Real Projective Plane

Real Projective Plane ℙ²(ℝ)

The real projective plane is the space of all lines through the origin in ℝ³. It can be constructed by identifying antipodal points on S². Like the Klein bottle, it cannot be embedded in ℝ³ without self-intersection. This is Boy's surface — a famous immersion discovered by Werner Boy in 1901.

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Construction

ℙ²(ℝ) = S² / (x ~ −x)

Identify antipodal points on the 2-sphere

Key Properties

  • • Euler characteristic: χ = 1
  • • Non-orientable
  • • Compact, closed (no boundary)
  • • H₁(ℙ²; ℤ) ≅ ℤ/2ℤ (torsion!)
  • • Contains a Möbius strip as a subset