Klein Bottle

Klein Bottle

The Klein bottle is a closed non-orientable surface that cannot be embedded in 3D space without self-intersection. This is a figure-8 immersion — in true 4D, the surface never touches itself.

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Figure-8 Immersion

x = (a + cos(½ v) sin u − sin(½ v) sin 2u) cos v

y = (a + cos(½ v) sin u − sin(½ v) sin 2u) sin v

z = sin(½ v) sin u + cos(½ v) sin 2u

u, v ∈ [0, 2π], a = 3

Key Properties

  • • Euler characteristic: χ = 0
  • • Non-orientable, no well-defined interior
  • • No boundary (closed surface)
  • • Cannot be embedded in ℝ³ (only immersed)
  • • Equivalent to two Möbius strips glued along their edges