We have a circular edge that simply connects to itself:

There is not yet any surface for the ant to walk on, which means:

      no over     no under

Since over/under refers to place/placed, this pair translates to:

      nowhere     none there

These are equivalent but not equal. The edge has thereby self-separated and self-combined. Now we have a twisted edge, which is a Möbius strip:

In the Möbius strip, there is a surface between the twisted edge. Now the ant can walk over the surface, but it also ends up walking under the surface:

       over = under

This is the elementary form of the subject $.

In order to distinguish over from under (place from placed), we must strip the Möbius strip: We disentangle the twist (Möbius-without-strip) from the edge (strip-without-Möbius) and get the cross-cap (marked 1 in the figure):

In the cross-cap, there is a twisted surface delimited by an ordinary edge:
— The twist is the wavering line (which looks like the mouth of a carnivorous plant).
— The edge is the bottom circle (which looks like a gray belt).
By being disentangled, the twist becomes explicit and distinguishes over from under:

       over / under   (cover)

The cross-cap covers the distinction without cancelling it, by sustaining its permeability. This is the fantasy $ ◊ a.

If we chose to cancel the distinction, we could erase the edge by mounting a disc over it (marked 2 in the figure). The resulting combination 1+2 is called the cross-capped disc [1]. When “distinction” is cancelled, it turns into “this thing shown” and becomes impermeable:

        over // under   (discover)

The following operations are equivalent:
— Mark “this cover”
— Discover “this thing shown”
— Mount a “disc over” the cross-cap
They all lead to the Master-Signifier S₁.

But there is another option: One can maintain the distinction by marrying one cross-cap to another cross-cap, which yields a Klein bottle:

In the Klein bottle, there is a virtual edge that erases itself by attaching one twist to another twist. Over and under are thus attached:

          over + under

This is the symptom S(Ⱥ). The place is the feminine subject and what is placed there is the unconscious:

          of her + unheard

Beware that the symptom is ordinarily disguised by commonsense:

          of her? offer!        unheard? endear!

          of her? lover!        unheard? endure!

Psychoanalysis can only work through uncommonsense by attaching the symptom back to the initial twist of the edge:

          of her? nowhere…        unheard? none there…

Işık Barış Fidaner is a computer scientist with a PhD from Boğaziçi University, İstanbul. Admin of Yersiz Şeyler, Editor of Žižekian Analysis, Curator of Görce Writings. Twitter: @BarisFidaner

Notes:

[1] The literature (including Lacan and Žižek) often confuses the cross-cap with the cross-capped disc and thereby automatically erases the edge and cancels the distinction. See “The Möbius Strip is an Island”

The cross-cap image is from Lacan’s Seminar 9 (French Edition, page 441).