The Möbius strip (MS) is rightfully famous for its single surface: When you stand on the MS, there seems to be an inaccessible “other side” under the ground where you stand. But if you walk along the MS for a sufficient duration, you eventually reach the “other side”, which means that it’s not really inaccessible: You can access the “other side” by sacrificing some of your time. In fact, this also means that there is no “other side”: There is only a single surface. You can explore the whole MS by traversing a distance that encompasses two whole lengths of the MS. There is an edifying lesson in this parable: The initial appearance of a disturbing non-identity is revealed to be an illusion that you can overcome by spending enough time following a pathway by means of which you will eventually reach the identity of the One-single-surface. However, this story does not cover the truly distinctive property of the MS: its unorientableness.
What is left unmentioned about the MS is that it also has a single edge: When you stand on the MS, there seem to be two edges that delimit your pathway, on your left and on your right. When you walk along the MS, the distinction between your left and right sides is constantly conserved as what orients your direction. This conserved orientation is symbolized by the shape of the equal sign ‘=’ which we use to denote a relation of constancy. The promise of constancy in the equal sign ‘=’ encourages the quest for the One. This promise orients your march along the MS.
You hear about the parable about the illusory “other side” and you intend to traverse the MS. You embark on your voyage to explore the promised constant identity of the One-single-surface. But this is what happens: After some time, you realize that what marks your “left side” and “right side” have switched. Notice that this switching never happens in the present time, in the here and now; instead it “will have happened” when you realize it after the fact. This is when you lose your orientation. Let’s symbolize this moment of realization of the switching by the shape of the cross sign ‘✕’. At this moment, you realize that these “two sides” are in fact parts of the same single edge. The promise of the constancy of the One is broken and you fall into the gap between the single edge. Your sacrifice for the identity of the One-single-surface is itself sacrificed and rendered meaningless.
So what is this single edge of the MS? Let me call this edge, the coastline of the MS. This makes the MS an island.
The coastline of the MS is the cut of the empty set that Rothenberg describes here (despite her particular focus on the MS, she never mentions its single edge):
The simple addition of a formal property, the empty set, which has no substance in and of itself, negates the state of sheer being that attends each thing-as-such. It does so by establishing a minimal point of orientation – like making a small cut in a sheet of paper. Once this cut is added, then “things” can bear some minimal relation to each other – they all have a relation to this minimal point of orientation. The “cut” of the empty set creates a vector, and with this stroke, things precipitate into a world of identities, properties, and relationships – as objects. (The Excessive Subject)
To conceive this cut as a self-splitting, let’s derive the MS from a circle. The circle consists of a “single edge alone” that connects back to itself. Let’s symbolize a local small segment (an arc) of this single edge alone by the shape of the minus sign ‘-‘. To derive the MS from the circle, a self-split occurs in the single edge alone, turning it into a single edge combined with itself. We symbolized the local semblance of orientation on the MS by ‘=’, and the eventual loss of this orientation by ‘✕’. This means that the MS emerges from the self-splitting of a single edge, and the space of the One-single-surface of the MS emerges in the gap between this single edge combined with itself.
The self-splitting and self-combining single edge of the MS, in other words, the cut of the empty set, is the “para-” dimension of our praxis signified by the Freudian term “parapraxis”. The symptom of the divided subject is located in the gap between the single edge of the MS. This cut marks what Milner calls a “paradoxical class” , in other words, a “cluster” . The cluster of the MS is an island with a coastline. Several cluster-islands can be bridged to one another by means of signifiers. In this way, signifiers can configure the symptoms of several subjects. This is the symptomatic conception  that works by means of decipherment (by exploring significations) .
On the other hand, when a Master-Signifier is introduced, the “cluster” becomes a “class”, it’s classified. One of the islands is identified to be the Mainland. The Mainland of the Master-Signifier basically consists of the monuments to the dead. By symbolic mourning, it reaffirms the melancholic staging of our losses. This is a political move that disavows the loss of orientation (parapraxis) in the cross sign ‘✕’, in order to restore the promise of constancy heralded by the equal sign ‘=’ that orients and gives direction and meaning to our sacrifice of time that is supposed to carry and elevate us towards the identity of the One-single-surface. At this point, we are no longer exploring symptomatic configurations; instead, we begin to extract “useful” resources in the name of the Mainland of the Master-Signifier. This is the fetishistic “conception” which works by means of decryption (by extracting hidden meanings). The scare quotes around the word “conception” indicate that the appearance of “usefulness” is often deceptive (due to the disavowal of the parapraxes).
As a final note, I will comment on the transformation of the MS into the cross-cap. According to Žižek, there is a cut in the MS only in an implicit sense, whereas the cross-cap makes the cut explicit . What Žižek overlooks, along with everyone who sticks to the parable of the One-single-surface, is that the cut of the empty set is already present in the MS in the form of its single edge, which I called the coastline of its island. The cross-cap does not make a cut. There is already a cut in the MS. So what does the cross-cap do?
What really happens in the transformation of the MS into the cross-cap is, unfortunately, not a constructive forward progress towards a higher level of more complexity like Žižek portrays it. Instead, what really happens in this transformation is, ironically, the annulment of the cut of the empty set: Topologically speaking, a cross-cap is obtained by taking an MS and mounting a disc over its single edge . By this operation, the cross-cap eliminates the single edge of the MS, destroying the coastline of the island. To make the cross-cap, the disc “caps” the “cross” (‘✕’) of the MS. Without the coastline, there can be no island, no ground, no cluster, no class. The cross-cap is not bridgeable, not configurable by means of signifiers. While the coastline of the MS is a self-splitting self-combination (the place of the combinatorial within the spatial ), the cross-cap is an un-combination. The cross-cap incapacitates the dialectical movement of decipherment. By failing to notice the cut of the empty set in the single edge of the MS (along with everyone who sticks to the edifying parable of the One-single-surface), Žižek fails to see this destructive aspect of the cross-cap.
Işık Barış Fidaner is a computer scientist with a PhD. Admin of Yersiz Şeyler (Placeless Things) blog, Admin/Editor/Curator of Žižekian Analysis, and one of the admins of “Žižek and the Slovenian School” group on Facebook. Twitter: @BarisFidaner
 Jean-Claude Milner (1983) Les noms indistincts.
 See “Decryption and Decipherment”
 Slavoj Žižek (2019) Sex and the Failed Absolute.
 See this video: “Moebius strip and Cross-cap”
 See “Spatial and Combinatorial”