Shapes of Space: Projective Space

Projective Space

A real projective plane is constructed in 4-space by attaching the boundary of a disk to the edge of a Möbius band. We may also envision a projective plane as the set of points in the southern hemisphere of a sphere, including half the points of the equator. The equator points are each glued to the the opposite (antipodal) point on the sphere. (6)

It is hard to see how this figure includes a Möbius band, but if we look only at the equator and think about how the points are attached, the construction should be clearer.

The projective plane is a nonorientable surface. A Flatlander traveling across the "seam" of the equator returns home reversed in the same way as a Flatlander who journeys around a Möbius band or a Klein bottle.

We can create a projective 3-space by taking half of a hypersphere and gluing together opposite points of its boundary (which is spherical). What is surprising is that this space is not nonorientable! If you cross the "seam", you do not come back reversed, because you flip in two directions. You are both left-right reversed and top-bottom reversed--you are rotated 180 degrees, but you are the same as before, so projective 3-space is orientable.

Next: Hyperbolic Space

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