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Lacan and the Torus

14:28 / 04.06.04

In his text "Of Structure as the Inmixing of an Otherness Prerequisite to Any Subject Whatever" (1966/70) Lacan stated that "[...] one can show that a cut on a torus corresponds to the neurotic subject [...]".

What are we to make of this?

06:41 / 05.06.04

Umm, not much to go on here, eh?

What happens when one cuts a torus? That is, what sort of form or structure is the result? And, what does Lacan mean by the phrase "neurotic subject"?

Our Lady Has Left the Building

10:37 / 05.06.04

Is this the full article you had in mind DecayingInsect?

Dictionary.com gives us:

to·rus
Architecture. A large convex molding, semicircular in cross section, located at the base of a classical column.
Anatomy. A bulging or rounded projection or swelling.
Botany. The receptacle of a flower.
Mathematics. A toroid generated by a circle; a surface having the shape of a doughnut.

04:37 / 06.06.04

He's talking about it in the mathematical sense - Lacan has a whole line on weird geometric concepts as analagous to various psychic processes. I can't help much more without more context - and, annoyingly, the only reference I can think of that might help is Sokal and Bricmont's Intellectual Impostures, where they discuss L's use of maths. And I hate that book. But it might help.

05:55 / 07.06.04

Apparently there are two fundamentally different sorts of tori (is this the proper plural?), depending on whether or not there are an even or odd number of twist in the construction of the figure (zero, here, being even, I suppose).

Read an explanation about it here.

Now, judging by Lacan’s figure in the article, he is talking about a torus with an odd number of twists, namely, one half twist. This is, apparently, topologically equivalent to a Mobius Strip that is made out of paper. The link talks about what happens when you cut a strip with various sorts of twists.

The interesting thing about the Mobius Strip is that you start with a piece of paper that has two sides, but after you twist it, you end up with a figure that has only one side. So I think this is related to what Lacan is going on about regarding something that is “…at the same time both one or two.” However, when it is cut, you end up with a loop of paper that has an even number of twists, which, when cut again, ends up as two interlocked rings (try it out).

So, if the Mobius Strip is representative of the subject, and a cut along the Mobius Strip gives us a figure which is no longer both one or two, then perhaps this goes some way to helping you decipher what Lacan might be on about?

10:28 / 07.06.04

I've actually spent a little time on this, as I once worked with a Lacanian psychoanalyst who tried to explain some of these concepts to me and threw reading material my way.

Let me get some of the maths out of the way (the cat's iao has some of this wrong). I say "torus" to mean a hollow doughnut and "solid torus" to mean a solid one.

If you think of a torus as being traced out by a "revolving" circle, then the number of half-twists gives you different geometric structures on it (in some sense). These are all equivalent topologically. Same for the solid torus, where the twisting gives different geometry and equivalent topology. Both the torus and solid torus are topologically different from a Moebius strip (and from each other), although the mobius strip is contained in the solid torus.

Now, all this is much less useful than you'd think since, as Sokal and Bricmont point out, Lacan writes down lots of mathematical stuff that is simply nonsense from a math point of view. The idea, as far as I can tell, is to set up an interesting analogy that opens up new conceptual frameworks. So it could well be that we should regard tori and moebius strips as essentially the same when looking at Lacan, even though they aren't in mathematics. The combination of the technicality and the fluidity of meaning make it pretty hard to understand though.

10:41 / 07.06.04

Both the torus and solid torus are topologically different from a Moebius strip (and from each other), although the mobius strip is contained in the solid torus.

Eh? A Mobius Strip made of paper: if you consider the thickness of the paper, a ring that makes a mobius strip is topologically the same as solid tori.

11:38 / 07.06.04

I think you are using "topological" to mean something weaker like "homotopic", but even then it isn't true. For a start, a solid torus is orientable and a moebius strip isn't. A solid torus is a lot more like an annulus than a moebius band, if you think about it.

11:51 / 07.06.04

I tell a lie. They *are* homotopic, but not topologically the same, which is a subtle distinction.

13:12 / 07.06.04

Thanks to the people who have taken the time to reply.

I think I'm OK on the math and am coming to this more from the angle of an interested outsider to critical theory.

The text in which the quote occurs is up here.

It's intruiging that L. suggests that other other examples can be given but does not on that occasion go into details.

Can those details be sought elsewhere? Exegesis from the cognoscenti would be welcome!

Also it seems that there is going to be a book:

Lacan: Topologically Speaking

featuring articles by 'Lacanian Topologists'... some names are given ---is anyone familiar with their work?

D.I.

00:55 / 07.07.04

very loosely; lacan is always going on about the divided subject writen as the barred S or S with a 'cut' through it. the neurotic subject is writen $<>a. a as the cause of desire is the object the neurotic subject attempt to 'refind' in the fundamental fantasy writen $<>a above. this a whether of the fantasy or the drive itself is what is sort -however indirectly- in speech but fundamentally misrecoignized and speech is determinant and a is not. in other psycopathalogical structures the cut is not so grreat and thew drive might be enjoyed in a more direwct manner than through speech.... or so they say...