A slightly lesser-known fact: the complex projective plane double covers the 4-sphere via the conjugation map. i.e. identify two equivalence classes of complex

A slightly lesser-known fact: the complex projective plane double covers the 4-sphere via the conjugation map. i.e. identify two equivalence classes of complex 3-vectors if one is the complex conjugate of the other. The resulting space is the 4-sphere. Calabi's visualization: take our oriented ellipses in real 3-space (spanned by the real and imaginary part of our complex 3-vector). Complex conjugation means negating the imaginary part, so the result of identifying complex conjugates is throwing out the orientation of the ellipse. So here's a visualization of the 4-sphere: unoriented ellipses in 3-space. There's a 1-parameter family with varying eccentricity; for each eccentricity there's a manifold which is 4-covered by SO(3); the singular sets at each end are a real projective plane.