What does it show?
The Möbius strip, described in 1858 by the German mathematician August Ferdinand Möbius, is the simplest example of a non-orientable surface. The ant in the model walks without ever leaving the surface: after one tour it arrives exactly behind the point where it started; at the end of the second tour it is back where it began. The strip has no inside and no outside. Its edge, too, is a single closed curve.
In his seminars of the 1960s, in particular L’identification (1961–1962) and Problèmes cruciaux pour la psychanalyse (1964–1965), Lacan used the Möbius strip to show the structure of the subject. Conscious and unconscious, what is said and the saying, are not two separate sides but two moments of a single side, seen to be joined only once one has gone all the way round.
The “cut down the middle” button shows one of topology’s most surprising experiments: the strip does not fall into two pieces but becomes a single band, twice as long and with two full twists; this new band is now two-sided. For Lacan interpretation is such a cut: it does not break the surface apart, it transforms its structure.
Related essay: The Established Text