Topological surfaces
Topology studies properties that survive stretching but not tearing. A coffee cup is a torus; a Möbius strip has one side. Rotate these surfaces, switch rendering modes, and watch a point travel along them to reveal their hidden structure.
χ = V − E + F (Euler characteristic)
Möbius strip
A surface with only one side and one boundary curve. If you walk along it, you return to your starting point with your orientation reversed. It is non-orientable: you cannot consistently define “inside” and “outside.” Euler characteristic χ = 0, genus 0 (with boundary).
Klein bottle
A closed, non-orientable surface with no boundary. In three dimensions it must pass through itself, but in four dimensions it embeds without self-intersection. It can be made by gluing two Möbius strips along their boundary. Euler characteristic χ = 0, genus 1 (non-orientable).
Torus
The surface of a donut — the product of two circles, S¹ × S¹. Orientable, genus 1, Euler characteristic χ = 0. A flat torus can tile the plane; the video game Asteroids takes place on a flat torus.
Real projective plane (RP²)
The set of all lines through the origin in R³. Non-orientable, closed, and cannot be embedded in R³ without self-intersection. Cross-cap immersion shown here. Euler characteristic χ = 1.
Boy’s surface
An immersion of the real projective plane in R³ discovered by Werner Boy in 1901. Unlike the cross-cap, it has no pinch points — it is a smooth immersion with triple self-intersection. It has remarkable three-fold symmetry.