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The Moebius band

“The Moebius band,” Tupelo said, “has unusual properties because it has a singularity. The Klein bottle, with two singularities, manages to be inside of itself. The topologists know surfaces with as many as thousand singularities, and they have properties that make the Moebius band and the Klein bottle both look simple. But a network with infinite connectivity must have an infinite number of singularities. Can you imagine what the properties of that network could be?”
Fantasia Mathematica by Clifton Fadiman

Alice in Wonderland was originally to have been entitled Alice’s Adventures Underground. But why didn’t Carroll keep this title? Because Alice progressively conquers surfaces. She rises or returns to the surface. She creates surfaces. Movements of penetration and burying give way to light lateral movements of sliding; the animals of the depths become figure on cards without thickness. All the more reason for Through the Looking-Glass to invest the surface of a mirror, to institute a game of chess. Pure events escape from states of affairs. We no longer penetrate in depth but through an act of sliding pass through the looking-glass, turning everything the other way round like a left-hander. The stock market of Fortunatus described by Carroll is a Moebius strip on which a single line traverses the two sides. Mathematics is good because it brings new surfaces into existence, and brings peace to a world whose mixtures in depth would be terrible: Carroll the mathematician, or Carroll the photographer. But the world of depths still rumbles under the surface, and threatens to break through it. Even unfolded and laid out flat, the monsters still haunt us.
Lewis Carroll by Gilles Deleuze