🦋🤖 Robo-Spun by IBF 🦋🤖

(All parts in English)

This elaboration on the surface, I justify its necessity; it is obvious that what I am giving you is the result of a reflection. You have not forgotten that the notion of surface in topology is not self-evident and is not given as an intuition. The surface is something that does not go without saying. How are we to approach it?

Starting from what, in the real, introduces it, that is to say what would show that space is not that open and contemptible expanse as BERGSON thought. Space is not as empty as he believed; it harbors many mysteries.
Let us set out certain terms from the start. It is certain that a first essential thing in the notion of surface is that of face: there would be two faces or two sides. That goes without saying if this surface, we immerse it in space.

But in order to appropriate for ourselves what the notion of surface can become for us, we must know what it delivers to us from its dimensions alone. To see what it can deliver to us as a surface dividing space by its dimensions alone suggests a beginning that will allow us to reconstruct space otherwise than we believed we had the intuition of it.

In other words, I propose that you consider it more evident, by virtue of imaginary capture, simpler, more certain because linked to action, more structural, to start from the surface in order to define space — of which I maintain that we are little assured — let us rather say define place, than to start from place — which we do not know — in order to define the surface.
You may moreover refer to what philosophy has been able to say about place. The place of the Other already has its place in our seminar.

To define the face of a surface, it is not enough to say what is on one side and on the other, all the more so since that is not at all satisfying, and if something gives us Pascalian vertigo, it is indeed those two regions into which the infinite plane would divide all space. How is this notion of face to be defined? It is the field over which a line, a path, can extend without having to encounter an edge. But there are surfaces without edge: the plane at infinity, the sphere, the torus, and several others which, as surfaces without edge, are practically reducible to a single one, the cross-capormitreorbonnet, figured here:

fig.1

The cross-cap in scholarly books is this, cut so that it can be inserted onto another surface:

fig.2

These three surfaces, sphere, torus, cross-cap, are elementary closed surfaces to whose composition all other closed surfaces can be reduced. I shall nevertheless call figure1 the cross-cap. Its true name is the projective plane of the theory of RIEMANN surfaces, of which this plane is the basis. It brings in at least the fourth dimension. Already the third dimension, for us ‘depth psychologists’, poses enough of a problem for us to consider it as little assured.
Nevertheless, in this simple figure, the cross-cap, the fourth is already necessarily implicated. The elementary knot,
made the other day with a piece of string, already makes the fourth dimension present.

There is no valid topological theory without our bringing in something that will lead us to the fourth dimension. If this knot, you want to try to reproduce it by using the torus, by following the turns and detours that you can make on the surface of a torus, you could after several turns come back upon a line that closes up like the knot above. You cannot do it without the line cutting itself.

Since on the surface of the torus you will not be able to mark that the line passes above or below, there is no way to make this knot on the torus. It is, on the other hand, perfectly feasible on the cross-cap. If this surface implies the presence of the fourth dimension, it is a beginning of proof that the simplest knot implies the fourth dimension.
This surface, the cross-cap, I am going to tell you how you can imagine it. That will not thereby impose its necessity, for us, as established. It is not without relation to the torus; it even has with the torus the deepest relation.

The simplest way to give you this relation is to remind you how the torus is constructed when it is decomposed in polyhedral form, that is to say by reducing it to its fundamental polygon. Here, this fundamental polygon is a quadrilateral. If you fold this quadrilateral back onto itself, so that what is a here joins to a’, you will have a tube by joining the edges:

→
If one vectorizes these edges by agreeing that only vectors going in the same direction can be joined to one another, the beginning of one vector being applied to the point where the other vector ends, from then on one has all the coordinates for defining
the structure of the torus. If you make a surface whose fundamental polygon is thus defined by vectors all going
in the same direction on the basic quadrilateral:

If you start from a polygon defined in this way, that would make only two edges, or even only one, you obtain what I materialize for you as the mitre[fig.1]I shall return to its function of symbolizing something and it will be clearer when this name serves as a support.In section, with its jaw-mouth look, it is not what you think.

This: is a line of penetration by virtue of which what is in front, below is a hemisphere; above, the wall passes by penetration into the opposite wall and comes back to the front. Why this shape rather than another? Its fundamental polygon is distinct from that of the torus:

toruscross-cap
A polygon whose edges are marked by vectors of the same direction, and distinct from that of the torus, which starts from one point
to go to the opposite point — what kind of surface does that make?

From now on problematic points of these surfaces emerge. I introduced surfaces without edge to you with respect to the face. If there is no edge, how define the face? And if we forbid ourselves as much as possible from plunging our model too quickly into the third dimension, where, where there is no edge, we would be assured that there is an inside and an outside.
That is what this surface without edge par excellence, the sphere, suggests.

I want to detach you from this uncertain intuition: there is what is inside and what is outside. Yet for the other surfaces that I enumerated, this notion of inside and outside slips away. For the infinite plane, it would not suffice.
For the torus, intuition seems to cling sufficiently in appearance because there is the inside of an inner tube and the outside.
Nevertheless, what happens in the field through which this outside space traverses the torus, that is to say the space of the central hole,
there lies the topological nerve of what made the torus interesting and where the relation of inside and outside is illustrated by something that can affect us.

Note that up to FREUD, traditional anatomy, somewhat Naturwissenschaft, with PARACELSUS and ARISTOTLE,
always counted among the orifices of the body the sense organs as authentic orifices. Psychoanalytic theory,
insofar as it is structured by the function of libido, made a very narrow choice among the orifices and does not speak to us of sensory orifices as orifices, except by reducing them to the signifier of the orifices first chosen. When one has made of scopophilia
a scopophagy, one says that scopophilic identification is an oral identification, as FENICHEL does.

The privilege of the oral, anal, and genital orifices holds us here in this: these are not really the orifices that open
onto the inside of the body. The digestive tube is only a passage-through; it is open to the outside. The true inside is the mesodermic inside and the orifices that introduce into it indeed exist in the form of the eyes or the ear, of which psychoanalytic theory never
makes mention as such, except on the cover of the journal La Psychanalyse.

That is the true scope given to the central hole of the torus, even though it is not a true inside, but rather that it suggests to us something of the order of a passage from inside to outside. This gives us the idea, which comes upon inspection of this closed surface: the cross-cap. Suppose something infinitely flat moving on this surface:

passing from the outside¹ of the closed surface to the inside² in order to continue farther on the inside³ until it arrives at the line of penetration where it will re-emerge on the outside⁴ from the back. This shows the difficulty of defining the distinction inside-outside, even when it is a matter of a closed surface, a surface without edge.

I have only opened the question; this is not to propose a paradox to you, it is to remind you that what is important
in this figure of the mitre is that this line of penetration must be held by you as null and void.
One cannot materialize it on the board without bringing in this line of penetration, because ordinary spatial intuition
requires that one show it, but speculation takes no account of it.

This line of penetration can be made to slide indefinitely. There is no reflection of one surface onto the other,
nothing of the order of a seam, there is no possible passage. Because of this, the problem of the inside and the outside
is raised in all its confusion. There are two orders of considerations regarding the surface: metric and topological.

One must renounce all metric consideration — indeed from this square I could give the whole surface —
from the topological point of view that makes no sense. Topologically the nature of the structural relations constituting
the surface is present at every point: the inner face merges with the outer face, the definition of the surface determining all its points and its properties.

To mark the interest of this, we are going to evoke a question never yet posed concerning the signifier:
does not a signifier always have as its place a surface?

That may seem a bizarre question, but it has at least the interest, if it is posed, of suggesting a dimension.
At first glance the graphic as such requires a surface. Insofar as the objection may arise that an upright stone,
a Greek column is a signifier and that it has a volume. Well then, do not be so sure, so sure that you can introduce
the notion of volume before being quite assured of what concerns the notion of surface.

Especially if, putting things to the test, you notice that the notion of volume is not graspable except starting from that of the envelope. No upright stone ever interested us for anything other, I will not say, than its envelope,
which would be to fall into a sophism, but for what it envelops. Before being volumes, architecture was made by mobilizing, by arranging surfaces around a void.

What are upright stones for? To make alignments or tables, to make something that serves by the hole there is around it. For that is the remainder with which we have to deal.

If, grasping the nature of the face, I started from the surface with edges in order to make you notice that the criterion failed us with surfaces without edge, if it is possible to show you a fundamental surface without edge, where the definition of the face
is not forced, since the surface without edge is not made to resolve the problem of the inside and the outside,
we must take account of the distinction between a surface ‘without’ and a surface ‘with’: it has the closest relation
with what interests us, namely the hole which must be positively brought in as such into the theory of surfaces.

This is not a verbal artifice. In the combinatory theory of general topology, every triangulable surface, that is to say composable of small triangular pieces that you glue to one another, torus or cross-cap, can be reduced by means of the fundamental polygon to a composition of the sphere to which would be adjoined more or less toric elements, cross-cap elements, and pure hole elements indispensable represented by this vector looped back onto itself.

Cross-cappure hole

Can a signifier, in its most radical essence, be envisaged only as cut: ><, in a surface?
These two signs: greater-than: >, and less-than: <, imposing themselves only by their structure of cut inscribed on something
where there is always marked, not only the continuity of a plane on which the sequence will be inscribed, but also the vectorial direction
where this will always be found again.

Why has the signifier in its bodily incarnation, that is to say vocal, always presented itself to us as of discontinuous essence? We therefore had no need of the surface: discontinuity constitutes it, interruption in the successive is part of its structure.
This temporal dimension of the functioning of the signifying chain that I first articulated for you as succession,
has as a consequence that scansion introduces one more element than the division of modulatory interruption; it introduces haste that I inserted as logical haste. It is an old work: Logical Time.[Écrits p. 197]

The step I am trying to make you cross has already begun to be traced: it is the one where discontinuity is knotted with what is the essence of the signifier, namely difference. If that on which we have made pivot, we have ceaselessly brought back this function
of the signifier, it is to draw your attention to this: even in repeating the same, the same by being repeated is inscribed as distinct.

Where is the interpolation of a difference?

– Does it reside only in the cut? It is here that the introduction of the topological dimension beyond temporal scansion interests us,

– or in that something else that we shall call ‘the simple possibility of being different’, the existence of the differential battery that constitutes the signifier and by which we cannot confuse synchrony with simultaneity, at the root of the phenomenon.

Synchronicity such that, reappearing the same, it is as distinct from what it repeats that the signifier reappears, and what may be considered as distinguishable is the interpolation of difference, insofar as we cannot posit as foundation of the signifying function the identity of ‘A is A’, namely whether the difference is in the cut,
or in the synchronic possibility that constitutes signifying difference.

In any case, what repeats itself as signifier is different only in being able to be inscribed. It remains no less true that the function of
the cut matters to us first and foremost in what can be written. And it is here that the notion of topological surface must be introduced into our mental functioning because it is only there that the function of the cut takes on its interest.

Inscription bringing us back to memory is an objection to be refuted. The memory that interests us, we analysts,
is to be distinguished from an organic memory, the one — if I may put it so — that, to the same ‘suction’ of the real would respond by the same way for the organism to defend itself against it: the one that maintains homeostasis, for the organism does not recognize the same
that renews itself as different. Organic memory ‘same-orizes’. Our memory is something else: it intervenes
as a function of the unary trait, marking the unique time, and has inscription as its support. Between stimulus and response,
inscription, printing, must be recalled in terms of Gutenberg printing.

The first draft of psychophysical theory against which we revolt is always atomistic; it is always
to impression in surface schemas that this psychophysics takes its first basis. It is not enough to say
that this is insufficient, before one has found something else.

For if it is of great interest to see that the first theory of relational life was inscribed in interesting terms
that translated — only without knowing it — the very structure of the signifier under the masked forms of the distinct effects
of contiguity and continuity, psychological associationism, if it is good to show that what was recognized and misrecognized as signifying dimension was the effects of signifier in the idealist world-structure from which this psychophysics
never detached itself, conversely what has been translated by Gestalt is insufficient to account for what happens at the level of vital phenomena, by reason of a fundamental ignorance that is expressed by the rapidity with which one takes for certain evidences that everything contradicts. The supposed good form of the circumference that the organism would stubbornly,
on all levels, subjective or objective, seek to reproduce is contrary to every observation of organic forms.
I would say to the Gestaltists that a donkey’s ear resembles a funnel, an arum, a Möbius surface.

A Möbius surface is the simplest illustration of the cross-cap: it is made with a strip of paper
whose two ends are glued together after twisting it, so that the infinitely flat being that walks on it can follow it
without ever crossing an edge. This shows the ambiguity of the notion of face. For it is not enough to say that it is a one-sided surface, with a single face, as some mathematicians formulate it: a formal definition is something else.

It remains no less true that there is coalescence for each point of two faces and that is what interests us. For us who do not content ourselves with calling it one-sided on the pretext that the two faces are present everywhere, it remains no less true
that we can manifest at every point the scandal for our intuition of this relation to two faces.
Indeed on a plane, if we trace a circle turning clockwise, on the other side,
through transparency, the same arrow turns in the opposite direction.

The infinitely flat being, the little figure on the Möbius strip, if he carries with him a circle turning around him
clockwise, this circle will always turn in the same direction, so that on the other side
of his starting point, what will be inscribed will turn clockwise — that is to say in the opposite sense to what would happen
on a normal strip, on the plane, where on the other face it turns in the contrary sense — it is not inverted.
That is why these surfaces are defined as non-orientable and yet it is no less oriented.

Desire, from not being articulable, we cannot for all that say that it is not articulated. For these little ears
in the Möbius strip, however non-orientable they may be, are more oriented than a normal strip.
Make yourself a conical belt, turn it inside out: what was open below is open above.

But the Möbius strip, turn it over: it will always have the same shape. Even when you turn the object over,
there will always be the recessed bulge on the left, the swollen bulge on the right. A non-orientable surface
is thus much more oriented than an orientable surface.

Something goes even further and surprises mathematicians who send the reader back with a smile to the experiment,
which is that if in this Möbius surface, with the help of scissors, you trace a cut at equal distance from the most accessible points of the edges — it has only one edge… — if you make a circle, the cut closes, you realize a cycle, a lacing, a Jordan closed curve. Now this cut not only leaves the surface whole, but transforms your non-orientable surface
into an orientable surface, that is to say into a strip of which, if you color one of the sides, one whole side will remain white, contrary
to what would have happened earlier on the whole Möbius surface, everything would have been colored without the brush changing face.

The simple intervention of the cut changed the omnipresent structure of all the points of the surface, I was telling you.
And if I ask you to tell me the difference between the object before the cut and this one, there is no way to do it.
This is to introduce the interest of the function of the cut.

The quadrilateral polygon is originary for the torus and the bonnet. If I have never introduced the true verbalization of this form: ◊, punch, desire uniting the S to the (a) in S◊a, this little quadrilateral must be read as: the subject insofar as marked by the signifier
is properly, in fantasy, cut of a.

Next time, you will see how this will give us a functioning support for articulating the question,
how what we can define, isolate from demand as the field of desire, on its ungraspable side,
can by some torsion be knotted with what, taken from another side, is defined as the field of the object(a).

How can desire be equal to (a)?

That is what I have introduced, and what will give you a useful model right into your practice.