopment of two thicknesses is a symetrical point of the process.

Continuous transformation

This is the operation founding the equality of surfaces in topology. Two surfaces are

said to be identical when one can transform one into the other by continuous transformations in

the domain of plungings.

It is defined by the existence, always possible, of a tangent on a curve that varies in a

continuous fashion.

Unilateral or Bilateral

Said of a surface depending on whether it has a single edge or two edges. This is the

concept Moebius brought to light in discovering the strip that bears his name.

This ribbon of Moebius is unilateral; one can also say that it is non-orientable.

4

Concise Bibliography of Topological Resources

Jacques Lacan, Published Texts and Seminars.

"La Politique de l'ignorance," Recherche, no. 41 publications C.N.R.S., September 1980.

Jean-Claude Pont, La topologie algéirque des origines á Poincaré, (Bibliotheque de

philosophie contemporaine) PUF, 1974.

C.P. Bruter, Topologie et Perception, (Recherches interdisciplinaires), Editions

Maloine-Doin, Paris 1974.

Payot, Les mathématique de l'imaninaire, Bibliothèque scientific, Paris 1970.

Martin Gardiner, La magie of paradoxes, Bibliothèque "Pour la science," diffusion Belin,

Paris 1980.

Martin Gardiner, The Ambidextrous Universe, New York, 1964.

"Des mathématiques avec un fil et une aiguille," in Pour la Science, a translation from

Scientific American, no. 113, August 1980.

Course of Pierre Soury, transcribed by Jeanne Lafont. Chaine, noueds, surfaces. Textes

et travaux de l'ecole de la Cause freudienne, Paris 1981 (out of print).

Jean-François Chabaud, Le noeud dit du fantasme, Weber 1984.

1

The Ordinary Topology of Jacques Lacan*

By Jeanne Lafont

Translated by Jack W. Stone

*Orignally published as La topologie ordinaire de Jacques Lacan, Point Hors Ligne, Paris, 1986.

2

In 1679, Leibniz defined a new branch of mathematics under the rubric "analysis situs."

The literal translation of this term, "study of place," situates this new discipline at the origin of

topology.

However, topology is not truly given body until Euler develops his first theorem, in 1750.

This theorem establishes a constant relation between the summits, surfaces, and edges of a

convex solid; for example, the Platonic solids and the volumes of our everyday experience, like

the pyramid, the cube, and the parallelepiped . . .: he thereby proposes some new solutions to

some very old geometrical problems.

This theorem, the first of its kind, provoked arguments, and many mathematicians sought

to define its limits.

In the perspective of these labors, in 1861, Moebius discovered the figure that would pass

his name on to posterity: the Moebius strip. Unilateral surfaces were created, and would in one

way or another devour, subsume under their laws, whole branches of mathematics. In 1874,

Felix Klein and Shläfly put forth the idea that the space of projective geometry is Moebian.

From here on, no one will speak of functions without reference to topology . . .

Thus, when Bourbaki, in 1948, newly formulates the ensemble of mathematical

discoveries, they (il) will enumerate three, or more precisely two, to which a third group will be

added: the structure of order, the structure of group, and the topological structures in reference to

which they add in a note "that they escape the limits of the blueprint (épure)."1

The work that we propose here is situated at this "limit of the mathematical blueprint." It

is a matter of studying the principal topological structures, beginning with the advances of

Jacques Lacan in this domain

1 Bourbaki, note 3 in "Architecture des mathématics," an article in Les Grands Courants de la mathématique,

presented by F. Le Lionnais (new edition, Albert Blanchard, Paris 1962).

3

Full-twist of the little spoon.

Chapter 1: Space, Structure

"In the beginning was space."

This paraphrase of both Saint John and Goethe puts the notion of space in relief; we will

put it in the introduction of our study of Lacanian topology.

4

A little experiment will help us to grasp this notion. Take a small spoon and suspend it

from a ribbon fixed at its top. This ribbon will materialize the tie of the spoon, our experimental

object, in space.

If we give the spoon one complete turn around a vertical axis, it will reassume its initial

position, while the ribbon, initially flat, has now taken on a helicoidal twist that reveals the

effected operation.

After two turns, then three, the ribbon shows a double twist, then a triple one . . .

Thus, by glancing at the initially flat ribbon, we can know the exact number of complete

turns effected by the spoon.

If we now give the spoon a turn, in the direction of a the hands on a watch for example,

and if, maintaining it rigorously parallel to itself, we make it pass over the vertical part of the

ribbon and it is returned to its first position at the bottom of the ribbon, we notice that the ribbon

no longer shows any trace of the turn. Although the spoon has at no time changed its

orientation, its movement alone has annulled the initial revolution (Cf. the photos at the

beginning of the chapter).

This experiment shows us several things: to start with, it allows us an effective approach

to the notion of space. The relation between the spoon and the ribbon is that of an object to its

space. If you take a book in which you have printed the same movements, you will not see space

appear; space is misrecognized in usual manipulations of objects. The spoon is an invariant

object plunged into space. The relations between the movements of revolution (the turns) and of

translation structure and define this space.

General topology is the study, the science, of these spaces and their properties.

It is not a question, as in classical Euclidean geometry, or even in that of Lobetchevsky or

Riemmann, of producing a system of calculations and notations allowing us to situate an object

and its movements in space. It is a question of describing space itself, while taking the

invariance of the object into account. We must indeed remain conscious of this change in

perspective in approaching topology, particularly that utilized by Lacan. Thus, Lacan could say,

in RSI: "all space is flat; there are mathematicians who have spelled this out for us (l'avoir écrit

en toutes lettres)."2 How are we to understand this remark?

Space in itself does not encompass the dimension of depth, the famous third dimension. It

is only for an object plunged into space, depending on how its movements unfold in time, that

there is a before and an after, and consequently a front and a behind. Topologists have

classically taken recourse, for manipulating this perception and its illusions, to the metaphor of

the ant.3

Imagine an ant offered the same system of perception as a man reduced to its size. This

animal walks along the surface of a Moebius strip, a flat surface of two dimensions, which is

defined in this way in relation to its immediate environment. On the other hand, the horizonal

point where the strip curves back on itself, and hints at its twist, always in relation to its

immediate environment, is perceived as a depth. This depth is measured by the time that it will

take for the ant to get to the point of the twist, and it will never attain it, since, as soon as it

arrives there, another horizon will present itself, always as a third dimension, as a depth.

The "flat" is defined by the surface of a picture enclosed by an edge, and space is defined

by the perception of depth. It is a question of a horizon that we know is not a limit, but,

2 Jacques Lacan, Séminaire du 14 janvier 1975, published in Ornicar n. 3.

3 An ant that we find diabolically represented in some pictures on the cover of issues of Quarto (the revue of la

cause freudienne in Belgium).

5

topologically, of the time that it takes to get there.4 Clinically, this reality of our space is essential

for conceiving of the state of the world experienced by the psychotic, of which one says that it is

without limit, in other words, without a temporal dimension. Time, from the point of view of

topology, is the dimension of space considered as flat, as a surface. Thus, in our experiment

with the little spoon, the relations between the movements of revolution and translation structure

this space which is ours, and define it. There exists, moreover, a property of our ordinary space

thus presented that leads to differences in space, differences in structures. To demonstrate this,

we are going to take an obligatory detour. We are going to try to demonstrate what is, according

to us, an error of Jean-Marc Lévy-Leblond, which will bring us back to the little spoon already

evoked.5 Let us cite, to start with, a fragment of his procedure: "the astonishing thing is the

completely particular role played by the revolution of two turns." If we redo this experiment

again, after an initial revolution of a single completed turn, we will quite clearly find the ribbon

affected by a turn in the opposite direction from the first. Without a supplementary revolution,

and by simple parallel movements of the spoon, it is possible to modify two by two the number

of the complete turns exhibited by the twist in the ribbon.

"There is a difference of an essential nature between the rotations involving an even

number and those involving an odd number. Thus, one can say that it is not sufficient to give an

object a complete turn of either three or five . . . bringing it back exactly into the same conditions

of relations with its spatial environment. To do this, we must effect two (or four) complete turns;

the double turn is found to be the basal unity."

Jean-Marc Lévy-LeBlond continues: "There we have a fundamental property, at once

extraordinarily simple and perfectly enigmatic, of the space where we live. At our level, this

property, to my knowledge, has no other manifestations than of permitting certain tricks to

professional magicians. On the other hand, it has a considerable importance in microscopic

physics. Quantum theory founds the existence of two distinct classes of fundamental particles on

this property: 'bosons,' described for mathematical beings by a single revolution left unchanged;

and "fermions," which require, on the contrary, a double revolution to recover their initial

description. If we now recognize that the 'fermions' obey the 'Pauli' exclusion principle, which,

applied to electrons, explains what is essential in the properties of ordinary matter, we see that it

is no longer a question of simple academic curiosity. In truth, it is precisely the experimental

discovery, then the theoretical understanding of 'fermions' at the quite esoteric level of

fundamental quantum physics, that has allowed us to go back to the source of their strange

characteristics, found finally in the unexpected difference in nature between a single and a

double turn."

What is at stake in this experiment is not meager and, as it is presented, it seems to

involve a mistake. The error is situated at the level of the naming of the particularity of our

ordinary space, and, in this argument, the Moebius strip is grossly misunderstood.

4 In L' Empire des lumières, in Traverses, September 1978.

5 Jean Claude Terrasson has published a quite illuminating text on this question in the revue Littoral n. 5, June

1981.

6

Half-twist of the little spoon.

If we again take up our demonstration, we must notice the difference between the full-

twist and the half-twist. J.M. Lévy-Leblond speaks of the revolution of two turns, when it is, in

fact, a question of a half-twist. A turn of the little spoon gives a full-twist to the ribbon, that is,

7

two half-twists. The translation in space, above the place where the spoon is connected to the

ribbon, always modifies the ribbon by two in two half-twists.

The little spoon can make three complete turns: this gives three full-twists of the ribbon,

that is, six visible half-twists, and three translations, to return to the first position of the ribbon.

On the other hand, if we give the spoon a half-turn, after the translation in space, the

ribbon takes on a complete twist. Thus it presents, at the end of the operation, a half-twist in the

opposite direction from that created by the first half-turn. For the turn of the little spoon, the

basal unity is a half, as it is for the ribbon, but the unity is 1 for the translation. It is impossible,

in ordinary space, to bring forth, by translation, a half-twist. It is here alone that we find the

radical difference between even and odd. Three half-turns are fundamentally different, for the

ribbon and for the materialization of space that it effects, from two or four half-turns.

This experiment makes appear with striking clarity the Moebian space, or, more

precisely, the space of the projective plane. If the space in which the ribbon is plunged were

Moebian, a translation would make a half-turn disappear. For translation also, the basal unity

would be the half-turn. One can demonstrate this by looping a Moebius strip through the little

spoon.

We see how these relations structure space, rendering it materializable, while it most

often escapes our perception. The specific object of topologists is this notion of space and the

relations that structure it.

It is in this measure that topology concerns psychoanalysis. Psychoanalysis is, in fact, a

study of structure disencumbered of a singular, substantified psychic object. The general

tendency of thought is to give body, subjectivity, to one or more of the concepts of our

discipline, for example, the subject or the unconscious. The subject is not the object of

psychoanalysis, just as the ant and the little spoon are not the objects of study of topologists.

They are only interested in their appearances, their trajectories, and in the possibilities that allow

them to describe a particular space. So it goes for the soul, a notion that serves as an exemplary

representation of this tendency toward subjectivation: for the topologist, the study of its depths

only puts in play questions of passages through space.

Moreover, topologists have defined a notion of space that is identical with the structure

utilized by the human sciences. Topology proves to be the study of the structure in play in these

sciences, as will be demonstrated after the fact by the discoveries of Jacques Lacan.

We can show this by drawing on the ethnographic studies of Lévi-Strauss, the importance

of which for the introduction of structuralism into the human sciences is well known.

This is the case in the study Lévi-Strauss conducts on the matrimonial system of the

"Kasieras" (Who has the right to marry whom?).

He defines the structure of this system starting with a division of the group of

matrimonial classes into two patrilinear halves, a division that is itself "perpendicular," he says,

"to the division into two matrilinear halves."

The word "perpendicular" does not at all refer to a precise mathematical definition, but

rather to an intuitive vision of space, to a schema that would represent this structure on two

perpendicular lines. Moreover, without entering into the details, Phllippe Courrèges has shown

that the true mathematical concept that would be pertinent here would be that of "product."6

We can also cite the work of Henri Pradelles on the kinship system of the Trobriands (a

classic object of Malinowski's studies). He shows that the concept of a "duality of cuts" is the

6 Phillippe Courrèges, in Anthropologie and Calcul, U.G.E., Coll. 10/18, Paris, 1971.

8

best approach to this particular ethnological structure.7 Without resorting to a forced analogy,

these examples show how, from the origin of the work of the structuralists, the conceptual

necessity of bringing in topology comes to light. At its limit, as soon as a "schema" takes on an

explanatory or even a didactic value, topology intervenes as an epistemological foundation to the

knowledge (connaissances) brought forth by this schema. It is wholly to Lacan's credit to have

sought to pinpoint (cerner) this specificity of topology and to have indicated what its usage could

be in the human sciences.

As for Lévi-Strauss, he did not seek exactitude in this domain, too conscious of the

distance that separated his work from mathematical formulations. For instance, in the preface of

The Raw and the Cooked, he writes: "It better that no one scruple over the very loose

acceptations that we give to terms such as symmetry, inversion, equivalence, homology,

isomorphy . . . We utilize them to designate large packets of relations of which we perceive

confusedly that they have something in common." The allusion to a "confused" perception is

appropriate: it appears to us, in fact, not only that topology is confusedly present in the work of

Lévi-Strauss, but that it is also a putting to work of this confused perception of structures.

Saying that he does topology without knowing it is in no way impertinent, if we consider,

apropos of common themes like the Oedipus complex, all of the work of Lacan, which consisted

in clearing up this confusion from a certain perspective.

Also, the aim of our work is not to know how mathematicians came to be interested in

this study of space and spaces, although renowned topologists like Poincaré speak with respect

of the "geometrical intuition" that allows us to confusedly perceive these "large packets of

relations" . . . On the contrary, it is a question of situating, beginning with what is at stake in

psychoanalysis, the topology of spaces. For this, we will take support, not from the requirements

of mathematical discourse, but from the necessities internal to analytic discourse. Topology

clarifies the notions on which the psychoanalytic treatment reposes. The psychoanalyst has a

means for establishing what is at stake in psychic suffering by recourse to topological structures.

We must now approach this topology directly.

Treating of the notion of space, Topology is interested neither in the metrical nor in

proportions. Based on this fact, equality is defined as the possible trajectory from one

presentation to another. Two figures are called identical if it is possible to pass from one to

another by a continuous transformation.

From this perspective, these objects are the same (we already grasp the importance of the

drawing).

7 Cf. the article of Charles-Henri Pradelles of Latour, Littoral, nos. 11-12, February 1984, Paris, "la Parenté

trobiandaise reconsidérée."

9

A surface like the disk can vary continually without modifying its structure.

However, at a certain moment there is rupture and a passage is effected from one

structure to another.

This transformation has only been possible by the slow, continuous preparation of the

surface.

Thus, on this disk, one part of the surface can pass beneath itself, and it is possible to

make reappear the portion that has been slipped beneath. We have thus created a line of

intersection. This line of intersection signals the passage of the structure of a submerged surface

to an immerged structure. We have radically changed a domain, passing from submersions to

immersions.

The hole that we designate at point A can then be reduced to a point: this is equivalent to

the phenomenon of the twist. This surface can be seen as a disk that can be twisted back on

itself. (The twist has a quite distinct status that will be the object of a whole chapter.)

10

By this phenomenon of intersection and of disappearance of the hole, there is an obvious

transformation of the structure.

It is intuitively perceptible that we have changed its space (we are already in the space of

the projective plane). We can then make the interior line evolve into this point A. We thus

obtain the immersed disk, constructed as an interior eight.

This exercise clearly shows why topology has been called "rubber geometry."

This example allows us to render easily sensible the game of transformations in topology.

There exists a whole dialectic between the preparation of a surface by continuous transformation

of its drawing and the brusque appearance of an event, of an act, of an operation; of a changing

of the structure of this surface or of its space of submersion. This dialectic, between the

continuity of the identical to the identical and a structural rupture, is essential to our approach to

topology. It allows us to understand what conditions the completely particular relationship to

time and to the scansion that we know in the treatment. Often, an interpretation only has an

effect after a long series of sessions that have done no more than make the presentation of the

symptom evolve, without modifying its structure.

Once we have set forth these preliminaries, we can take on the study of topological

spaces. We will then see how the Borromean knot will formalize the recourse to surfaces. Not

until the end, in a supplement that will take the form of an index, will we give the mathematical

definitions (signaled in the text by an asterix) that serve in the elaboration of this presentation of

the topology proper to Jacques Lacan.

11

Chapter 2: The Moebius Strip

At issue is a physical object that can be easily constructed. It suffices to take a strip of

paper and to stick its ends together while impressing in it the movement of a twist. We thus

obtain, starting with an ordinary rectangular surface, a surface that presents several paradoxical

phenomena. This object, held so easily in the hand, nonetheless opposes in diverse ways our

habitual experience of physical objects.

Effecting a half-turn on the strip that we started with, before joining it end to end, is a

very simple sleight of hand, which, let us insist, subverts, properly speaking, our everyday space

of representation.

This operation brings to light a number of different paradoxes:

After the sleight of hand we have just described, the topside and underside of this strip of

paper are found to be continuous. The common usage of "head or tails" is subverted. The

topside and the underside are continuous with one another. A little gentleman or an ant that

walked along one of the sides of this surface would find himself upside down on the other side

without even perceiving this incongruity. A finger that follows the surface of the strip will be

found, after a complete turn, and without having been lifted, without crossing the edge, on the

underside of its point of departure. After a second complete turn it will return to this point, on the

topside.

Only a temporal event differentiates the topside and the bottom side, which are separated

by the time it takes to make a supplementary turn. The dichotomy between the two notions,

underside and topside, only reappears at the price of the intervention of a new dimension, that of

time. Time, as continuity, makes the difference between the two faces. If there are no longer

two measures for the surface, but only an edge, time imposes itself as accounting for the strip.

The existence of a single edge is essential, since one of the topological definitions of the

Moebius strip is supported by this paradox.

It has only one edge: we have joined the two extremities of the original strip, in reversing

their orientation:

The line AC of the original strip continues into BD. There is only one edge. It traces a

figure that resembles an eight that folds back on itself. Lacan gives it the name "double-buckle."

At the same time, to see this design appear, we must perform an operation that

topologists call a "putting flat." From the drawing of the first figure that still evokes a three

dimensional object represented in ordinary space (length, width, and thickness), with an illusion

12

of depth, one passes to a two-dimensional drawing, written on a sheet of paper, put flat. Depth is

then marked by a crossing of the line over itself, an "above-beneath" (dessus-dessous). The

discontinuity of the line

does not evoke its interruption, but the passage under the line, at a moment in its trajectory. This

above-beneath is necessary for the illusion of depth to disappear. There remains, as a trace of

depth in this drawing of the putting flat, only this above-beneath.

Again, there is the necessity of writing a temporal moment. It is marked on the trajectory

of the line. Thus the conventions of drawing give to the putting-flat the status of a writing.

For example, the dashes evoke the continuity of a line, hidden, however, from the gaze of

the reader by a surface:

There is a problem: on a paper surface, there has to be, to draw this Moebius strip, which,

it also, is only a surface, to represent depth, a third dimension, let us say.

13

These points of above-beneath, these dashes, are the condition for the Moebius strip to be

representable on the surface of a sheet of paper, without bringing in the conventional evidences

of perspective. Topologists have in this way obtained an entirely readable drawing, that is, one

that does not make a call to the imaginary.

Moreover, although it is a physical object that can be constructed by hand, we no longer

establish on the drawing more than a single measurable dimension. It is indeed dimension that is

put into question by the Moebius strip. It straddles (est á cheval) 1 and 3 dimensions. This

paradox is insoluble.

Topologists sometimes represent the Moebius strip in a drawing with a base of straight

lines, which multiplies the above-beneaths and makes them dashes. They are perfectly readable

on this drawing:

This is how Moebius draws it for the first time in a scientific publication,1 with straight

lines; he calls it a "unilateral surface" (from unus: one, and latus, lateris: flank, side).

It is a surface with only one face. This single edge, which describes a double-buckle,

encloses a surface with a single face.

Let us make this paradox felt by drawing a pencil that passes through the Moebius strip.

1 Ornicar, nos. 17-18, Spring 1979, "Moebius, la première bande," introductory text by J. A. Miller.

14

It passes through the Moebius strip as it would any surface, but the strip still has only one

face. Locally, at the place of the pencil, there are two faces, but the whole (l'ensemble) of the

strip, as continuous, has only one face. This articulation between "part" and "whole" is entirely

new. The Moebius strip allows a subversion, in relation to (par rapport) habitual conceptual

space, of the rapport between the parts and the whole. The whole, manifestly, is not always

equal to the sum of its parts . . . Analysis makes appear in parts another dimension that does not

enclose the whole. The pencil allows us to define again, at a given place on the strip, the two

faces of an ordinary surface that does not, however, know the Moebius strip.

Between the static and dynamic points of view there necessarily exists an element that

disappears. Such a notion has its importance for establishing, in the unrolling of the signifiers,

repetition and scansion.

We are now going to show another essential paradox: the cutting of a Moebius strip,

along its length, produces a surprising effect that also has served for a definition of this famous

strip. This cut does not produce two pieces detached from one another; it describes the path of

an interior eight with a single turn and destroys the structure of the strip.

15

It remains a single strip, two times longer and bearing four half-twists, but which, this

time, has an underside and topside. It is a bilateral surface with two edges, resembling an

ordinary strip. The essential characteristics of the Moebius strip have disappeared.

This disappearance of the Moebian structure by means of the cut, without destroying the

physical object in its unity, allows us to reduce the Moebius strip to its cut. Moreover, since one

only makes a single turn with the scissors, the cut defines a path which is that of the interior-

eight (in dashes in the drawing), thus we somehow displace into the movement of the scissors

itself the characteristics of the Moebius strip. In the drawing, if we pay attention to the

discontinuous movement that edges the space of the cut, we can see again in the void born of the

cut a Moebian surface. Lacan gives a demonstration of it in L'Etourdit. Here, it is only a matter

of a "monstration," of an effort to evoke the spatial support of the Moebius strip. Let us note

before concluding that it is on this paradox that Lacan bases one the central notions of the

analytic cure: that of interpretation. The axiom "interpretation is the cut," allows us to discern

how this type of intervention on the part of the analyst discovers the desire of the analysand,

masked in his own dire.

The fact that the cutting of a Moebius strip makes a strip with four half-twists appear,

which is not Moebian this time, introduces a final characteristic of the Moebius strip. To create a

Moebius strip the number of half-twists must be odd. In Moebius's original notes, the first strip

drawn is a strip with three half-twists. And Lacan, in the course of his theoretical elaboration,

makes use more and more of the strip with three half-twists. This pregnancy of the Moebius

strip with three half-twists can be explained, but it remains surprising.

In fact, at the moment of the putting-flat, in the drawing where one effects it with straight

lines, it appears that the strip with one half-twist is drawn as a strip with three half-twists of

which one is toward the left and two are toward the right, or visa-versa. Thus, the three is

already present; we therefore prefer to draw a strip with three half-twists leftward or rightward,

identical in any case:

The presentation of the material, physical object that is the Moebius strip, can appear a

little unordered: the presented phenomena have not found their place in a formalized and

complete theory. It is not impossible that the problems posed will remain without a solution.

The character of subversion of everyday space that the Moebian surface puts to work cannot, in

fact, be reduced insofar as it is a question of a real that precisely has not yet found its sense.

16

On these multiple paradoxes Lacan suspends different notions, depending on whether he

wishes to reunify two separate concepts, or wishes to represent a certain type of relation between

two notions. Often, for example, he supports a concept by one definition of the Moebius strip;

then, by making use of another, he overturns our understanding of this concept. In doing so, he

give a logical leap to notions identical to the topological leap that consists in seeing in a drawing

the path of an interior eight, the putting-flat of a Moebian surface, or an illusion of perspective.

Thus he is lead to consider the drawing itself of the Moebius strip as a writing that

situates a real, that writes a matheme. A drawing is a matheme in the sense that it is transmitted

as it is, beyond the different effects of sense that it can produce. This notion of a writing gives

birth to a usage that Lacan expresses crudely, apropos of the Borromean knot, in his seminar

R.S.I: "we must use it stupidly,"2 which is to say, somehow, we must not too much concern

ourselves with the problem of topology's epistemological status.

Let us take up again some different usages. Saussure had supported the dichotomy

between signifier and signified and the force of their relations, nonetheless arbitrary, with the

two sides of a sheet of paper. Lacan took up the same metaphor when, in the seminar on

Identification, he supports two laws of the signifier with the Moebius strip: "A signifier cannot

signify itself" and another aspect of this law: "a signifier represents a subject for another

signifier."3 In question here is a symptomatic Lacanian topological practice. The Moebius strip

in fact subverts this signifier-signified opposition inscribed on the two sides of a sheet of paper,

since the topside and underside continue one into the other. The temporal turn, the additional

turn that we must make to the underside to return to our point of departure on the topside, allows

us to redefine some relations between signifier and signified, which, if they still remain arbitrary,

are nonetheless marked by this paradox. We will begin with the following commentary: locally,

at each instant of our progress on the strip, two sides are distinguishable. To this extent the

signifier and the signified are opposed, but in fact their difference is only supported by a

temporal factor. A signifier signifies something at a given moment, in a certain discursive

context, but one cannot give a signifier its signified at the same instant. The signifier never stops

slipping to the underside and, finally, once a complete turn has been effected, it is another

signifer, at the place that at this time defines the first. A signifier only ever returns to another

signifier; it represents a subject, for another signifier.

A contrario, a signifier cannot signify itself. This law is intuitively perceptible in the

repetition of a signifier; it is never anodine, nor deprived of sense. Thus, in the sentence "a man

is a man," it is felt that between the two words "man" there is a splitting (partage) of an identical

signified. A meaning (signification) of the sentence imposes itself, whether it be a matter of a

tautology, or whether it be a matter of differentiating apropos of the term "man" the general

concept and the isolated individual. The first "man" is not the same as the second. Between the

two is necessarily inscribed a difference, a space. Lacan supports this difference with the line of

the interior eight. The word is repeated, the buckle closes on itself, but however small the space

left, there is always between the two circles the space of a Moebius strip, and, because of this,

one finds, at the center of these two circles, a void. If the interior eight is seen in space, the

interior of the circles is empty. At this place, there is no surface.

Not only is there no join between signifier and signified, as in a relation "A=B," of

mathematical or logical equivalence, but their relationship is constructed around a void, which is

2 Séminaire de Jacques Lacan du 17 dec. 1974, RSI, published in Ornicar n. 2.

3 Séminaire du 9 mai 1962, Identification, unpublished.

17

that of reference. It is necessarily only for another signifier that a first signifier means (vouloir

dire) something, and the meaning (signification) is always marked by the void it encloses.

In this part of Lacanian teaching, the status of the Moebius strip is defined as "a model of

a transcendental aesthetic"; later, he speaks of it more simply as "an intuitive and im-agin-ative

support."4 He uses it in this way to illustrate the trajectory of repetition.

In the seminar on "the logic of the fantasy," the "topology of the return of repetition"

(2/15/67) is inscribed by Lacan on the line of the interior eight. This parallel is supported by

several traits that Lacan brings to light one after the other.

The doubling of the circle, which buckles itself after the second turn, leaves a trace: that

of a crossing, an above-beneath. It is also what allows the subject to exist. Thus in the

repetition of an act, of a behavior or a symptom, there is a trace: "what is repeated in the

repeating is found at the origin, this trace, which based on this fact, from then on, marks the

repeated as such" (2/15/67). This trace resembles the trace left by the line in its return over

itself.

This buckle is also the drawing of the putting flat of a Moebius strip; between the two

circles extends the Moebian surface. The retroaction of one buckle over the other still delimits a

difference between the one and the other, a space. On this difference, Lacan supports the

progressive effect of repetition. Although it repeats, the element is not the same; this makes felt

the progressive effect of what one calls regression. However, regression, because it is a

repetition, is precisely not the same thing as what it repeats. Between the repeated and the

repeating, there is the Moebian space, inasmuch as it reveals an element that is unmeasurable,

and uncountable, but present structurally as a fundamental support, although it remains ignored.

It puts us on the path of this "one-in-addition, one-too-much (un en plus, un en trop)" that we

forget to count because it is only defined starting from the void and time. In question is what

Lacan describes as "this unmeasurable element that is called the one-in-addition, the one-too-

much--desire" (2/15/67). This drawing thus illustrates the material on which analysis will

operate: repetition, manifestation in the cure of a desire. The act of the analyst will aim at

making this space felt.

Let us remark that beginning with this drawing of the interior eight a notion finds itself at

once decomposed into diverse acceptations (regression and progression) and unified as a

concept. The multiplicity of readings of a concept is accorded its true richness; it is not that a

concept has several senses, it is that it is the unique representative of a complex material

analyzable in several effects.

In this line of the interior eight, we read repetition and the difference of the repeated from

the repeating. Once we have recalled its "put-flat" aspect, we can evoke the point of self-

crossing as a stroke (trait) of recognition. Finally, this drawing reveals the ignored space of the

surface of the strip, which is related to desire. This manner of bringing an "ignored" to light,

which was nonetheless always there in its effects, is parallel to the unconscious's mode of

existence. Thanks to the cure and its apparatus, there is repetition as putting-flat. It is for the

analyst to read there, thanks to a certain immersion, unconscious desire, until then ignored as

space . . . The relationship exposed in this fashion between topology and psychoanalysis--is it

still metaphoric, or is it a question of an "intuitive support"?

Beginning at this degree of rapprochement something breaks down in the formulation. In

fact, it is the status of topology as intuitive support that is put in question. To pose this

utilization of the Moebius strip as metaphor, or even as didactic, seems to me unacceptable.

4 Séminaire du 15 fev. 1967, La Logique du phantasme, unpublished.

18

Lacan tends to reduce the metaphor; it is not necessary to pose it because there is an equivalence

between the one and the other. Between topology and the analytic experience are established

some relations that the words "intuitive support" do not define, or, rather, the intuition refers to

topology as the mode of approach of this geometry. Henri Poincaré, the great topologist of the

beginning of the twentieth century, defines it as follows:

"What interests us in this 'analysis situs' (a name given to topology at the beginning of its

existence) is that it is here that the geometric intuition truly intervenes."5 He later adds that this

intuition is of another nature than "the algebraic intuition."

Intuition, under the pen of Lacan, refers to the qualities proper to topology as a global

apprehension of space. Psychoanalysis, as the bringing to light of the structure of the

speakingbeing, stages the space itself wherein topology develops (enchaine) its phenomena.

It is in this context that Lacan links one of the absolutely essential notions of the analytic

practice to another paradox of the Moebius strip. As we have already sketched out, he founds

interpretation, the analytic act par excellence, on the cutting of the Moebius strip.

At the center of the Moebius strip, in the direction of its length, one can with a single cut

(trait) of a pair of scissors, trace an interior eight that divides it without cutting it into two pieces.

The structure of the surface changes without modifying its material, its physical consistency.

This cut is the act.

At the same time, at the moment of the utilization of the scissors, because it describes in

its progress a circle, one can say that the signifier is equal to itself. The act is thus equivalent to

its sense. This equality can be exemplified by this sentence: "the fact that I walk signifies that I

walk" (2/15/67). Such a condensation defines the true act. In this moment of the cut, because it

persists in tracing a double-buckle, the subject of this act remains divided. Lacan comments as

follows on the sentence just cited: "For the fact that I walk to become an act, the fact that I walk

must signify that I walk as such or that I say it as such." The act is in itself the double-buckle of

the signifier. "One could say," Lacan continues, "and this would be to deceive oneself, that in its

act the signifier signifies itself; we know that this is impossible. It is no less true that it is as

close as possible to this operation" (2/15/67). In the act, if the subject is equivalent to his

signifier, he remains no less divided. The Moebius strip in its cutting illustrates this series of

paradoxical relations; it allows us to evoke these different paradoxes of the act.

To the extent that it is analytic, the act must be situated in language. It finds its efficacy

on the side of the signifying equivoke. Thanks to this, a single proffering of the signifier can

make felt two turns, two times the opposition signifier/signified. One sees how this operation

detaches the signifier from the signified, to make the signifying chain appear as enclosing a void,

a space, that of desire as unnamable. The trajectory of the pair of scissors creates a void that, as

we have seen, is again a Moebius strip. Interpretation is the operation of the cut; it points to

desire. It is situated in the "field of desire,"6 in the space left between the two circles of the

double-buckle. However, this cut has changed the topological structure of the strip. After the

cut, we have a strip with two faces: if the act is repetition as interior eight, it remains no less true

that it produces some effects of structure. It makes appear the space of the desire of the subject,

all in destroying this space at the same moment. The cut in time where it is effected shows the

surface of the strip. It is a question of this time of which Lacan says that it is ignored and

uncountable as such before the operation. However, this operation has made the structure of this

space disappear: an effect of fading, opening and closing of the unconscious, missed encounter,

5 Henri Poincaré, Dernières pensées (Paris: Bibliothèque scientific, Ernest Flammarion, 1913).

6 Jacques Lacan, "The Four Fundamental Concepts of Psychoanalysis."

19

aphanisis; there are for the subject always effects of this order. The space shows itself in

disappearing.

Lacan supported the analytic situation with the interior-eight, sometimes as a perimeter

rolling back on itself, sometimes as a surface with one face, sometimes as a trap of duration,

sometimes as place of a paradoxical cut. In the same order of idea, the usage made by Lacan of

the cross-cap7 also holds to the paradoxes that the Moebius strip lays out for us.

The Moebius strip keeps, in fact, in our space, this status of a representative

(représentant) of the unrepresentable. This paradoxical function is a necessity, because of the

debility of our perception and of our intuitive imagination of space.

It is useful to recall, before bringing in other topological objects, that only the Moebius

strip is really constructable and manipulable as a unilateral* object. Thus this object allows for

the representation of an abstraction knotted to a real. It allows us to acquire an assurance on

which Lacan will draw later when he introduces the Borromean knot.

7 Cf. Chapter 4.

21

Two linked torii

Chapter 3: The Torus

In the text of the "Rome Report" of 1953, "Function and Field of Speech and Language,"

we find under the pen of Jacques Lacan a reference to the topology of the torus.

To speak of the nature of the subject proper to the unconscious, Lacan situates himself

thusly:

"When we wish to attain in the subject what was before the serial games of speech and

what is primordial in the birth of symbols, we find it in death, from where its existence takes all

that it has of sense. It is as desire for death, in fact, that it affirms itself for others; if it is

identified with the other, it is in freezing him in the metamorphosis of his essential image, and no

being is ever evoked by it except among the shadows of death."

"Saying that this mortal sense reveals in speech a center external to language is more

than a metaphor and manifests a structure; this structure is different from the spacialization of the

circumference or the sphere where one is pleased to schematize the limits of the living being (du

vivant) and its mean (milieu): this structure corresponds instead to the relational group that

symbolic logic designates topologically as a ring."

"In wishing to give to it an intuitive representation, it seems that, rather than to the

superficiality of a zone, it is to the three dimensional form of the torus that we must take

recourse, inasmuch as its peripheral exteriority and its central exteriority constitute a single

region."

This citation is accompanied by a note from 1966 that reminds us of a usage of topology.

Our attention should be drawn to several terms that enclose some notions essential to the

psychoanalytic cure. At issue in this "what is primordial to the birth of symbols" is what Freud

calls the identification with the father of the primal horde, with the archaic father, with the dead

father. At the dawn of the birth of the subject there is identification. This is the great question

that the torus and its topology allows us to pose in clear terms.

From the "Rome Report" in 1953 to his séminaire of 1976, Lacan refines the

formalization of this question. At first, he speaks of an "intuitive representation," and then the

model gathers so much strength and conviction that once again the formula "topology is

structure" imposes itself. Lacan does not have to articulate the relations between identification

22

and the turning inside-out of the torus; the one is equivalent to the other. But we are getting

ahead of ourselves.

The torus gives us an apt representation of this relational group for which the center and

the exterior are one and the same space. The surface of the torus envelops an interior space and

detaches it from the exterior, at the cost of a center that remains exterior. A torus is defined as a

surface without edge, and is equivalent in this sense to the sphere, but its center is empty. The

best physical approximation is the buoy or the inner-tube. A ring can also figure it if one takes

into account the material sameness of a cup with its handle.

The lines drawn in this representation are the folds in the surface. We must imagine a flattened

buoy.

We can obtain a torus by combining a circle with a circle (the Cartesian product of S

S).

One circle is called the soul of the torus, the interior, or more precisely, the emptiness in

the interior of the torus. The other is a little circle or meridian circle.

We can construct a torus from a cylinder: it suffices to start by transforming it into a

handle by curving it lengthwise and joining its ends.

23

The torus can also be constructed from two crowns (couronnes): it suffices to join them

along their edges.

We can represent this operation beginning with a rectangle, on which the arrows indicate

the directions of the joins: we first obtain a cylinder, and then, by joining its ends, we obtain a

torus.

These latter drawings allow us to define the torus as an edgeless surface that two cuts do

not make disappear and do not divide (at issue are two dual cuts* that meet at a single point).

24

In this instance, Lacan defines the torus as an organization of the hole.1 He expresses

himself as follows: "These two holes isolated on the surface of a sphere are those which, lined

up with each other (rejoins l'un à l'autre) and then very much extended and conjoined, have

given us the torus." Around the interior hole "buried in the surface," and the axial hole, which

Lacan calls the "current-of-air hole," a surface is organized. It is a holed interior that misses the

center.

The torus, an edgeless surface, delimits an interior and an exterior with this particularity

of having an "exterior" center. This center is holed; it allows for a knotting.

Lacan utilizes this surface structure in reflecting on the great question of identification.

In 1976, he expresses himself in this way: "What relationship is there between our having to

admit that we have an interior, which one calls what one can, psychism for example--we even

see Freud write endo-psychism, and it does not go without saying that the psyche be endo, that

this endo must be endorsed--what relation is there between this interior and what we currently

call identification?"

Identification is an answer to a question: how does something from the exterior become

interior, exterior, and still central?

Let us recall that for Freud there are three identifications:

--The primordial identification, said to be "with the dead father"; this identification

results from the love dedicated to the father; it is responsible for the introduction of the symbolic.

--The identification with the unary trait, where it would not at all be a question of love;

Freud puts this identification at the foundation of the constitution of masses. Lacan illustrates it

with the example of the Führer's mustache.

--The identification that implies a participation: it is pinned to the term hysteric, the

identification with the desire of the Other.

How does the torus account for identification? This object offers a support that allows us

to perceive the implications of this term that has become commonplace without, however,

becoming more explicit. Lacan begins by drawing on a very particular phenomena of toric

transformation, "the turning inside-out." But before taking on this notion, we must clarify

certain terms that Lacan utilizes in reference to this surface, which allow us to establish the

relation that unites desire and demand.

Let us designate with letters desire (d) and demand (D). On the surface of the torus exists

a trajectory following a meridian circle. It closes itself in a buckle. On the other hand, if this

trajectory around the torus misses its starting point, closing itself without intersecting itself, the

loops are multiplied and the trajectory also completes a long longitudinal turn. It also turns the

length of the torus's soul.

1 Séminaire on L'Identification, May 23, 1962.

25

This trajectory encircles the torus's central hole in a manner that we could call

pointillistic. In doing so, it describes a supplementary turn around the hole. This additional turn

is forgotten; and anyhow, how are we to count it?

The turns succeed one another and are counted; they are identical, without the possibility

of counting the additional turn completed around the central hole. Here is illustrated demand and

its fundamental repetition, a repetition effected in the misrecognition (méconnaissance) of its

expression of a misrecognized albeit essential desire. We have thus defined the "one-too-many"

(l'un-en-trop): this forgotten circle of longitude, which is properly speaking what Lacan names

desire.

This progress allows a fundamental aspect of the misrecognition of desire to appear, one

summed up by the importance of the central hole: demand repeats itself and designates the object

as lacking. Described in this way, this object is always missed, with a missing that is nonetheless

structural, tied to the progress of demand and necessary to its repetition. This central hole is also

in communication with the exterior, and Lacan utilizes these properties to define two distinct

positions of desire in relation to demand.

On the one hand, it is "beyond" demand--"it transcends it, goes farther, and is in this

regard eternal." Demand, in articulating desire to the conditions of language, expresses itself

through signifiers that betray its true aim. In a way, the missing is fundamental to demand; the

figure of the object (a) is profiled in the central void. Lacan later describes this object as taken

between the three rounds of the Borromean knot, RSI. Remember that these three rounds are

torii; they have the consistency of the cord, "these are gut-torii (tores-boyaux)."

26

On the other hand, desire is "within"; the central void communicates with the exterior.

Demand recalls the radical "lack in being" that subtends desire. "Desire hollows itself within, in

that, as an unconditional demand for absence or presence, it evokes the radical lack in being in

the three figures of the nothing, which founds the demand for love, of hatred, which tends to

deny the being of the other, and of the unsayable, which ignores its request." We see appear in

this sentence the three passions that Lacan situates at the level of being and not of the object. It is

a question of hatred, love, and ignorance.

27

The turning inside-out of a torus

The demand that circles, borders the "lack in being," the nothing of the universe, creates,

by its repetition itself, a surface separating an interior and an exterior. This structure accounts

for the birth of a subject of the unconscious. The within of demand also introduces us to what

there is in it of a knotting to the other. Love, hatred, and ignorance concern the other in his being.

Remember that the subject who enters analysis puts himself in the position of "he who is

ignorant" of what he says.

Lacan supports the neurotic dialectic between the subject and the Other with the knotting

of two torii. In this knotting, the desire of the one is isomorphic with the demand of the other,

and the central void serves only for the knotting of the two torii.

(Cf. the photographs.)

28

At issue is an essential articulation for entering the problematic of identification. An

object demanded by the other, the mother, the primordial Other, finds itself in the position of

object of desire for the subject. This articulation allows for a new envisioning of mother-child

relations, which are relations of dependence, certainly, but do not belong to a symbiotic

confusion or infra-verbal communication.

The signifiers that become unconscious are tied to signifiers witnessing to the moment of

access to language. Here is fixed the structure of the fantasy, within demand, which is the mode

of appearance of the Other. The fundamental fantasy situates (cerne) the moment of separation

from the real experience, linked to the present demand of the other, and its hallucinatory

revivification. It constitutes the separation between the object that fills and the sign that

inscribes at the same time the object and its absence. At issue is the putting in place of the

conditions of speech, the structure of which gives meaning to the aphorism: "the unconscious is

the desire of the other."

Thanks to the notion of the turning inside-out of the torus, Lacan once more makes his

thoughts clear. Turning the torus inside-out consists in making pass to the exterior the face

which was on the interior. This operation can be effected thanks to a cut, to a hole. We then

observe an astonishing phenomenon: the circles of demand and desire exchange positions. The

meridian circle becomes a circle through the soul of the torus. However, the central hole remains

the same.

At the physical level, the experience is simple, but its writing or drawing is very difficult,

because the conventional lines of the curves disappear. This operation brings to light the purely

conventional aspect of drawing.

At the end of the process, the torus remains the same, but its writing is different.

The photos depict the whole of the operation, but we are going to study in detail these

difficulties of the writing of it.

The turning inside-out illuminates how the drawing no longer maintains the illusion of a

representation of the real that escapes writing.

We must begin with the classic drawing of the torus in which the lines represent the

curvings of the surface: on this drawing we make a cut.

For the sake of greater simplicity, a hole will suffice.

29

A hole in the surface, a rupture, is of another nature than the central or interior holes of a

torus.

Then we begin to pull the surface through the hole.

We turn the torus inside-out like a glove, or, better yet, like a poncho and its lining.

We see that the space of the central hole is going to become the internal space.

It is a question of a turning inside-out that remains in the domain of plungings

(plongements). In general, mathematicians turn the torus inside-out at the cost of an intersection

of the surface and enter into the domain of immersions.* The trajectory then passes through the

Klein bottle, which is a unilateral continuous surface, and therefore puts interior and exterior in

communication. The cut is here much more economical for operating the same suppression of

the edge.

Moreover, in the field of psychoanalysis, the cut has different operatory dimension than

the intersection, since it refers to interpretation in the cure and, more generally, to the act of

speech.

We can also make the following monstration:

The hole is open to show again what becomes of space. We can materialize this outcome

by knotting a cord to the initial torus, which is what the photographs show.

We then obtain another writing of the torus:

This drawing is represented as a sphere with a tunnel, with two openings, but then the lines that

represent it are not creased. Lacan calls this presentation the "torus-cudgel" (tore-trique). By a

rotation of a quarter of a turn on the part of the observer, we fall back into the classic drawing of

30

the torus. At issue is the same torus that we began with, but now its internal face is on the

exterior.

In these representations, we find again the importance of the twist. The subject of

perception makes a quarter turn, as is symbolized by an eye in these drawings:

We see how the twist is an extrinsic characteristic of the surface, which only appears to

an external gaze. (This question will be addressed in a whole chapter of this study).

Let us only note that it is a question of a quarter turn, of a half of a half twist. We have

seen how the half-twist is the unit of counting in our space (Cf. the little spoon).

A thread that we have now represented in the following drawings materializes the

transformation of the meridian circle into the circle of the soul. A circle of demand becomes a

circle of desire.

31

This process accounts for identification, for the transformation of an object of love into a

trait of the Ego, a trait with which the Ego identifies, or, rather, identifies its desire.

The drawings also allow us an approach to the scenario of the turning inside-out of two

linked torii. It suffices to give to the thread the consistency of a cord or of a tube; we see how

one can turn a torus inside-out and see the torus linked on its interior.

Or, visa-versa, the interior torus, from the moment when one turns inside-out the torus

that encloses it, will be knotted to the first, as the place of the neurotic dialectic with the Other.

(Cf. above).

We see now why a mechanism like this is important for accounting for the process of the

development of mother-child relations, and how identification is an outcome of this attachment.

We remember, for instance, how Melanie Klein established the mourning necessary for the

separation from the primary object and the structuring role that it brings into play in the

"depressive position."

It is possible for us now to return to the three Freudian identifications, to envisage them

in terms of three scenarios of linked torii turned inside-out;

--a single cut, a single turning inside-out.

--a cut in the torus that we have just come to in our drawings and a turning inside-out;

--a cut in each torus, and two turnings inside-out.

With these schematizations, Lacan tries to support diverse Freudian identifications. He

asks the following question:2 "How are we to designate in a homologous fashion the three

identifications distinguished by Freud: hysterical identification, the loving identification said to

be with the father, and the identification that I will name neuter, which is neither the one nor the

other, the identification with a particular trait, with a trait that I call no matter which, with a trait

that is only the same? How are we to distribute these three inversions of torii, homogenous in

their application (practique) and which, moreover, maintain the symmetry of one torus with

another?"

In the following seminar, Lacan takes up this problem in different terms: he endeavors to

knot these identifications with the function of the unconscious.

It is in fact appropriate to situate the unconscious and its effects of speech in this

problematic. "The torus can be cut into a double Moebius strip, and it is this that gives the image

of the link between the consciousness and unconscious."

Elsewhere Lacan declares: "consciousness and the unconscious are supported and

communicate by a toric world."

Toric space has many relations with the Moebius strip, but this not immediately felt.

By wrapping a Moebius strip with four half-twists all the length of its ring as a doubled

covering (revêtement), we in fact find again a unilateral Moebius strip with a half-twist,* which,

moreover, founds what there is of a hole. The hole is Moebian inasmuch as, if a surface has a

topside and an underside, it has two holes, that in the topside and that in the underside. The

Moebius strip, since it joins the underside and the topside, is a hole.

2 Seminar of November 16, 1976, Ornicar nos. 12-13, Le Seuil.

32

Some such layout is primordial for the functioning of the unconscious, for it allows us to

situate its principal characteristic. It is a question of a hole, but of a different nature than the

axial hole in the torus, although it entertains with it some particular relations.

How can a unilateral Moebius strip be cut from a torus?

The torus is a continuous, bilateral, orientable surface without an edge. One cannot pass

from the interior to the exterior without crossing an edge, a frontier, without creating a

phenomenon of rupture. It is therefore mathematically impossible to inscribe a unilateral and

non-orientable Moebius strip on its surface. To inscribe it, there must be a particular process

with specific events, which we are going to trace: one first makes a double-buckle shaped cut, an

interior eight which is also seen to be equivalent to the edge of a Moebius strip.3 Thanks to this

process, one turns around the axial hole twice.

One then obtains a strip with four half-twists, even numbered, and thus a bilateral strip

with two edges (materialized here by a different writing of two edges.)

With a slight transformation of the cut, we can establish a half-twist, often forgotten. The

twist at the edge of the fold accounts for the axial hole in the torus.

In the first drawing, this half-twist is in fact reduced to a point. This reduction owes to

our difficulty with the mental representation of the torus itself. Without this half-twist nothing

any longer distinguishes the torus from the sphere. The half-twist is here the expression of the

structure of the torus. However, the sticking together of these two half-twists one on top of the

other allows us to create a Moebius strip. By these two half-twists, the torus "cut out" in this way

is revealed to be the "combining as two thicknesses" (revêtement à deux feullilets) of the

Moebius strip. The following drawings illustrate the process step by step.

Let us start again beginning with the cut:

Because of its suppleness, we can reduce the surface of the torus to the space bordering

the cut.

Two folded ends remain.

3 Cf. Chapter 2.

33

The surface crosses over itself.

We can then reduce the folds to the half-twist that they represent, then put the two

buckles en miroir, and bring them together. Finally, we see how a sticking together of these two

buckles, all the length of their surface, creates a unilateral Moebius strip with one half-twist.

This is what one calls the "combining as two thicknesses" of the Moebius strip.

Moreover, this trajectory is the reverse of the cutting of a Moebius strip. The latter

creates a bilateral strip with four half-twists. At issue is a strip that can be cut out of a torus.

This combining as two thicknesses is the operation accounting for how the unconscious

and consciousness "communicate by a toric world." This happens, for example, in what Freud

calls the "double inscription": the same memory is inscribed in an unconscious and conscious

chain. The Moebius strip gives us the structure of the signifying chain and the cutting effects of

certain words, as interpretation.

34

This relationship of the Moebius strip with the structure of the torus links a whole series

of questions to the moment of the first identifications and the learning of language (la langue). A

contrario, psychosis and its possible possible psychoanalytic treatment perhaps finds a new

direction thanks to these formulations. What, in effect, is interpretation in the psychotic

structure? Is it not rather the matter of a construction?

We might ask ourselves if it is a matter of a cutting of the sticking together of a

combination of two thicknesses? The structure itself of language (la langue) is laid out here (est

ici sur le métier), and topological objects give us the means of formalizing it.

37

Chapter 4: The Cross-Cap

The drawing of the cross-cap, as we find it in the teachings of Lacan, dates from 1890. It

appears for the first time in the work of Van Dyck.

This closed surface, without an edge and unilateral,* is an abstract object whose

mathematical definition preceded its representation; it is a presentation of the projective plane.

The projective plane is the space in which projective geometry is conceived. It is defined

by the adjunction of a point called "by convention" at infinity to the Cartesian plane (the three

coordinates x, y, z, give us its calculation).

Toward the 1880's, mathematicians like Felix Klein and Schäfli recognized in this

abstract object the quality of being unilateral, or non-orientable.*

Thanks to Van Dyck, we find the projective plane represented for the first time in the not

very explicit drawing below:

Speaking rigorously, this drawing is an immersion* of the projective plane into our

ordinary space. The projective plane itself must be conceived of as a space on the same basis as

our ordinary space.

The surface is continuous and accounts for the infinity of the space of the projective

plane (the term "continuous" means "without edge"). It is also unilateral, that is, its interior is

continuous with the exterior. It is a closed surface, but one that does not delimit space.

38

The lines of the drawings are not borders, but lines of curves and an intersection (the

vertical line in the middle of the egg in the photo).

The intersection is a phenomenon that escapes our everyday, intuitive perception. Its

difficulty arises from the fact that cross-cap as object is not physically realizable in our ordinary

space except at the cost of this intersection. We must remember that there is nothing

extraordinary about this, since it is a question of another space that should allow us to perceive

the laws of our ordinary space á contrario. We submit to these laws as if they were self-evident.

The cross-cap has the merit of bringing this evidence into question.

Thinking its structure requires an effort of the imagination that allows for particular

phenomena like the intersection or "double-line"; two surfaces intersect, but it is a question of an

abstract intersection that is situated nowhere. Lacan calls it a "pseudo-intersection."

Two surfaces intersect, passing through each other along an arbitrarily drawn line. If we

were to represent a little ant walking on one of its surfaces, it would follow its trajectory without

knowing that another surface had crossed the first.

The cross-cap is a sphere creased by a line of intersection. On the one hand, it is closed

like a sphere; the line is a line of the fold. However, at the top of the drawing, the cross-cap is

closed in reversing the two thicknesses. They intersect along a double line.

This structure possesses two points that are particularly difficult to think.

The first is that where this double line stops. The surface re-closes itself at this point.

39

When one cuts away this particularity alone, one obtains what we call the eight-cone: a

cone whose surface intersects itself. The line of the base of the cone describes a lengthened

figure eight.

The second point, which puts our imaginary to the toughest test so far, is the point of

departure of this line of intersection. It represents the point of the impossible to think, but not to

write. It is the point off the line (point hors ligne), the point where one passes locally from a

situation where two surfaces are posed one over the other to the situation where the same two

surfaces intersect. One could say that this point sums up the set of the characteristics of the

cross-cap, transforming a bilateral object (two thicknesses posed one over the other) into a

unilateral object (these two thicknesses exchanging places).

This intersection has as its consequence the putting in continuity of the external face with

the internal face.

If we consider the cross-cap as pure surface without thickness, the interior of the sphere

communicates with the exterior. In the same manner, the Moebius strip puts the topside in

continuity with the underside.

We can now write the different trajectories possible for our ant: the four possibilities are

presented as follows:

__________ topside in front

40

underside behind

. . . . topside in front

_ _ _ _ topside behind

Lacan calls this figure a bonnet croisé, a translation of "cross-cap."

He also at times employs the term "bishop's miter." He then alludes to the real form of a

bishops miter creased toward the middle, a miter of which the bottom would be closed in a

spherical manner, and of which the intersection would not be a banal effect of sewing, but

effectively the passage from behind of what was in front and visa-versa.

front profile

The Boy surface is another immersion of the projective plane, constructed on a Moebius

strip. While the cross-cap is constructed from a Moebius strip with one half-twist, the Boy

surface is constructed from a Moebius strip with three half-twists, as is shown in the following

drawing: it is not a question of a demonstration, but only of a drawing that renders perceptible

the construction of the Boy surface.

41

But let us return to the definition of the projective plane itself.

In mathematics, the projective plane is first of all a structure of organization such that

each point of the sphere is associated with another diametrically opposed point.

Hence, a point situated at place (a) will correspond, will be identical to, a point situated at

a place diametrically opposed to it (a'); it's the same for (b) and (b'). We in this way describe a

circular relation, a structure where the trajectories entangle themselves in a star.

Since we are also dealing with a continuous structure, there is no possibility of marking

our starting point.

We can construct a Moebius strip beginning with this definition. We start by giving a

simple strip with two faces on which we make correspond to one point a diametrically opposed

point. We then effect a series of constructions that transform the simple strip into a Moebius

strip.1

The following drawings show this construction in reversing the trajectory. The reader

will better follow the disappearance of the Moebius strip and the fate of the point marked A on

the two edges of the cut.

1 Cf. Chapter 2.

42

The problem of twists, of an even number, which maintain the two faces of the simple

strip, will be dealt with in the next chapter. It is a question of a characteristic that does not

modify the intrinsic nature of the surface.

On the cross-cap, this equivalence between diametrically opposed points creates an

entanglement that still leaves an unimaginable point at the center. Lacan situates it thusly: "This

circular relation must be perceived as a sort of rayed intersection concentrating the exchange of a

point with the opposed point on the single edge of this hole, and concentrating it around a vast

central intersection that escapes our thought."2

As we can guess from the preceding drawings, one can construct a cross-cap from a

Moebius strip. A contrario, one can define a Moebius strip with a cross-cap; it is a holed cross-

cap.

To obtain a cross-cap, we must first crease the Moebius strip lengthwise. Then, at the

place of the twist, whether it is to the right or to the left, there is a problem.

The two sides of the fold must cross each other (the dotted numbers on the drawing

indicate their continuous movement onto the underside of the surface); the double line or

intersection inscribes this structure of the crossing of surfaces:

2 Seminar of June 6, 1962, unpublished, L'Indentification.

43

All that remains then is to close the whole (l'ensemble) of the surface, which is to reduce

the hole to a point.

Before the closing of the hole, this object is a Moebius strip; after this same closing, it is

a cross-cap. This is why one speaks of the cross-cap in reference to a Moebian space.

This point remains irreducible. As soon as it disappears as a hole, what becomes of the

surfaces is a mystery, since nothing any longer makes a border.

This is what permits Lacan to turn the question inside out. The hole, this hole, becomes

the construction's point of departure: the cross-cap is an organization of the hole. "It conjures

away the hole."3

"It is a surface that in some way has taken the place of the hole, a surface where one

divines it, although the important thing for the structure of the hole remains the central point,

where the line of pseudo-intersection begins" (L'Identification 5/23/62).

This manner of making the cross-cap appear gives to its surface an altogether particular

dimension. This central point, in fact, if we can subsume all the history of mathematics under

the "point at infinity," is also what permits Lacan to make a new usage of this topological object.

He is interested in the cut and the place of this point-hole in the effects of the cut.

This usage of the cut completely subverts mathematical discourse. The cut is an

operation. One can never say enough about the originality of this step.

This operative quality of the cut leads us understand Lacan's interest in the Borromean

knot (whose definition is supported by this operation of cutting). But we will come back to that.

We must first present in detail this cut on the cross-cap, whose characteristics permit an

economic and synthetic formulation of the analytic experience.

The cut has a particular relation with the central point of the cross-cap, as can be shown

by this series of drawings: a simple cut opens the surface without separating it.

But it suffices to make this cut turn around the central hole to divide the surface:

3 Seminar of May 23, 1962, unpublished, L'Indentification.

44

One then obtains two pieces:

--a Moebius strip that can be either left-handed our right-handed when it is unfolded (it is the

strip folded lengthwise).

--a disk, which possesses this particular point of the cross-cap. (The numbers refer to the number

of thicknesses).

It is a question of a bilateral disk, with a double line, as is shown in the following

drawing: it seems also to be related to an eight-cone, constructed on an interior eight.

The bilateral disk, with two distinct faces, bears the point essential to the cross-cap's

surface. The double quality of this object, by definition bilateral and with two faces, but

nonetheless bearing the point essential to the unilateral structure of the cross-cap, gives a

particular status to this disk.

Intuitively, one indeed perceives that it suffices, starting with the disk, to reclose the

interior eight on itself to once more obtain a cross-cap. It is this disk that Lacan identifies with

the object (a):

"It is in articulating the function of this point that we can find all kinds of felicitous

formulas that permit us to conceive of the function of the phallus at the center of the constitution

of the object of desire" (L'Identification 6/27/62).

45

The point of the impossible to think, but not to write, the point-off-the-line (point hors

ligne), is identified by Lacan with the .

For Lacan, the cross-cap is the "topological support that we can give to the fantasy"

(6/27/62).

The fantasy is a cutting of the cross-cap that detaches an object without specular image,

the central disk, from a Moebius strip.

To be more rigorous, we should add that this cut created the Moebius strip and the disk

bearing the central point.

The usage Lacan makes of this cross-cap figure is original; the cut is an operation that

does not function to underscore a definition but to provoke a transformation formalized as such:

at issue is the constitution of the fantasy. The formulation of the fantasy is written with this

remainder, the object (a), detached from a Moebius strip, which represents the barred subject on

the basis of this loss. We see how the operation of the cut synthesizes, sums up, the definition of

the subject in relation to the object, at the same time as their relations. The latter are formalized

elsewhere as "separation-alienation," written with the matheme: a.

We can now refer to the Schema R, drawn by Lacan several years before he defined it in

a note of 1967 as the laying out of a projective plane:

"Perhaps it will be of interest to recognize that enigmatically, but perfectly readably for

those who know what follows, as is the case for those who claim to be supported by it, that what

the schema R lays out is the projective plane."

"Specifically, the points of which it is not by chance (nor for fun) that we have chosen the

letters to which they correspond mMil and which are those that give the framework for the only

worthwhile cut in this schema (the cut miMI) well enough indicating that this cut isolates in the

field a Moebius strip."4

The Schema R is therefore a putting flat of the projective plane or the laying out of the

cross-cap in which two cuts are made: one is situated at the place of the line of intersection in the

classic drawing. This cut permits the delimitation of a surface where four points are

establishable and can then be assimilated to a square. On this square, the cut of the fantasy

follows the lines delimiting "the field of reality," marked R on the Schema.

4 "On the Possible Treatment of Psychosis," Écrits

46

The following drawings will show these two cuts and their outcome on the cross-cap.

The square of Schema R takes the form of a pouch, of a sphere, from which one

withdraws a portion.

One can then write on the surface the letters of the Schema R.

The cross in dotted lines evokes as much the re-closing of the cross-cap as the the twist in

the Moebius strip, miMI.

(Moreover, on the trajectory of the cut called "of the fantasy," there is a displacement of

the above-beneath.)

Based on this, the field of reality in the Schema R is directly readable on the cross-cap, as

the cut in the form of an interior eight. In question is an operation that effects the fundamental

fantasy: at once separating and joining a subject and an object. At the same time, the fantasy

constitutes the framework of our perception of reality.

We here see again the utilization of the Moebius strip as an operation of the cut that

permits us to unveil structure. It renders possible a laying out that belongs only to speech. We

see how topology synthesizes, sums up, diverse aspects while taking them up in a same

synchronic, structural, access to meaning. It thereby permits us to bring to light the operation,

the functioning, of concepts between themselves, and it is in this that it is structure.

Fundamentally, it is a question of a taking into account of the operative effects of speech

as interpretation.

The transference is then the space in which this cut operates. (Let us say, rather, that one

must add a conception of the transference.) It is also on the figure of the cross-cap that Lacan

establishes the structure of the transference. The analytic situation is thus articulated by the same

object in the measure where its operation brings to light the fundamental fantasy.

In seminar XI, Lacan speaks from the place of the analyst in the transference,5 and says

how he situates it in the central place of the construction.

To support his development, Lacan takes up again his celebrated formula dating from ten

years earlier: "the unconscious is structured like a language." He poses the unconscious as an

effect of speech on the subject. It is then that is posed the question of situating in this

formulation the Freudian discovery of the importance of sexuality. Lacan makes use of the

Moebius strip and, hence, of the cross-cap, to knot these two aspects of the unconscious, that

which Freud brings to light in his study of "jokes" (les mots d' esprit) and that of the repression

analyzed in his "Studies on Hysteria."

To show the importance of this question of the articulation of discourse to sex, Lacan

begins by supporting his views a contrario on Jung's reflections. He shows how, to effect this

knotting, which Freud leaves as is, Jung comes to consider unconscious sexuality as a

5 Séminar XI, Quatre concepts fondamentaux de la psychanalyse Le

47

remanaging of archaic thought. (As has since been shown by Levi-Strauss's discoveries, it seems

that the signifier came into the world beginning with the sexual difference.) In this way, Jung

founds the notion of the archetype and then sacrifices to the recognition of the original, sexuality

as material of the unconscious. If sexuality is present in the unconscious, it is in the name of its

original importance in the constitution of language.

We know that Freud always refused this consequence, holding as essential the sexual

reality of the libido. We should remember that he went so far as to break with Jung, long

considered his successor.

Lacan takes this articulation of sex to speech up again and makes use of the Moebius strip

and of the space that it founds as a point of departure for making perceived the points of

conjunction and disjunction of the one with the other. He thus makes a first conceptual

distinction: he speaks of a field of unconscious development to refer to its language aspect,

distinguishing it from the reality of the unconscious, which is sexual. The Libido is point of

crossing that can be drawn on the surfaces enclosed by an interior eight: it is a question of the

point of junction between the two fields.

For the desiring subject, it is beginning at the point where he desires that the connotation

of reality is given to these perceptions. Hallucination shows it a contrario.

Due to the defiles of the signifier and their discontinuous nature, the demand articulated

in signifiers always leaves a remainder. Desire is metonymic and runs beneath the chain.

However, it is only it that insures the cohesion of the discontinuous elements that are the words.

It is for a desiring subject that the sentence closes itself on a sense.

Lacan adds that this point of junction "libido" is inscribed in the transference: one must

beware of conceiving of this formulation as subjectivized, representing an individual in his

functioning.

At issue is the analytic situation. For this drawing in which one lobe hides another,

depicting the Moebius strip, is here taken entirely in a surface. What the putting in place of the

transference effects is the crossing of this hidden lobe to the front, along a line of intersection.

48

It is a question already of the cross-cap, the structure of which we recognize.

Lacan gives this description of the operation:

"You can obtain the cross-cap beginning with the interior eight. Unite two-by-two the

edges as they are presented here, with a complementary surface, and close it. Somehow, it plays

the same complementary role in relation to the initial eight as a sphere does in relation to a circle,

a sphere that would enclose what the circle already would offer to contain. And indeed this

surface is a Moebian surface, its topside continuing its underside. There is a second necessity

that arises from this figure: to close its curve it must somewhere traverse the preceding surface,

at this point here, along the line that I reproduce."

Lacan adds: "This image allows us to figure desire as place of a junction of the field of

demand where are presented the syncopes of the unconscious with sexual reality."

"All of that depends on a line we will call desire, tied to demand."

"What is this desire? Do you think it is there that I designate the incidence of the

transference. Yes and no. You will see that the thing does not go without saying, if I tell you that

the desire in question is the desire of the analyst."

For the unconscious to unveil itself, for interpretation to operate with its effects of

alleviating psychic suffering, sedating conflicts, for the cut of speech to operate, the analytic

situation in its entirely is required, and precisely the desire of the analyst, which founds the

transference.

If this desire is noted at the central point of the cross-cap, it is because represents the

analyst as object (a), marked by the cut from the object and its loss.

The transference is founded on the desire of the analyst, on the cut that, for him, has

signaled the advent of desire and the detachment of the object.

The cross-cap is introduced as a space made necessary by the formulation of the cure. It

is important to take the time to establish this passage from the topology of illustration to the

equivalence between space and structure.

For that, it helps to clarify the formulation of the transference.

Lacan writes on the interior eight the places of the different notions we have taken up.

49

He situates the analyst on the line between (I), the point of fascinating identification, and

(a), the object, at the point of departure of the line, where he writes (T), basal point of the

transference. He then insists on this beyond of identification, of which we must remember that it

signals, for analysts on the other side of the Atlantic, the end of analysis. This beyond is defined

by the relation to and the distance of the object (a) from the big idealizing (I) of identification.

We again find the letters of the Schema R and the contour of the cut the fantasy puts in

operation. To let appear in the cure the place of the object is also to permit the fantasy and the

cut that it puts into operation between the subject and the object to be put in place. It is by this

operation that the subject can identify with what he desires.

We thus see how the cross-cap is a structure fundamental to the formulation of analysis

for Lacan, above all because it allows the showing of the operation effected by speech. It gives

support to the analytic act by marking the conceptual points between which it operates.

What does a psychoanalyst do? Lacan gives a first answer here.

Later, with the Borromean knot, he will give another theory of this set. But one can

already remark that he then situates the cut at a place still more central to the constitution itself of

the concepts in play.

The Klein Bottle

Chapter 5: From the Specular to the Non-Specular

There exists in general topology a phenomenon that, although patent, is still usually

misrecognized: the left-handed or right-handed twist in topological objects.

It is no exaggeration to affirm that that science was nonetheless created around this

phenomenon. For example, Galileo moves the twist of the sun toward the earth from the exterior

to the interior.

Moreover, life displays this twist at key points of its appearance. So it is for the double

helix of DNA, which accounts for the physico-chemical structure of chromosomes, but so it also

is for the umbilical cord, which is a triple braid (torsade) made up of an artery and two veins.

However, as a group, physicians, like mathematicians, care little for this phenomenon;

they evacuate it totally from their definitions, an oversight that is no doubt not usually due just to

chance.

Indeed, the importance of the mirror and of the reversal it effects in this set of facts is

primordial.

For the mirror, the object and its reversed image are identical. The right-handed twist

becomes a left-handed twist, but they are perceived as being the same.

Moreover, man finds in the reversed mirror image an illusion so primordial to his identity

that this right-left difference remains unrecognized.

Man is his reversed image, his enantiomorph, his specular image.

Lacan had to come along, with the importance he knew to give to this identification with

the image, to undo this feeling of identity between left-handed right-handed twists.

More precisely, it is thanks to the formulation of an object (a), without a specular image,

that we can conceptualize (penser) the image and its reflection in their originality and in their

effects.

Let us clarify the topology of this action (agencement): there are images that have an

entiomorphic image, which is to say a specular image.

Thus the Moebius strip presents a right-handed or left-handed twist:

In the same manner, there is a right-handed trefle knot and a left-handed trefle-knot,

depending on whether it is the edge of a Moebius strip with three half-twists to the right or to the

left:

When I say "there is," I refer in fact to a topological given: two objects are said to be

different on the condition that it is impossible to pass by continuous transformation from one to

the other. In this domain, where the forms are as supple as rubber, it is not everyday that we find

two forms of which we know with certainty that they are different. Now it is impossible to

transform in a continuous fashion a left-handed Moebius strip into a right-handed one. The

direction (sens) of the twist insures a radically different existence for each.

The mirror inverts the object it mirrors along an axis of vertical symmetry of right and

left (and not, for example, of bottom and top . . . )

This letter becomes this sign ; a right-handed twist becomes left-handed.

On the other hand, whenever the object has a vertical, internal axis, the image in the

mirror is identical. The letter A becomes A; it is not transformed by its mirror reflection.

This is the case for the body of man, which seems to have an axis of vertical symmetry.

(Let us remember that this is an illusion of representation, because . . . the heart is to the left, or

to the right in the mirror, the appendix to the right . . .).

In these right-handed and left-handed pairs of objects, the symmetry is only apparent and

in fact creates an untraversible barrier. Never will a left-handed Moebius strip become a right-

handed one. They are totally different, the one from the other, even if they have the same

properties. The twist creates a symmetry in relation to the mirror that differentiates them totally.

It is apropos of this relation of the object with its image that Lacan founds the imaginary

relation constitutive of the Ego (Moi). Man looks at himself in the mirror and recognizes himself

in the reflection he glimpses.

In 1958, in the schema L, Lacan defines the condition of the subject starting from this

relation.

Thus: the subject is a participant in this discourse insofar as he is pinned to the four

corners of the schema: "(S), his ineffable and stupid existence.

-(a), his objects.

-(a') his ego, or rather what is reflected of its form in the objects.

-and (A), the place from where can be posed for him the question of his existence."1

This relation between (a), the objects, and (a'), the ego, is the first articulation that Lacan

works out concerning the problem that occupies us. We see that the "ego" already has a

definition that calls for some developments.

Between (a) and (a'), all of the imaginary relation is in play. At issue is a particular space

on which Lacan places his patent. "Imaginary" does not mean false, unless its falsity is structural

and necessary to the establishment of the discourse in which the neurotic finds his place. The

psychotic suffers from its failure (carence), from the non-installation of this imaginary relation.

The mirror stage is in fact fundamental to the installation of the imaginary couple. Man

is alienated in his own image, not recognizing the twist.

That identificatory haste (précipitation), which, as we know, unleashes a jubilation for

the child, has multiple structuring aspects. Its failure allows us to establish some elements of the

clinic of the psychoses:

In fact, the unity both of the subject and of the object is structured in the specular

relation.

Schizophrenia, with its array of clinical manifestations that can be summed up under the

rubric "fragmented body" (corps morcelé), is one consequence of the impossible unity of the

subject. Different instances cut into the body itself of the psychotic. "The body itself is all-

important": it makes use of the (a), the (a'), and the (A), and seeks despite everything to give a

consistency to discourse.

The specular relation structures the unity of the object: on this basis, Lacan speaks of the

paranoiac structure of knowledge (connaissance) . . .: "what constitutes the Ego and its objects

1 Ecrits of J. Lacan, D'une question préliminaire á tout traitement possible de la psychose, in Ecrits, Seuil, 1966.

under the attributes of permanence and of identity and of substantiality, in brief, in a form of

things very different from what we know of the gestalts of the animal world."2

Although it is possible to evoke schizophrenia when this relationship fails, paranoia

results from the formalization itself of the unity of the object.

For the paranoiac, (a) and (a') only support their difference on themselves. The subject

and his image only find support for their separation on themselves, whence the precipitation of

this separation into a mortal rivalry. The object is fixed in an image frozen in the mirror where

the subject can only read the agressivity of a semblable, the little other.

His own unifying image is the enemy, whether it is a question of neighbors who wish him

ill and spy on him, or if this hatred, supposed in others and misrecognized, throws him into

infinite recriminations. Others are depositories by their gaze of this image where he can only be

alienated. To separate himself from it, his only possibility is to kill it, to invoke its death, so that

he can exist as a subject in his truth as speakingbeing (vérité de parlêtre).

Here, death is the equivalent of a call to the symbolic. Lacan formulates it as follows:

"the imaginary couple of the mirror stage by what it manifests of the against-nature, if we must

relate it to a prematuration of birth specific to man, finds itself appropriated as the base of the

imaginary triangle that the symbolic relation can in some fashion cover over.”

"It is, in fact, by the gap that this prematuration opens in the imaginary and where the

effects of the mirror stage swarm that the human animal is capable of imagining himself

mortal."3

Death holds the functional place of the symbolic, because, in this false identity of the

object with its image, if the mirror illusion is not sustained, the identity of the one is achieved at

the price of the death of the other. Thus, there exists a struggle to the death between man and his

double, between man and his shadow, which accounts for his difficulties (échecs) with the

illusion of identity, such as is produced in the mirror.

The symbolic relation, the possibility of articulating a signifying chain, is supported by

an object without a specular image, of which the prototype is the disk bearing the point

detached from the cross-cap.

There exist, in fact, objects without a specular image, which Lacan defines with the term

object (a): the breast, the feces, the gaze, and the voice.

The cross-cap also accounts for their articulation with the point . Objects without a

specular image are images without a double. So it is with the sphere, related by Lacan to the

breast,4 and the torus, related by Lacan to the scybale, the "feces." These are objects whose

internal axes of symmetry make it so they are their own inverse; objects somehow coming before

the right-left distinction, the creation of the twist.

2 Jacques Lacan, L'Agressivité en psychanalyse, in Ecrits, Seuil, 1966.

3 Ecrits of J. Lacan, D'une question préliminaire á tout traitement possible de la psychose, in Ecrits, Seuil, 1966.

4 Sémiaire du 26 mars 1969, unpublished. D'un Autre à l'autre.

Remember that it is possible to apply a cut to the torus that produces a bilateral Moebius

strip with four half-twists, right-handed or left-handed, depending on the trajectory of the cut

around the central hole. We thus find the possibility of a gyration, toward the right or the left,

which we do not find on the sphere. In a certain differentiation of structures by their gyration, an

increasing complication can account for the impression of a progress often ratified as such in the

clinic. It is not a question of producing an explanatory principle for this, but of showing how the

structures themselves produce this illusion.

The two other structures on which Lacan supports the two objects (a'), the gaze and the

voice, are the cross-cap and the Klein bottle. But these surfaces go beyond the left-right

distinction; they envelope it, we could say, not as a being without sex, asexual, but as a being

with two sexes. They bring into view the set of the possibilities of gyration to the left and to the

right.

The recourse to the Platonic myth of the androgyn is not anodyne. There is, in fact, in

these structures a resistance to representation, or an effect of obscenity which is related,

existentially speaking, to the fascination or disgust provoked by the evocation of the genital

organs, above all in the figure of hermaphroditism.

We will see the role of the , the phallic symbol, in this place.

Lacan supports the object (a) that is the gaze with the cross-cap.

The cross-cap is an object without a specular image because the Moebius strip from

which it is constructed loses its right-left disparity. As we have seen, the twist is replaced by an

intersection, as soon as we fold the Moebius strip lengthwise. It is on the cross-cap, remember,

that Lacan supports the cutting of the fantasy that detaches an object.

The gaze is an object that falls, disappears as such, in the jubilatory assumption of the

mirror stage.

In the same register are situated two points of structure: on the one hand, the words that

pertain to thought are in the register of the gaze and vision. On the other hand, the cross-cap

allows us to establish the function of the point . For Lacan subsumes all of the objects (a)

under the term .

It is not at all surprising that this point , if theoretically establishable in all of the

structures, is particularly so beginning with the cross-cap.

The object (a) in the scopic drive is the most evanescent, which is not unrelated to the

place of vision in the mirror stage.

Moreover, the cutting of the cross-cap detaches a right-handed or left-handed Moebius

strip, depending on how it is unfolded, and a single object (a), the disk--more precisely an

immersion of the disk, a disk provided with a line of intersection, as is shown in these drawings:

The numbers on the flattened surfaces of the disk refer to the order of the mounting of the

triangles in the eight-cone.

We must then interiorize the loop of the eight and look at the cone from above, having

given our gaze a quarter-twist:

The right-left disparity has no role in these occurrences.

On this disk is situated the point insuring the cross-cap structure.

It is appropriate to evoke here the three modalities of the drive established in grammatical

terms by Freud: the passive, the active, and the pronominal (the middle voice of ancient Greek).

Not that these differences are recovered, but rather that language translates according to its

means a disparity situated elsewhere topologically. Because the unfolding of these three objects,

after the cutting of the fantasy--their being plunged into our ordinary space--produces a left-

handed Moebius strip, a right-handed Moebius strip, and a bilateral disk.

The imaginary couple of the mirror stage leaves in the hollow (en creux) the place

marked by the point , which allows us to subsume within this organization called the cross-cap

all of the objects (a).

Beyond the imaginary identification with his image, the subject is also identified with a

third term, "that of the ternary imaginary, where the subject identifies with the opposite of his

living being, which is nothing other than the phallic image whose unveiling in this function is not

the least scandalous aspect of the Freudian discovery" (Traitement).

According to Freud, the imaginary function presides over the investment of the

narcissistic object. We have shown that "the specular image is the channel taken by the

transfusion of the libido of the body toward the object."

"But insofar as a part (of the libido) remains preserved from this immersion,

concentrating in itself the most intimate aspect of autoeroticism. Its 'plunging' (en pointe)5

position in the form predisposes it to the fantasy of an outmodedness where is achieved the

exclusion where it is found the specular image and the prototype it constitutes, for the world of

objects. It is in this way that the erectile organ comes to symbolize the place of jouissance, not

as itself, nor even as an image, but as a part lacking in the desired image."6

This quotation reminds us that at the moment itself when the structures of the object (a)

are diversified, the point remains central to the functional organization of this object, which

differentiates the drives.

On the sphere and the torus, the phallus must be sought in the Other, present with all the

carnal weight of a body in the organization of the drive. On the cross-cap, it is in the right place

(de droit), we will say.

Finally, it must be positioned on the Klein bottle.

The object (a) remains a point of articulation between fantasy, drive, and the "passions of

being" (love, hatred, or ignorance). Excluded from the specular image, it nonetheless constitutes

a hole in the organization of the Ego; it permits a knotting with something of the Other, from the

exterior.

We have already evoked the sphere, the torus, and the cross-cap. It is apropos of the

problem of the mirror and of the twist that we introduce the Klein bottle, a topological object

associated by Lacan with the voice.

It is a question of a sphere on which a tunnel becomes a handle: (see photo)

5 In haute couture, "en pointe" refers to a plunging neckline (décolleté) (Tr.).

6 Subversion du sujet et dialectic du desir.

We can also describe it as a bottle whose base joins its neck, producing an intersection in

the form of a circle:

This structure makes appear a space whose interior is continuous with the exterior. Only

a trajectory differentiates them. Since it retains a central hole, it is associated with the torus.

Topologists show how a torus can be rolled as a double surface around a Klein bottle.

They also say that it constructed from two Moebius strips joined along their edge, but two

strips with different twists:

(The torus is also reconstituted from two Moebius strips, but with an identical twist.)

The Klein bottle can in theory be represented by a sphere to which are added two

intersection like that of the cross-cap.

Is this based on the fact that we can position the point on the Klein bottle, in a doubling

of castration, that of the subject and the Other?

Lacan considers, in fact, that "nowhere is the subject more interested in the Other than by

this object." Clinically, moreover, remember that the voice is the object Lacan puts at the center

"of the relations between the sadist and the masochist." He develops it thusly: "Masoch

organizes things so as to have no more speech; he signs some contracts enjoining him to have

nothing more to say. The sadist tries to de-complete the other by withdrawing speech from him

and imposing his voice on him."7

By way of these objects (a), the body is present by its orifices. The holes of the organism

offer their edge to structures accounted for by topological structures. They are organizations of

the hole, and they give form to the space of the hole.

Hence, the voice takes up two bodily orifices in the same structure, the ear to hear and the

mouth to speak.

The gaze puts in place a structure so particular that we have to make use of the joke to

account for it: it is a wonder "that cats have two holes cut in their skin precisely where their eyes

are." (We recognize here one of Freud's examples in his study on jokes). 8

The torus encloses

the mouth and the anus in the same organization.

These are the two orifices of the same hole: the gut or the digestive tract. Also at issue is

the physical structure of the set (l'ensemble) of the body: ectoderm, mesoderm, endo . . .

.

7 Seminar of March 26, 1969, unpublished, D'un Autre á l'autre.

8 Sigmund Freud, Jokes and Their Relation to the Unconscious, translated by James Strachey (New York: W. W.

Norton & Co., 1963), p. 59 (Tr.).

The sphere constructs the hole on its denial. The breast completes the mouth of man's

child. The psychic organization of the subject then refers to a totality in which the breast

detached from the mother forms a part.

The coupling two-by-two of objects (a) finds another raison d'être. The cross-cap and

the Klein bottle are structures of unilateral space. They put interior and exterior in continuity.

The voice and the gaze are on the body, by the quite specific ways of sensorial organs. At issue

are the only passages of exterior space into mesodermic space. Only a unilateral, Moebien

organization of space can account for this. 9

The four objects (a) are without specular image, because they are holes, henceforth

specifically organized.

There are thus two kinds of objects dividing human knowledge: those to which the mirror

gives an identity and a substantiality that is only the reflection of the Ego, and those that plug up

the hole, organizing it, veiling it. The failure (l'échec) of this function makes castration anxiety

well-up or spring forth.

Clinically, Lacan came to distinguish between two imaginaries: "the true and the

false."10

The false imaginary pertains to the necessary illusions of the mirror, the true to the

fantasy, to desire, to anxiety.

This is why Lacan always opposed i(a), the image of the other, and the object (a).

9 Seminar of May 16, 1962, unpublished, L'Identification.

10 Seminar of June 13, 1962, unpublished, L'Identification.

The so-called knot of the fantasy.

1

Chapter 6: From Surfaces to Knots

In his topological advances, Lacan moved from a usage of surface structures to knots.

This movement was problematic to the extent that there is no mathematical conception

that englobes these two parts, although they are very much linked in general topology.

Thus the edge of a Moebius strip with three half-twists is seen to be knotted in a trefle-

knot: on a manipulation (pliage) of this knot is constructed another immersion of the projective

plane known as the "surface of Boy,"1 the structure of which is shown in the following drawings:

This surface of Boy puts in place what is called a "triple point." When three surfaces

intersect, they define a point: when there are two, they define a line (as we have seen).

This triple point also appears in a mounting of surfaces on the Borromean knot:

1 Drawing of J.C. Petit, published in Pour la Science, January 1981.

2

How are the knots and the surfaces linked?

Several responses, or rather several trajectories, allow us to account for this liaison and

for Lacan's progress from one to the other.

The surfaces and knots treat of some possible articulations between the elements of a

structure.

The surfaces with their spatial representation put in question, even to work, the schematic

conception of structure, which is already at work in the labors of Levi-Strauss. The notion of

space is fundamental.

Knots, on the other hand, are supported exclusively by the operation of the cut.

On the surfaces, Lacan makes an operative use of the cut. In this respect, he does

something new. This usage is essential to how Lacanian topology serves psychoanalysis. The

cut accounts for the analytic act: essentially, for interpretation.

We must now insist on the subversive aspect of this usage of the cut on surfaces.

In mathematics, the cut serves some definitions of surfaces; care is taken to differentiate

them so as to classify them and number them (cf. genre*).

Lacan completely subverts this logic. He draws on the wish to work with the surface, to

produce a history thereof, to cross over it in a movement, to permit speech, because it emerges

from the a-temporal laying out of structures.

This subversion leads Lacan to knot theory. Knot theory puts this operation to work. A

knot is defined negatively by the necessity of the cut:

A knot is all the interlacings of threads that must be cut to make it disappear. The cut

also leads to knots topologically. It is cuts on surfaces that create knots.

The result of the cutting operation is a knot:

Hence, a median cut on the Moebius strip leaves the strip whole but with four half-twists.

This strip cut again separates into two pieces, albeit knotted. The photographs make this

quite clear.

On the Moebian strip, a cut that encircles the surface twice separates the surface into two

pieces, but they are knotted.

3

Of these two pieces, one is a Moebius strip, with one half-twist, as at first, and the other

is a strip with four half twists. We create two objects, one bilateral, the other unilateral.

This operation evokes the so-called cutting "of the fantasy" produced (opérée) on the cross-cap.2

On the other hand, the knotting evokes the so-called knot of the fantasy: there is an interior eight

and a round: an edge of a Moebius strip and an edge of a disk (see photo).

2 Cutting of the cross-cap, cf. Chapter 5.

4

Cutting of a strip with four half-twists.

In other words, the fantasy as a cut on the surfaces, or knot, is at the heart of this passage

from surfaces to knots.

5

On the Moebius strip with three half-twists, an interior-eight-shaped cut produces two

pieces knotted like this:

There are two above-beneaths too many for the knot of the fantasy. What is surprising is

that the real of our everyday space is maintained despite the impossibility of arriving, by a

cutting of a surface, at the so-called knot of the fantasy.

Let us take up again the problem beginning with the twist in surfaces that are the

expression of our everyday space and of the characteristics proper to it. We have seen from a

variety of perspectives the importance of the twist in the topology of surfaces and how they

allow us to account for the analytic experience. The twist plays a primordial role here, since the

constitution of the object (a) and the misrecognition belonging to the constitution of the Ego is

brought into play around the difference between the specular and the non-specular. The twist

retains its fundamental place in this distortion.

However, the topology of mathematicians up to now evacuates this twist. A strip is

bilateral whatever the number of its half-twists, provided that this number is even. In the field of

psychoanalysis the twist of surface renders the cut operatory and creative of knots. Based on this,

the half-twist of a Moebius strip is exemplary:

It writes the fundamental difference of above-beneath.

There are two of them, just as there are two Moebius strips, depending on the directions

of the twist:

The drawing of the interior eight accounting for them brings in an above-beneath and a

beneath-above. The Moebius strip is here once again exemplary.

6

Moreover, although the above-beneath writes the direction of the twist as right or left, it

the also dispenses with the concave or convex characteristic of a surface. In this sense, it (and

the knot) proceeds from a phenomenon of the simplification of surfaces.

Thus, when we retrace the history of the cutting of a Moebius strip with one half-twist,

we see how this history, centered on the twists, is subject to complications.

The cut creates a second edge and multiplies the half-twists by two; we remove the

forward buckle (boucle de devant), which makes one half-twist disappear from one of the

branches of the eight, and leaves two on the other: the half-twists are displaced.

We suppress the self-crossing in unfolding the lower buckle: two half-twists disappear

but we cannot truly flatten the unfolded branch.

7

The self-crossing, "the above-beneath," is equivalent to two half-twists. We displace

some of the half-twists and we transform one twist, which can be called convex, into a concave

twist, which changes the direction of the twist.

Here we see four events susceptible of affecting a half-twist: right, left; before, behind.

The above-beneath lets one of these oppositions fall:

At this stage, if we think of it terms of a movement along threads, there are only two

possible outcomes: the passing of one thread under another--which can be the same at another

moment of its movement (self-crossing)--or above it, while a surface twists to the left, to the

right, before or behind.

Based on this, the history of the same cut on a Moebius strip can be retraced more

simply.

We only retain the history of its edges and their knotting:

8

Thus, if the knot is a simplification of writing, for the fantasy the two writings have their

raison d'être. In fact, it is a symbolic operation and an imaginary function.

The cutting of the cross-cap gives us a Moebius strip the direction whose twist is not

specified.

In the place of the twist, there is a line of intersection.

Here, the surface gives rise to a difference without effect.

On the other hand, the disk left by the cut keeps its pertinence, in relation to the simple

round of the knot, since it is constructed on an interior eight.

Let us remember that it bears the remarkable point :

The so-called "knot of the fantasy" is also formed by two threads, a disk-shaped simple

round and the interior eight of a Moebius strip. Their places are interchangeable:

9

The one becomes the other, and visa-versa.

The exchange writes, at a level internal to the knots, the effects of the mirror on twisted

surfaces. The fantasy is an imaginary function.

We see how this knot is situated on the cross-cap, as a cut before the twist, an effect of

the symbolic constitution of the object (a) and with the knots that englobe specular effects

themselves. It is in fact a knot whose multiple forms can offer a near symmetry.3 Moreover,

"there are" two, a left-handed and right-handed one, we will say, although it is no longer here a

question of a half-twist: the one is the specular image of the other.

Thus, from the twist in surfaces to knots there is a writing, a simplification, but also a

radical change in nature, a leap from the imaginary to the symbolic that cannot be evacuated.

If the cut creates knots on the surface, they are knots formed from two threads. They are

all Borromean. The Borromean quality is trivial to them. Whatever the knotting, since there are

only, two consistencies, it suffices to cut a single thread for the knot to disappear. The

Borromean quality begins at three.

The negative function of the cut on the knots contributes to the definition as an operation

a contrario. "A knot is an interlacing (enrelacs) that a cut can make disappear."

The Borromean knot plays an exemplary role here in that it is the effect of the most

simple knotting. One cut and, whatever the number of threads in play, the knot disappears.

A contrario, it is necessary to speak of surfaces to bring the cut into operation, before

three consistencies.

The omnipresence of the number three in this passage owes to structure. The symbolic

makes three, allows for counting and naming. Before it, as we have seen, there is a slippage, a

constant exchange between the one and the other.

We can now, apropos of this topological work, speak of an effect that gives all of its

value to the Lacanian advance.

In Scilicet n. 2/3, the article titled "For a Logic of the Fantasy" develops an effect of

exclusion between "topos and logos." The articulation of the one to the other is impossible even

though it is real and, moreover, it presents us with the necessity of a writing (as third term: you

read this work; you do not hear it).

In fact, if "logos" refers (renvoie) to speech, "topos" is nothing other than the place of the

body. The subject is the effect of this exclusion.

Why, then, does the place of the body find itself excluded from speech? We say to begin

with that this exclusion owes to the fact of its being not-two. Let us begin (Prenons acte) by

3 Cf. the book of Jean-François Chabaud: Le noeud dit du fantasme.

10

noting that the unity of the body is opposed to the system of logic, which is defined starting with

the dyad, under the rubric of the principal of identity, of non-contradiction or of bivalence. There

has to be a proposition "x f of y."

The place of the body can only be translated (se traduire) into the order of language by a

series of disjunctions from which the unconscious acts: "the field of formations of the

unconscious with which the psychoanalyst is concerned is, however, that of the compromise

formations that, in the mode of denial (dénégation), allow it to retranslate place into logic," into a

series of divisions:

the division man-woman,

that as object of desire (a) and the set of the effects of Demand (A),

that of the body and jouissance, the division between knowledge and truth. In this way of seeing

things, working with surfaces is a challenge for speech. It is situated in the order of the body, of

space, and not in that of the metonymy proper to speech, to the "signifying chain."

The concepts are no longer points of reference; they are shown in a simultaneous vision,

a space proper where our corporal perceptive possibilities contribute. The drawing puts to work

the scopic space in its relation with the imaginary of representation. Manipulation brings into

play the skill of the hands, what is concrete in the movements of the body.

In his final seminars, Lacan no longer spoke of a topology. The position of the work in

this field of place and its characteristics entails the loss of the possibility of knotting a discursive

thread.

The real of the drawing and of its effects of representation that unfold there take their

place in a laying out of the atemporal structure of words. The beginning and the end of a

sentence are not present. All of the threads are permitted, and all are particular, individual,

subjective.

In the same way seeing things is situated Jacques-Alain Miller's warning in his

"Commented Table of Graphic Representations" adjoined to Lacan's Écrits.4 He expresses

himself thusly: "If it is true that perception eclipses structure, a schema will infallibly lead the

subject to forget in an intuitive image the analysis that supports it."

"It is the role of symbolism to interdict the imaginary capture."

Although the drawings are "graphic representations" of an analysis, they stage this

imaginary capture. In Lacan's work, it is the operation of the cut that truly transforms these

objects represented into a topology of the subject.

There is no longer an occultation of the symbolic in topology because this presence of an

operation evokes, properly speaking at the interior of geometrical intuition, the furrowing of

speech.

Elsewhere, at the end of this warning, Jacques-Alain Miller refers us "for learning the

rules of the transformation of geometrical intuition into a topology of the subject" to the note,

added in 1966, to the Schema R. We have seen how this note puts in place on the Schema R the

cut essential to the structure of the cross-cap.

A contrario here we have what gives topology all of its worth in the approach to the

psychoses. The limit-effects our perception encounters are to the image of the world of the

psychotic. For a long time already, psychoanalysts of children have put drawing to work in their

sessions. The topological notions of edge, of interior-exterior, of cut, of envelop, are constantly

present there, as well as spaces where these notions are not differentiated.

4 Jacques-Alain Miller, postface to the Écrits of Jacques Lacan.

11

We have shown by the intermediary of the cut that structure itself induces this passage

from the writing of surfaces to knots, when one passes from two to three dimensions.

In a still more general way, we can mark this passage in the work of Lacan as he himself

defined it. In his famous trilogy "real, symbolic, and imaginary," he recognizes that he began

with the imaginary in the optical schema, then he took up the symbolic, and finally the real. The

knot is real, and the real only appears as a third dimension. From the moment when three rounds

of thread are in play, there are no longer specular effects. There is only a Borromean knot.

In this domain the cut is already a symbolic effect as operation, because it brings into

play, by its effectuation itself, the dimension of language (la langue). The cut that counts,

psychoanalytically speaking, is the interior eight, the edge of the Moebius strip, whose

relationship with the creation of sense in language we have seen.

The omnipresence of the number three owes to structure. At issue is nothing other than

the Oedipus complex. The Borromean knot articulates the place given its mythical version in the

Oedipus complex; it clarifies this three, "papa, mama, and me," which functions from the

beginning.

In relation to this beginning, the psychotic does not find his place and the neurotic only

finds it at the cost of the imaginary castration from which he suffers. It is to the extent that this

castration becomes symbolic that the neurotic finds a path toward jouissance.

1

The Borromean Knot

Chapter7: The Borromean Knot

The Borromean knot is a certain way of knotting loops of thread.

Mathematicians have only very recently begun to occupy themselves with the part of

human experience constituted by the art of knots. This discordance is astonishing. On the

carpenter's chain at the origin of Egyptian mathematics, knots were already utilized to mark

measurements. Every year, after the rising of the Nile, it was necessary to re-measure the fields,

and they relied on knots to do so. This is to say that the knot was part of a very ancient,

practical, realistic, concrete human experience, of which mathematics is a theoretical and logical

reflection. It was not until the 20th century that mathematicians began to take an interest in

knots. They then defined different forms of knots and chains, according to their qualities of

knotting and unknotting. The Borromean knot retains a particular place in these studies.

It must be remarked that is an abuse of language to speak of the Borromean knot to

designate this chain of three threads:

A knot is in fact formed by single thread that follows a particular enough trajectory not to

be reducible to a simple round. On the other hand, as soon as there are several threads in play,

one speaks of a chain.

We speak of a Borromean knot to designate a Borromean chain.

2

Among all the chains existent or imaginable, the Borromean knot holds a place apart,

because the threads constituting it are held together by a knot, or rather an operation of knotting,

which does not go without reminding us of a knot in the strict sense, called a "trefle knot" (in

white in the drawings below).

What is the Borromean quality of a chain?

"A Borromean chain is a chain such that if one cuts any one of the rings, all are

unlinked." One can represent the chain in a way that puts the accent on the possibility of

multiplying to infinity the number of distinct threads:

In this representation, called "a generalized Borromean chain," it is clear that the central

loop, in the form of a crescent, can be multiplied: the number is not relevant to the Borromean

quality of the knot. It suffices to cut any one among them to undo the knot.

The following drawings show how a Borromean knot of this type is undone, whichever

loop is cut:

There are three ways of cutting any Borromean knot, and they introduce two aspects

essential to the Borromean knot: the number three and the putting flat (which is to say, the

manner of representing a knot or a chain by a drawing).

For the representation of the generalized chain, there are three different drawn forms

(tracés) found regardless of the number of loops one adds at the center:

Another representation exists, the one used most often by Lacan, which presents in its

drawn form itself the identical function of all of the rounds, as regards the Borromean quality.

3

The three rounds of thread each plays the same role vis-à-vis the other two: two loops are

posed one over the other, and the third ties them together, all three, in a Borromean manner. It

suffices, in fact, to cut any one of them for the whole thing to be undone.

The following drawing shows how a single round (in white) links the two others by an

alternation of above-beneaths:

It is in the name of this homogeneity of functions in the drawing that Pierre Soury

preserved for the three the role of unity in the classification of Borromean chains. The more

specifically mathematical concern of his course of 1980 was "to demonstrate the exemplarity of

the three looped Borromean in the classification of chains." He added that the Borromean knot

played a central organizing role in the Milnorian classification of chains. This tells us how

important this three-ringed knot is.

Beginning at three, it should be recalled, the Borromean quality becomes pertinent.

If we knot together, in any manner, only two rings, the Borromean quality is in every way

verifiable. This is what logicians call the trivial: however one approaches it, it is verified:

On the other hand, in a Borromean chain with four rings, subgroups always appear,

whether one and three, or two and two, as is shown in the following drawings.

Subgroups, one and three:

(the subgroup of one is in white)

4

Subgroups, two and two:

There are many ways to present a Borromean knot:

This presentation, classic in the realm of analytic discourse, has the interest of showing

the functional equivalence of the rounds to one another: it writes the homogeneity of three

consistencies:

5

The armillary presentation, on the other hand, called this because it evokes the

presentation of the moon and the stars by the armillary spheres of the middle ages, creates a

distinction between the knot and the and the schemas that designate sets having intersections and

unions:

We often forget the above-beneath is not a point of intersection between two lines as in

the Venn-Euler diagrams. For chains and knots, this above-beneath is the letter of the imaginary

dimension.

We must add that in the domain of the topology of chains and knots a round can always

be represented by an infinite line: a circle can, by convention, close itself at infinity . . . Whence

the many images of the Borromean knot that Lacan finds in iconographies throughout time:

Lacan makes a particular use of this possibility, since he makes of it the writing of

existence. But that introduces us into a more complex reflection, which we will attempt to

clarify (mettre á plat).

6

The analytic use that Lacan makes of the Borromean chain is more explicit in the

Seminar RSI, if only because a drawing supports it. The trilogy of the real, the symbolic, and the

imaginary is taken up again by Lacan throughout this seminar and related to the Borromean knot.

Though Lacan speaks of chance in evoking the coming into play of the Borromean knot in his

reflections, this knot nonetheless acquires a necessity therein that must be commented on. The

Borromean knot, or the three-ringed Borromean chain, comes to write the relations exchanged

between the three registers of the real, the symbolic, and the imaginary. They write its "common

measure."

This drawing is neither a graphic representation, nor a schema, but a topological writing.

Just as with a writing, this drawing makes consist, and exist, what is at issue in analytic practice.

Lacan presented this drawing beginning with his Séminaire of September 17. It is at first

approach astonishing, and retains all of the piquancy (sel) of this astonishment, especially in the

spaces delimited by the lines of the drawing. Indeed, at issue is neither an intersection, nor a

surface; there is nothing delimited or even measurable about these spaces. They can vary to

infinity without transforming the structure of their relation. In his course, Pierre Soury takes on

the question of the legitimacy of such a writing with the help of the following designs:

The absence of certain terms is revealed to be symptomatic of a given organization. For

example, the drawing evincing courtly love does not include the term "phallic jouissance":

Courtly love is a vision of the world organized around the structural refusal of the sexual

realization of love. The absence of phallic jouissance in this form of the Borromean knot results

7

in the complete masking of the "object (a)." The imaginary, which is in a median position, is the

operator of the knotting: courtly love is literary figure, a poetics.1

The question of how to qualify the simultaneous writing, if we dare this definition, of the

little letters, in the interstices of the Borromean knot, still remains mysterious. The drawing is

suggestive enough, in its continuous transformations, for having marked the places of absence, of

voids. At issue is a way of clarifying the clinic in its infinite variety; is it a question of

illustrating what in structure usually remains invisible despite its organizing role?

Lacan allows us to glimpse the direction (sens) of his development, although he councils

us to "use it stupidly." He writes: "I have found only one way to give a common measure to the

terms Real, Symbolic, and Imaginary, which is to knot them together in a Borromean knot." He

adds later: "I have been captured by the Borromean knot," or again:

"The three rounds came to me like a ring to a finger"; "I have always known that the knot

incited me to announce of the symbolic, the imaginary, and the real something that homogenized

them." It is only a question of finding out how to count them, from the moment that the count

begins at three.

Beyond the surprise that such comments can give rise to, we should keep in mind that

Lacan demands of the Borromean knot to explain, to formalize, some relations that are not

written elsewhere. The knot does not illustrate the relations between the terms, it creates them.

To pose the question, or stop at the imprecision of what the words write in the schema recalled

above, does not take into account the creative aspect of Lacan's development. The effort of this

seminar RSI is to say, name, write, and create the words appropriate for speaking of the relations

entertained between the symbolic, the real, and the imaginary. In this attempt, Lacan calls on us

to re-situate the terms of Freudian research as points of illumination that should help us

conceptualize this writing. The written words then find, by their placement between the above-

beneaths of the Borromean knot, a new structural dimension: "one triplicity is doubled by

another triplicity."

Let us return first to the homogenization between the three registers created by the

Borromean knot. In their knotting, the real, the symbolic, and the imaginary have the same

function: they can be counted as three. Until then, in fact, there is no necessity of naming only

three registers, rather than four or only two. The Borromean knot brings in this necessity.

Indeed, Lacan says that other dimensions can be invented; this does not prevent the Borromean

knot from always bearing the mark of the three.

1 Seminar RSI, published in Ornicar, #2 to 5, Le Seuil, 1974-75. This commentary takes all of its citations from

these four issues.

8

He gives to each round, the unity, the "one" that is the "common measure." He gives as

three "ones" the real, the symbolic, and the imaginary, which nonetheless are only to be

understood in their relations, their knotting.

Lacan seeks proofs for the existence of the Borromean chain as a foundation of thought

and above all of sense. It is a question of showing that three is the necessary figure for posing an

existence that does not make an image . . . As he says throughout his seminar, the words are

contaminated by the reflection made on them. Henceforth existence and image are terms

echoing what is different in the three registers.

With the Borromean knot is posed the question of the creation of sense and of its

relations with the unconscious and the symptom. We cannot evoke this problem, make sense

with these words in speaking of sense and of its birth, without at the same time making intervene

what is in question in this research.

We can compare the Lacanian advance on these problems to those problems of nuclear

physics where the instrument of measurement, the electron microscope, itself modifies the field

of the experience. The examiner only sees the instrument of measurement in these experiences,

except for infinitesimal variations revealing the existence of the object of study beyond the

instrument.

Thus, Lacan speaks in terms defined by words that measure at the same time their

definitions and the gap between the words and this definition: "If I state in speech (parole) that

the consistency of these three rounds is only supported by the real, it is indeed because I make

use of the separation (écart) of sense permitted between RSI as individualizing these rounds.

The separation is there, supposed taken at a certain maximum. But what is the maximum

separation allowed from sense?" These formulations give us the impression of a thought that

chases its own tail, that does not situate its object, and that even loses its words and its sense.

Beginning in his written introduction added to his oral course, we are given this impression of

"futility" or "debility": "It is the type of problem I encounter at every step, without looking for it,

I must say, but the measure itself of the effects I say cannot fail to modulate my statement (dire).

If we add the fatigue involved in this statement, we are still not relieved of the duty to account

for it, on the contrary. A marginal note may be necessary to complete the circuit elided in the

seminar; it is not the touching up that is 'futile' here, but, as I stress, the mental itself, to the

extent that it exists."

How does Lacan define the relations between the three rounds?

Let us try to summarize the diverse definitions given in the Seminar. It is a matter of

pinpointing the relations entertained between the real, the symbolic, and the imaginary, which

are defined respectively by existence, the hole, and consistency.

The three terms R, S, and I hold together, make three, by the adjunction of the imaginary

to two others, "for the triad of the real, the symbolic, and the imaginary only exists by the

addition of the imaginary as third." This is why it is appropriate to begin by taking up the term

consistency.

Consistency, which Lacan indicates to be equivalent to the imaginary, is necessary for the

knot to be, and to be three, since a knot only begins at three. A certain material is necessary, and

for Lacan the material is imaginary, inasmuch as it is pegged to the body. In Lacan's teaching,

the imaginary refers to the problematic of the image in the mirror. Lacan amply developed the

structuring function for the subject of the appearance of its image in the mirror, the image of a

little other with which the child identifies in a "precipitation" signaling its entry into the

9

symbolic. At issue is a knotting of three registers. In this image, the child recognizes the object

of the desire of his mother. He identifies with it, and clothes himself with it, and by this means,

brings into consistency the symbolic hole, equivalent to the lack presented by the gaze of the

mother.

In this way the knotted three is made to consist: "consistency, to designate it by its name,

I mean by its correspondence, is of the imaginary order." Each of these three rounds, by their

knotting itself, has its own consistency, because thought, sense, is only revealed, is only said, by

way of the articulation of three registers. To think the real, there must be a peel of the imaginary.

Lacan speaks of the cord: "They have a consistency that I am indeed forced to call real, and

which is that of the cord."

To say the symbolic, the recourse to the imaginary is just as necessary. Lacan then refers

to what is implied by the organizing term. In this perspective, what Freud calls the "dead father"

is an imaginarization of the symbolic.

Just as Lacan says "real to the second power," we must say "imaginary to the second

power" to evoke the consistency of the imaginary: "consistency for the speakingbeing (parlêtre)

is what is fabricated and invented. In this case, it is the knot inasmuch as one weaves it together,

but it is precisely not inasmuch as one has woven it together it that it exists. This existence is

what corresponds to the real."

Existence defines another aspect of the relation between the three rounds.

At the moment of knotting, the consistency, the material of the three rounds must

necessarily enter or exit a hole. We will come back to this.

In a Borromean chain, two rounds are posed one over another, without relation, free from

each other, and a third knots them: there is always for two of the three rounds a third that realizes

the knotting (in white in the following drawings):

Lacan defines this third as existing functionally to the two others. Ex-sist means, more

precisely, situating itself elsewhere, although a presence is necessary to the two others as a point

of support, of limit, of knotting.

Lacan, in L'Etourdit,2 shows the necessity of a point of exclusion for sense. The

universal only poses itself from a point that it excludes: "there is no universal that must not be

contained by an existence that denies it."

2 L'Etourdit, in Scilicet 4, Le Seuil 1973.

10

We are again at the heart of the problem of sense. One of the functions of the Borromean

knot is to show how what is excluded is necessary, or how the tie is made by a third, beginning

with the two that are not knotted.

"The existent is what turns around the consistent and makes an interval."

Lacan makes the term existence correspond to the register of the real. By definition, the

real is what is not symbolized, what is outside of what makes sense. However, he does pinpoint

what is conceivable: "Existence is only defined in effacing all sense."

In the schema in the first seminar of 1974, Lacan situates existence in each consistency.

In their drawings themselves, he shows that the function of a round is necessary to the knotting

of the two others: Lacan draws, parallel to each round, an open line that he defines as follows: "I

propose to symbolize by an intermediary field what exists to the real of the hole; this

intermediary field is given by the opening of the round as an infinite line isolated from its

consistency."

As for the notion of the hole, its equivalence varies in the course of the seminar: it is at

first real, then symbolic. Lacan explains that the Borromean knot allows us to distinguish the

hole from existence: existence is made of this infinite line that knots two other rounds. "One of

the consistencies is not knotted to the other, does make a chain with it," whence the existence of

a third and the non-reciprocity of the passage of one of these consistencies into the hole offered

to it by the other. For each round, there is the necessity of a hole in its consistency to permit the

knotting, but this hole is differentiated precisely from the third that enters and exits it as the

"operator of knotting."

At the beginning of this Seminar, the distinction is not made and Lacan situates the hole

on the side of the real: "Whence the correspondence with the hole that I try at first with a real

that will later find itself conditioned by existence." At this moment, in fact, he seeks the

definition of what is not consistency in the rounds and that allows for their knotting. Later, he

says: "We are lead to pose that the hole is of the symbolic order, which I have founded on the

siginifier." But to think this hole there has to be a peel of the imaginary, or at least the elements

of an image that allow us to elaborate it: the topology of the torus then imposes itself.

The torus is a holed sphere, a transformation that encounters the obstacle of another cord

supposed to consist. The central hole of the torus is imagined starting with the knotting of

another torus in this hole: the image of linked torii:

This image supports the metaphor of the hole. It brings a consistency supporting the

contradiction of the not having: the hole has a consistency that is not imaginable. The edge of

the hole is imposed as the representation of the hole itself.

"It suffices to imagine the circle as a consistent cord to see that the within and the without

are exactly the same thing. There is only a single within; it is what we imagine as the inside of

the torus, but the introduction of the figure of the torus consists precisely in our not taking count

of it". The torus is a surface without a hole, without a rupture, around a central hole.

11

As for the hole itself, "no one knows that it is a hole," Lacan adds. Nonetheless, analytic

thought puts the accent on the hole, "although this accent plugs it up." Well before its last

topological developments, Lacan spoke of the object (a) as the stopper in a hoop net . . .

Topology is the only way to approach this question of the hole and leads to some notions

that are not simple. From the start, there is a a structural, essential , difficulty to topology.

Lacan even speaks of an aversion to it, reading there the trace of an original repression. "Why

not see in the aversion that it manifests the trace of primary repression itself?" This is a way to

fold the unconscious over the symbolic, and, more precisely, thanks to the reading permitted by

the Borromean knot, of making the unconscious "what exists to the symbolic." We can then

differentiate what constitutes the hole, primary repression, of which we will never know

anything. This original repression, to extend the metaphor, is the symbolic at the second power,

the symbolic hole in the symbolic. Death refers to the hole in the imaginary. And all of the

orifices in the body, inasmuch as they are interchangeable, constitute what makes a hole in the

real: "the inner-tube torus” (tore-boyau).

This is how the relations or non-relations between these three rounds of the Borromean

knot are summed up. Consistency, as imaginary, existence, as real, and the hole as equivalent to

the symbolic form three terms that we must only use in remembering all of the seminar RSI.

Freudian concepts can now be taken up again at the interior of this writing.

For instance, the unconscious can now be situated in relation to the symbolic, the

imaginary, and the real as what exists to the symbolic, which is necessary for the knotting of the

real and the imaginary. Likewise, the "phallus" is necessary as "existent, to the knotting of the

symbolic and the imaginary, testified to by 'sense.'" There is nothing astonishing about all sense

being able to be definitively lead back to phallic signification. "The phallus" is its support, as

existent . . .

"Phallus" and "Unconscious" are Freudian notions, derived from analytic experience.

They find a place in this writing that transforms the enumeration of their definitions into a

structural positioning.

In conclusion, we should hold on to the idea that the Borromean writing of Freudian

notions allows for a simultaneous, paradoxical, and illuminating reading, in relation to what can

be heard, as the unfolding in the metonymic duration of a discourse, by a psychoanalyst in his

armchair.

The Borromean knot offers us a support that is neither a model, nor an illustration, of

how the truth is wedged, suspended, at variable points of which the place can nonetheless always

be marked (est pourtant toujours repérable).