Take a rectangular strip of paper, make a half-twist, and attach the ends. The resulting surface, a Möbius strip, has a number of surprising properties. It is one-sided: If we attempt to color one side of the paper "continuing as far as we can" without lifting our pencil or brush, no uncolored paper remains when we finish. Cutting a Möbius strip along the center line yields a single twisted strip, not two linked strips.
The boundary of a Möbius strip is a single closed loop, abstractly a circle. The boundary of a disk is also a circle. Abstractly, we can glue a Möbius strip to a disk, joining their circle boundaries and obtaining a one-sided surface with no boundary. To a topologist, this surface is the real projective plane. Boy's surface is a model of the real projective plane in Euclidean three-space.
Although Boy's surface "cuts through" itself, or has self-intersections, there is a unique tangent plane at each point of the abstract surface. With one exception, each point of three-space where the surface crosses itself corresponds to two points of the abstract surface; accordingly, two tangent planes to Boy's surface pass through most self-intersections. (There is a single "triple point" near the top center where three points of the abstract surface coincide.)
Abstractly, a point of Boy's surface may be viewed as representing a line through the origin in three-space. Since every such line meets the unit sphere in a pair of antipodes, Boy's surface is obtained from a sphere by gluing every point to its antipode.
The parametrization shown was published by Robert Bryant and Rob Kusner in 1986.
Handmade in the UK by specialist picture framers using premium fine art paper with a gently textured surface and satin finish, high-quality wood, and an FSC certified off-white mat. Delivered ready for hanging.
