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Topology 🇺🇦 UK

🌀 Klein Bottle

The Klein bottle is a non-orientable surface with no boundary and no inside/outside — impossible to embed in 3D without self-intersection, it truly lives in 4D.

🔷 Parametrization

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📐 Topology facts

Euler χ
0
Orientable
No
Boundary
∅ (none)
= 2 Möbius
✓ Yes

ℹ️ Topology

The Klein bottle is a closed non-orientable surface. Its Euler characteristic χ = 0 equals the torus, but it is non-orientable unlike it. Cutting it along its axis yields two Möbius strips.

In 3D the surface must pass through itself — the self-intersection is an artifact of the projection from 4D. The two popular immersions are the standard (tube-through-body) and figure-8 (symmetric cross-cap).

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