🦋🤖 Robo-Spun by IBF 🦋🤖

(All parts in English)

Today we are going to continue elaborating the function of what one may call ‘the signifier of the cut’, or again the ‘inner eight’, or again ‘the lacing’, or again what I called last time the ‘Polish signifier’. I would like to be able to give it an even less meaningful name, in order to try to approach what it has that is purely signifying.

We have advanced on this terrain as it presents itself, that is to say in a remarkable ambiguity since, as a pure line, nothing indicates that it intersects itself, as the form in which I drew it there reminds you, but at the same time leaves open the possibility of this self-intersection.

In short, this signifier prejudges in no way the space in which it is situated. Nevertheless, in order to make something of it, we posit that it is around this signifier of the cut that what we call the surface is organized, in the sense in which we understand it here. Last time, I reminded you—because it is not the first time I showed it before you—how the surface of the torus can be constructed, around—and around only—a cut, an ordered cut, manipulated in this quadrilateral way, which the formula expressed by the succession of an a, a b, then an a’ and a b’, our witnesses respectively insofar as they can be related, attached to the preceding ones, in a disposition that we can characterize, in general, by two terms: oriented on the one hand, crossed on the other hand.

I showed you the relation, the relation, if one may say exemplary, to the first aspect, metaphorical…
and whose very question is whether this metaphor goes beyond, if one may say, the pure plane of metaphor
…the metaphorical relation, I say, that it can take from the relation of the subject to the Other, on condition that in exploring the structure of the torus we perceive that we can place two tori, insofar as they are linked one to the other, in a mode of correspondence such that, to such-and-such a privileged circle on one of the two, we have made correspond, for analogical reasons, the function of demand.

Namely that sort of circle turning in the familiar form of the coil which seems to us particularly apt to symbolize the repetition of demand, insofar as it entails this sort of necessity of closing upon itself. If it is excluded that it intersects itself after numerous repetitions as multiplied as we can suppose, ad libitum, for having made this closure, for having drawn the circuit, the contour of another void than the one it encircles: the one that we distinguished first, the one defining that place of nothingness whose circuit drawn for itself serves us to symbolize, under the form of the other circle topologically defined in the structure of the torus, the object of desire.

For those who were not there—I know there are some in this assembly—I illustrate what I have just said by this very simple form: by repeating that this loop of the winding of demand, which is found around the constitutive void of the torus, comes to draw what serves us to symbolize the circle of the object of desire, namely all the circles that go around the central hole of the ring. There are therefore two sorts of privileged circles on a torus: those that are drawn around the central hole, and those that pass through it.

A circle can combine both properties. This is precisely what happens with this circle drawn in this way:

[fig. 1]

I put it in dotted lines when it passes to the other side. On the quadrilateral surface of the fundamental polygon which serves to show in a clear and univocal way the structure of the torus, I symbolize here:

for using the same colors:
– from here to here a circle called the circle of demand [D],
– from here to here a circle called circle a [a] symbolizing the object of desire,
– and it is that circle [fig. 1]—which you see in the first figure—which is here drawn in yellow, representing the oblique circle, which could at a pinch serve us to symbolize, as the subject’s cut, desire itself.

The expressive, symbolic value, of the torus on this occasion, is precisely to make us see the difficulty—insofar as it is a matter of the surface of the torus and not of another—of ordering this circle here, yellow, of desire, with the circle, blue, of the object of desire. Their relation is all the less univocal in that the object is here fixed, determined, by nothing other than the place of a nothing which, if one may say so, prefigures its eventual place, but in no way allows it to be situated. Such is the exemplary value of the torus. You heard last time that this exemplary value is completed by this: that if we suppose it chained, concatenated with another torus insofar as it would symbolize the Other…

…we see that assuredly—this, I told you, can be demonstrated; I left it to you, this demonstration, to find yourselves, so as not to linger—we see that assuredly, by tracing thus the circle of desire projected on the first torus, on the torus that fits into it, symbolizing the place of the Other, we find a circle oriented in the same way.

[¹][²]
Remember, you represented opposite this figure [1]—which I shall do again if the thing does not seem too tedious to you—the tracing [2] which is a symmetrical image. We shall then have an oblique line, oriented from south to north, which we may call inverted, specular properly speaking. But the tipping by 90°, corresponding to the 90° fitting-into-one-another of the two tori, will restore the same obliquity:

In other words, after having effectively taken—these are very easy experiments to carry out, and they have all the value of an experiment—these two tori, and after having effectively made, by the method of rotating one torus inside the other that I indicated to you last time, this tracing, having recorded, if one may say, the trace of these two circles, arbitrarily drawn on one and from then on determined on the other, you will be able to see, in comparing them afterward, that they are exactly, relative to the circle that sections them, superposable one upon the other.

Whereby then this image proves appropriate to represent the formula that the desire of the subject is the desire of the Other.

Nevertheless, I told you, if we suppose, not this simple circle drawn in this property, in this particular topological definition: of both surrounding the hole and passing through it, but making it pass through the hole twice, and surround it only once, that is to say on the fundamental polygon, representing it thus [fig. 1], these two points here x, x’ being equivalent, we then have something which, on the tracing, at the level of the Other, presents itself according to the following formula [fig. 2].

If you wish, let us say that the realization of going around twice, which corresponds to the function of the object, and to the transfer, on the tracing on the other torus, twice over, of demand according to the formula of equivalence which is precious to us on this occasion, is to symbolize this: that, in a certain form of subjective structure:
the subject’s demand consists in the object of the Other, the subject’s object consists in the demand of the Other.

Recutting: then the superposition of the two terms after the tipping is no longer possible. After the 90° tipping, the cut is this one [fig. 2], which does not superpose upon the preceding form [fig. 1].

We recognized there a correspondence that is already familiar to us, insofar as what we can express of the neurotic’s relation to the Other insofar as it conditions, in the final term, his structure, is precisely this crossed equivalence:
– of the subject’s demand to the object of the Other,
– of the subject’s object to the demand of the Other.
One senses there, in a sort of impasse—or at least ambiguity—the realization of the identity of the two desires.

This is obviously as abbreviated as possible as a formula, and of course already presupposes an acquired familiarity with these references, which presuppose our whole prior discourse. The question therefore remaining open is that which we are going to address today, of a structure that would allow us to formalize in an exemplary way, rich in resources, in suggestions, that gives us a support for what our research points toward precisely, namely the function of fantasy.

It is for this end that the particular structure called the cross-cap or the projective plane can serve us, insofar as I have also already given you a sufficient indication of it so that this object may be, if not quite familiar to you, at least such that you have already attempted to deepen what it represents as exemplary properties.

I therefore apologize for entering, from now on, into an explanation which, for a moment, is going to remain very closely tied to this object of a particular geometry called topological geometry—non-metric but topological geometry—of which I have already pointed out to you as much as I could, in passing, what idea you should form of it, on the understanding that, after having taken the trouble to follow me in what I am now going to explain to you, you will then be rewarded by what it will allow us to support as a formula concerning the subjective organization that is the one that interests us, by what it will allow us to exemplify as being the authentic structure of desire, in what one could call its ‘central organizing function’. Of course I am not without reluctance at the moment, once again, of leading you onto terrains that may not be without tiring you.

That is why I shall refer for a moment to two terms that happen to be close in my experience, and that will give me the occasion first—a first reference—to announce to you the imminent publication of the translation made by someone eminent, who honors us today with his visit, namely Mr. De WAELHENS.

Mr. de WAELHENS has just completed the translation, of which one cannot sufficiently wonder that it was not done earlier, of Being and Time, Sein und Zeit, at least bringing to its point of completion the first part of the published volume, which you know is only the first part of a project whose second part never came to light.

So in this first part there are two sections and the first section is already translated by Mr. De WAELHENS, who did me the great honor—the favor—of communicating it to me, which allowed me myself to become acquainted with this part—still only half—and I must say with infinite pleasure, a pleasure that will allow me to grant myself a second one: namely finally to say, in this place, what I have had on my heart for a long time and what I have always refrained from professing in public, because in truth, given the reputation of this work, which I do not believe many people here have read, it would have looked like a provocation. It is this: that there are few texts clearer, indeed of a clarity and a concreteness and at last a directness, I do not know what qualifications I need to invent to add an additional dimension to the obviousness, than HEIDEGGER’s texts. It is not because what M. SARTRE made of it is indeed rather difficult to read that this takes anything away from the fact that this text of HEIDEGGER—I do not say all the others—is a text that carries within itself that sort of overabundance of clarity that makes it truly accessible, without any difficulty, to any intelligence not intoxicated by prior philosophical teaching. I can tell you this now, because you will very soon have the opportunity to notice it, thanks to the translation by Mr. De WAELHENS; you will see to what extent this is so.

The second remark is this, which you will be able to observe at the same time: assertions have circulated in bizarre little sheets, on the part of a professional dribbler, that my teaching is neo-Heideggerian. This was said with harmful intent. The person probably put ‘neo’ out of a certain prudence, since she knew neither what ‘Heideggerian’ meant, nor what my teaching meant either, and that sheltered her from a certain number of refutations, that this teaching of mine has truly nothing either ‘neo’ or Heideggerian about it, despite the excessive reverence I have for HEIDEGGER’s teaching.

The third remark is tied to a second reference, namely that something is going to appear—you are going to be treated soon—which is at least as important—well, importance is not measured, in different domains, with a centimeter—which is very important too, let us say: it is the volume, which is not yet in bookstores, I am told, by Claude LÉVI-STRAUSS called La Pensée sauvage. Has it appeared, you tell me?

I hope you have already begun to enjoy yourselves! Thanks to the demands imposed on me by our seminar, I have not gotten very far, but I have read the magisterial opening pages by which Claude LÉVI-STRAUSS enters into the interpretation of what he calls ‘La pensée sauvage’, which must be understood—as, I think, his interview in Le Figaro has already taught you [Cf. Gilles Lapouge: Le Figaro Littéraire of 02-06-1962, p. 3.]—not as the thought of savages, but as, one might say, the wild state of thought, let us say, thought insofar as it functions well, effectively, with all the characteristics of thought, before having taken the form of scientific thought, of modern scientific thought with its status.

And Claude LÉVI-STRAUSS shows us that it is quite impossible to place such a radical cut there since thought that has not yet conquered its scientific status is already altogether appropriate to bear certain scientific effects. Such is at least his apparent aim at the outset, and he takes quite singularly as an example, to illustrate what he means by sauvage thought, something in which no doubt he intends to rejoin that something common there would be with thought, let us say such as—he emphasizes it—such as it bore fundamental fruits beginning from the very moment that cannot be described as absolutely ahistorical since he specifies it: thought beginning from the Neolithic era gives us—he tells us—still all the foundations of our place-setting in the world.

To illustrate it, if I may say so, still functioning within our reach, he finds nothing else and nothing better than to exemplify it under a form, no doubt not unique but privileged by his demonstration, under the form of what he calls bricolage [DIY tinkering/makeshift construction]. This passage has all the brilliance we know in him, the originality proper to this sort of abruptness, novelty, thing that tips and overturns banally received perspectives, and it is a piece that is assuredly highly suggestive.

But it seemed to me precisely particularly suggestive for me, after the rereading I had just done, thanks to M. de WAELHENS, of Heideggerian themes: precisely insofar as he takes as an example in his search for the status, if one may say, of knowledge insofar as it can be established in an approach which, in order to establish it, claims to proceed from questioning concerning what he calls ‘being-there’, that is to say the form at once most veiled and most immediate of a certain type of ‘being’: the fact of being, which is that particular to the human being.

One cannot fail to be struck, although the remark would probably revolt both of these authors alike, by the surprising identity of the terrain on which both advance. I mean that what HEIDEGGER first encounters in this inquiry is a certain relation of being-there to a being which is defined as utensil, as tool, as that something one has at hand, Vorhanden, to use the term he uses, as Zuhandenheit, for what is within reach of the hand. Such is the first form of link, not to the world, but to being, that HEIDEGGER sketches for us. And it is only starting from there, namely, if one may say so, in the implications, the possibility, of such a relation, that he will, he says, give its proper status to what makes the first great pivot of his analysis: the function of being in its relation with time, namely Weltlichkeit, which M. de WAELHENS has translated as mondanéité [worldliness]. Namely the constitution of the world as in some sense prior, prior to that level of being-there which has not yet detached itself within being, these sorts of ‘beings’ that we can consider as purely and simply subsisting by themselves.

The world is something other than the aggregate, the encompassing of all those beings that exist, subsist by themselves, with which we deal at the level of that conception of the world that seems to us so immediately natural. And for good reason, because it is what we call nature. The anteriority of the constitution of this worldliness with respect to the moment when we can consider it as nature, such is the interval that HEIDEGGER preserves through his analysis.

This primitive relation of instrumentality prefiguring the Umwelt, still prior to the surrounding milieu which is constituted, in relation to it, only secondarily—there is HEIDEGGER’s move, and it is exactly the same…
I do not think I am saying anything here that could be retained as a criticism which, certainly, after all that I know of Claude LÉVI-STRAUSS’s thought and statements, would seem to us the move most opposed to his own, insofar as what he gives as the status of ethnographic inquiry would arise only from a position of aversion with respect to HEIDEGGER’s metaphysical, or even ultra-metaphysical inquiry
…and yet, it is indeed the same one that we find in this first step by which Claude LÉVI-STRAUSS intends to introduce us to sauvage thought under the form of this ‘bricolage’, which is nothing other than the same analysis, simply in different terms, a scarcely modified lighting, a no doubt distinct aim, of this same relation to instrumentality as what both consider anterior, primordial with respect to that sort of structured access that is ours, with respect to the field of scientific investigation, insofar as it allows it to be distinguished as founded on an articulation of objectivity that is in some way autonomous, independent of what is properly speaking our existence, and that with it we no longer retain except that so-called ‘subject–object’ relation which is the point at which, to this day, all that we can articulate of epistemology is summed up.

Well then, let us say—for fixing it once—what our enterprise here, insofar as it is founded on analytic experience, has that is distinct with respect to both of those inquiries whose parallel character I have just shown you, is that we too seek here this status, if one may say, prior to the classical access to the status of the object, entirely concentrated in the ‘subject-object’ opposition.

And we seek it in what? In that something which, whatever may be its evident character of approach, attraction, in thought—as much HEIDEGGER’s as Claude LÉVI-STRAUSS’s—is nevertheless quite distinct from it, since neither one nor the other names as such this object as object of desire. The primordial status of the object, for, let us say in any case, analytic thought, cannot be and must not be anything other than the object of desire.

All the confusions with which analytic theory has burdened itself up to now are consequences of this: of an attempt—more than one attempt: of all possible models of attempts—to reduce what imposes itself on us, namely this search for the status of the object of desire, to reduce it to already known references of which the simplest and most common is that of the status of the object of science insofar as a philosophizing epistemology organizes it in the final and radical ‘subject-object’ opposition, insofar as an interpretation, more or less inflected by the nuances of phenomenological inquiry, can at a pinch speak of it as the object of desire.

This status of the object of desire as such always remains eluded in all the forms thus far articulated of analytic theory, and what we seek here is precisely to give it its proper status. It is on this line that the aim I am pursuing before you for the moment is situated.

So here are the figures [on the blackboard] where today I am going to try to make you notice what interests us in this surface structure whose privileged properties are made to retain us as a structuring support of this relation of the subject to the object of desire, insofar as it is situated as supporting all that we can articulate, at whatever level of analytic experience, in other words as this structure that we call the fundamental fantasy.

For those who were not there at the previous seminar, I recall this form, drawn here in white; this is what we call cross-cap or, to be more precise—since, as I told you, a certain ambiguity remains regarding the use of this term Cross-cap—the projective plane.

As its drawing here in white chalk is not enough, for those who have not yet grasped it, to make you represent what it is, I am going to try to make you imagine it by describing it to you as if this surface were here made of balloon rubber. To be even clearer, I am going to start from the base. Suppose you have two hoops like those of a ‘wolf trap’. That is what will serve us to represent the cut.

If we orient the two circles of the ‘wolf trap’ in the same direction, that means that we are simply going to close them one upon the other. If you have a cut that is made in this way and you stretch balloon rubber from one to the other, precisely if you blow into it and if you close the wolf trap, it is still within reach of the most elementary imaginations to see that you are going to make a sphere: if the breath does not seem sufficient to you, you fill it with water until you obtain this form here, you close the two semicircles of the wolf trap, and you have a half-full sphere, or a half-empty one.

I have already explained to you how instead of that one can make a torus. A torus is this: you put the two corners of this handkerchief [joined in the air] like this, and the two others from below like this, and that is enough to make a torus. The essential thing about the torus is there since you have here the central hole, and here the circular void around which the circuit of demand turns. That is what the fundamental polygon of the torus has already illustrated for you. A torus is not at all like a sphere.

Naturally, a cross-cap is not at all like a sphere either. The cross-cap, you have it here:

You must imagine it as being, for this lower half, realized like the half of what you did just now with the balloon rubber, when you filled it with water or with your breath. In the upper part, what is here anterior will come to pass through what is continuous, what is here posterior.

The two faces cross one another, give the appearance of penetrating one another since the conventions concerning surfaces are free. For do not forget that we consider them only as surfaces, that we can say that no doubt the properties of space as we imagine it force us, in representation, to represent them as penetrating one another, but it is enough that we take no account of this line of intersection, at any moment in our treatment of this surface, for everything to proceed as if we held it for nothing.

It is not an edge, it is nothing but something that we are forced to represent to ourselves, because we want to represent here this surface, as a line of penetration. But this line, if one may say so, in the constitution of the surface, has no privilege. You will say to me: What does what you are in the process of saying mean?…

X, in the audience

Does that mean that you admit, with Kant’s transcendental aesthetic, the fundamental constitution of space in 3 dimensions, since you tell us that, in order to represent things here, you are forced to go through something which in the representation is in some way awkward?

LACAN

Of course, in a certain way, yes. All those who articulate what concerns the topology of surfaces as such begin—it is the b-a, ba [ABCs] of the question—from this distinction between what one can call the intrinsic properties of the surface and the extrinsic properties. They will tell us that everything they are going to articulate, determine, concerning the functioning of surfaces thus defined, is to be distinguished from what happens, as they literally put it, when one immerses the said surface in space, namely, in the present case, in three dimensions.

This fundamental distinction is also the one I have constantly recalled to you, to tell you that we must not consider the ring, the torus, as a solid and that, when I speak of the void that is central, of the circumference of the ring, as of the hole that is, if I may say so, axial to it, these are terms that must be taken within this: that we do not have to make them function insofar as we are aiming purely and simply at the surface.

It nonetheless remains the case that it is insofar as—as the topologists put it—we immerse this surface in a space, which we can leave in the state of x—what is the number of dimensions that structures it? We are in no way forced to prejudge it—that we can bring out this or that intrinsic property at stake in a surface. And the proof is precisely this: that the torus, we shall have no difficulty representing it in the three-dimensional space that is intuitively familiar to us, whereas for this one we shall still have a certain difficulty, since we shall have to add the small note of all sorts of reservations concerning what we have to read when we attempt to represent in this space this surface.

This is what will allow us precisely to pose the question of the structure of a space insofar as it admits or does not admit our surfaces as we have previously constituted them. These reservations having been made, I now ask you to continue and to consider what I have to teach you about this surface, precisely insofar as it is with respect to its representation in space that I am going to try to bring out for you certain of its characteristics, which are no less intrinsic for all that.

For if I have already eliminated the value that we can give to this line, the line of penetration, whose detail you see illustrated here:

Thus we can represent it; you see that merely by the way in which I myself have already drawn it on the board, there is here something that poses a question for us: is the value of this point that is here [a] a value that we can in some way erase, like the value of this line? Is this point likewise something that depends only on the necessity of representation in three-dimensional space?

I tell you right away, in order to illuminate my point somewhat in advance, this point, as to its function, is not eliminable, at least at a certain level of speculation on the surface, a level that is not defined solely by the existence of three-dimensional space. Indeed, what radically does the construction of this so-called cross-cap surface mean, insofar as it is organized from the cut that I represented to you earlier as a ‘wolf trap’ that closes? Nothing is simpler than to see that this ‘wolf trap’ must be bipartite when it is a matter of the sphere, since it must indeed fold back somewhere, and that its two halves are oriented in the same direction. The terminus ad quos will therefore be distinguished from the terminus ad quem insofar as they must overlap along their length.

We can say that here we have the way in which the two halves of the border that must be joined in order to constitute a projective plane function, one with respect to the other. Here, they are oriented in opposite directions, which means that a point situated in this place, point a for example, will correspond, will be identical, equivalent to a point situated in this place at a’, diametrically opposite, that another point b situated here for example will relate to another point b’ situated diametrically.

Does this not incline us to think that given this antipodal relation of points, on this circuit oriented in a continuous way always in the same direction, no point will have privilege, and that, whatever our difficulty in intuiting what is at issue, we must simply think this circular antipodal relation as a sort of radiating intercrossing, if one may say so, concentrating the exchange of a point with the opposite point of the single border of this hole, and concentrating it, if one may say so, around a vast central intercrossing that escapes our thought and therefore in no way allows us to give a satisfactory representation of it?

Nevertheless, what justifies things being represented thus is that there is something that should not be forgotten: it is that these are not metric figures.

Namely that it is not the distance from a to A, and from a’ to A’ that governs the point-by-point correspondence that allows us to construct the surface by organizing the cut in this way, but only the relative position of the points, in other words, in a set of three points that are situated on the half—allow me the use of the term half that I am using on this occasion, which is already represented by the analogical reference I made here to the two halves of the border—it is insofar as on this border, on this line, as on any line, a point can be defined as being between two others, that a point c, for example, will be able to find its correspondent in point c’ on the other side.

But if we do not have a point of origin, a point ἀρχήν [arken]…
Τὴνἀρχήνὄτικὰι λαλοὑμίν [tenarkeno ti kailaloumin], as is said in the Gospel, which has lent itself to such difficulties of translation that a thinker from Franche-Comté [Raymond Ruyer] thought he should say to me:
‘That is indeed where one recognizes you! The only passage of the Gospel on which no one can agree, that is the one you took as the epigraph for a part of your Rome report.’
…ἀρχήν [arken] therefore, the beginning, if there are no such points of beginning somewhere, it is impossible to define a point as being between two others, for c and c’ are just as much between these two others, a and B, if there is no AA’ to mark in a univocal way what happens in each segment.

It is therefore for reasons other than the possibility of representing them in space that we must define a point of origin for this intercrossed exchange, which constitutes the surface of the projective plane, between one border that—despite the fact that it always turns in the same direction—we still have to divide into two.

This may seem very tedious to you, but you are going to see that it is going to take on greater and greater interest. I announce right away what I mean to say, I mean to say that this point ἀρχήν [arken], origin, has a quite privileged structure, that it is this point, it is its presence that assures to the inner loop of our Polish signifier a status that is, itself, quite special. Indeed, so as not to keep you waiting any longer, I apply this signifier, called inner eight, to the surface of the cross-cap. We shall see afterwards what that means.

Observe all the same that to apply it in this way:

This means that the line drawn by our inner-eight signifier comes here to go twice around this privileged point. There, make an effort of imagination… I am willing to illustrate it for you with something. See what that can make:

You have here, if you wish, the bulge of the lower half [a], the bulge of the left claw of the lobster claw [b], the bulge of the right claw [c]. Here, it goes into the other, it passes to the other side [d].

What does that mean? It means that you have, in sum, a plane that winds like this on itself:

then which at a certain moment passes through itself, so that this makes something like two kinds of shutters, or superposed flapping wings here, which are in sum, by the cut, isolated from the lower bulge, and at the upper level these two wings cross one another. It is not very inconceivable.

If you had interested yourselves in this object as long as I have, obviously it would seem little surprising to you, for in truth, the privilege of this double cut, that is very interesting. It is very interesting in this sense that, concerning the torus:

I have already shown you:
– if you make one cut [1] it transforms it into a band.
– if you make a second [2] that crosses the first, that does not fragment it for all that. That is what allows you to spread it out like a fine square.
– if you make two cuts that do not recross, on a torus—try to imagine that—there you necessarily put it into two pieces:

Here, on the cross-cap, with a cut that is a simple cut like the one that can be drawn thus:

you open this surface—amuse yourselves by drawing it, it will be a very good intellectual exercise to know what happens at that moment—you open the surface, you do not cut it in two, you do not make two pieces of it. If you make any other cut, whether it crosses or does not cross, you divide it.

What is paradoxical and interesting is that in sum here it is only a single cut throughout [‘inner-eight’ cut], and that nevertheless, by simply making it go twice around the privileged point, you divide the surface.

It is not at all the same on a torus. On a torus, if you go as many times as you wish around the central hole, you will never obtain anything but, in some way, a lengthening of the band, but you will not divide it for all that.

This is to make you notice that we are touching here, no doubt, something interesting concerning the function of this surface. There is moreover something no less interesting, namely that this double turn, with this result, is something that you cannot repeat a single time more. If you make a triple turn, you will be led to draw on the surface something that will repeat itself indefinitely, in the manner of the loops you make on the torus when you engage in the winding operation I spoke to you about at the outset, except that here the line will never rejoin itself, will never bite its own tail:

The privileged value of this double turn is therefore sufficiently assured by these two properties. Let us now consider the surface that this double turn isolates on the projective plane. I am going to point out certain of its properties to you.

First, it is what we can call a surface—let us call it that, for the sake of speed, among ourselves if one may say so, since I am going to remind you what that means—it is a left-handed surface [gauche can mean left/awkward], like a left-handed body, like anything whatsoever that we can define as such in space. I am not using it to oppose it to right, I am using it to define this, which you should know well, namely that if you want to define the winding of a snail, which as you know is privileged, dextrogyrous or levogyrous no matter, it depends how you define one or the other, this winding, you will find it the same whether you look at the snail from the side of its tip or whether you turn it over to look at it from the side where it begins a hollow.

In other words, it is that if one turns the cross-cap over here to see it from the other side, if we define here the rotation from left to right moving away from the central point, you see that it still turns in the same direction on the other side.
recto verso
This is the property of all bodies that are dissymmetrical. So it is indeed a dissymmetry that is at issue, fundamental to the form of this surface. As proof, below you have something that is the image of this surface thus defined on our double loop, in the mirror. Here it is:

[a][b]
We must expect that, as in every dissymmetrical body, the image in the mirror will not be superposable on it, just as our image in the mirror, we who are not symmetrical despite what we believe, does not at all superpose on our own support: if we have a mole on the right cheek, this mole will be on the left cheek of the image in the mirror.

Nevertheless the property of this surface is such that, as you see, it is enough to raise that loop there [a] just a little bit, and it is legitimate to make it pass above the other, since the two planes do not really pass through one another, for you to have an absolutely identical image [b], and therefore superposable on the first, on the one from which we started.

You see what happens: raise that very gently, progressively up to here, and see what is going to happen, namely that the occultation of this little dotted part situated here is the identical realization of what is in the primitive image.

This serves us to illustrate this property that I told you was that of (a) as object of desire, namely to be that something which is at once orientable—and assuredly very oriented—but which is not, if I may so express myself, ‘specularizable’. At this radical level that constitutes the subject in his dependence with respect to the object of desire, the function i(a), the specular function, loses its grip, if one may say so.

And all this is governed by what? By something that is precisely this point [the central point] insofar as it belongs to this surface. To clarify right away what I mean, I shall tell you that it is by articulating the function of this point that we shall be able to find all sorts of felicitous formulas that allow us to conceive the function of the phallus at the center of the constitution of the object of desire.

That is why it is worth our while to continue taking an interest in the structure of this point. This point, insofar as it is what is the key to the structure of this surface thus defined, cut out by our cut in the projective plane. This point, I must pause for a moment to show you what its true function is. This will of course still require a little patience from you. What is the function of this point?

What at this moment where we stop is manifest there, is that it is in one of the two parts into which—by the double cut—the projective plane is divided. It belongs to this part, which detaches itself, it does not belong to the part that remains.

Since it seems that you were capable earlier—I must at least infer it from the fact that no murmur of protest arose—of conceiving how this figure can pass to this one by simple legitimate displacement of the level of the cut, you are, I think, just as capable of making the mental effort to see what happens:

– if on the one hand we make this cut [a] cross the horizon of the lower dead-end bottom of the surface, thus making it pass to the other side as indicated by my yellow arrow,

– and if we make the upper part of the loop likewise cross the horizon of what is at the top of the cross-cap [b], this leads us without difficulty to the following figure.

The passage to the last [c] is a little more difficult to conceive, not for the lower loop as you see, but for the upper loop, insofar as you may perhaps have a moment of hesitation concerning what happens at the moment of crossing what here presents itself as the end of the line of penetration.

If you reflect on it a little, you will see that if it is on the other side that the cut is brought to cross this line of penetration, obviously it will present itself like that [c], that is to say as it is on the other side, it will be dotted on this side, and it will be solid, since according to our convention what is dotted is seen through transparency.

Nothing in the structure of the surface allows us to distinguish the value of these cuts, therefore of those at which we arrive here, but to the eye they present themselves as both entering from the same side of the line of penetration.

Is that very simple for the eye? Surely not. For this difference between, for the cut, entering from two different sides or entering from the same side, is something that still must show itself in the result, on the figure. And besides, this is entirely perceptible. If you reflect on what it is, what is henceforth cut out on this surface [d], you will recognize it easily: first, it is the same thing as our signifier. In addition, by the way it cuts out a surface, it cuts out a surface of which you feel very well—you have only to look at the figure—that it is a band, a band with only one border. I have already shown you what that is: it is a Möbius surface [S].

Now, the properties of a Möbius surface are properties completely different from those of this little turning surface [a] whose properties I showed you earlier by turning it over, mirroring it, transforming it, and finally telling you that it is that one that interests us.

This little ‘sleight of hand’ obviously has a reason that is not hard to seek. Its interest is simply to show you that this cut always divides the surface into two parts, one of which keeps the point at issue in its interior, and the other of which no longer has it. This other part, which is just as present there as in the terminal figure, is a Möbius surface. The double cut always divides the surface called ‘cross-cap’ into two:
– that something [a] in which we are interested and of which I am going to make for you the support of the explanation of the relation of S with (a) in fantasy,
– and on the other side, a Möbius surface [S].

What is the first thing I made you grasp when I gave you that little set of five or six Möbius surfaces that I threw across the assembly? It is that the Möbius surface, itself, in the sense in which I meant it earlier, is irreducibly left-handed; whatever modification you make it undergo, you will not be able to superpose on it its image in the mirror.

So there is the function of this cut and what it shows as exemplary. It is such that, by dividing a certain surface in a privileged way, a surface whose nature and function are completely enigmatic to us since we can scarcely situate it in space, it brings out privileged functions on one side, which are those I called earlier being specularizable, that is to say involving its irreducibility to the specular image, and on the other side, a surface which, although presenting all the privileges of an oriented surface, is not specularized.

For note well that of this surface one cannot say, as on the Möbius surface, that an infinitely flat being walking along will suddenly find itself on this surface on its own reverse side: each face is indeed separated from the other in this one. This property, of course, is something that leaves an enigma open, for it is not so simple. It is all the less simple in that the total surface, this is quite evident, is reconstructible—and reconstructible immediately—only from this one [a]. The most fundamental properties of the surface must therefore indeed be conserved somewhere, despite its more rational appearance than that of the other, in this surface. It is quite clear that they are conserved at the level of the point.

If the passage which, in the total figure, always makes it possible for an infinitely flat traveler to find himself again, by an exceedingly brief path, at a point that is his own reverse side, I say, on the total surface, if that is no longer possible at the level of the central surface fragmented, divided by the signifier of the double loop, it is because precisely something of that is conserved at the level of the point.

With this reservation, precisely, that for this point to function as this point, it has this privilege of being, precisely, uncrossable, except at the cost of making the whole structure of the surface vanish, if one may say so. You see, I have not even yet been able to give full development to what I have just said about this point. If you reflect on it, you will be able by next time to find it yourselves.

The hour is late and that is indeed where I am forced to leave you. I apologize for the aridity of what I have been led today to present before you, due to the complexity itself, although it is a complexity extraordinarily punctiform, that is the word for it.

That is where I shall take it up next time. I therefore return to what I said at the beginning: the fact that I could only get as far as this point in my presentation will mean that next Wednesday’s seminar—tell this to those who received the next notice—will be maintained with the design of not leaving too much space, too much interval, between these two seminars, for that space could be harmful to the continuation of our explanation.