BOY'S SURFACE   Click on this link for an applet allowing to manipulate the surface

 Surface studied by Werner Boy in 1902, Bernard Morin in 1978, Jean-Pierre Petit and Jerome Souriau in 1981. Werner Boy (1879 - 1914) : student of Hilbert (see here what Jean-Pierre Petit says about him). See also: en.wikipedia.org/wiki/Boy%27s_surface and Jean-Pierre Petit's article: www.cs.berkeley.edu/~sequin/CS294/IMGS/boysurface.htm

 Cartesian parametrization of Apery (1986), used for the images above: with . Cartesian parametrization of Bryant-Kusner (1987): where , , , with Cartesian parametrization of Morin-Apery (1987), ranging from the Roman surface (k = 0) to Boy's surface (k = 1): where . Algebraic surface of degree 6.

Boy's surface was discovered after the search of a model in of the projective plane that would not have other singularities than self-intersections along which the various surfaces would have a well-defined tangent plane (the Roman surface and the cross-cap, which were known, have cuspidal points).
Boy described his surface in a conceptual way, and it is only in 1981 that J.P. Petit and J. Souriau found a parametrization of it.

 Much as the cross-cap is obtained by twisting the edge of a disk into a curve with one crossing until the two parts coincide (so that the opposite points of the disk coincide)...  ...Boy's surface is obtained by twisting a disk into a curve with three crossings until the 6 parts coincide two by two (hence the order 3 symmetry of this surface)  Opposite, various animations explaining the construction (Bryant-Kusner parametrization):   Boy's surface has 3 orifices leading to tunnels that come together in the central part. Follow a path to understand that the surface has only one face. Opposite, an animation using the Morin-Apery parametrization, showing the deformation of the Roman surface into Boy's surface; the 3 cuspidal points of the Roman surface disappear as from . The self-intersection curve of Boy's surface is a twisted trifolium; the 3 tangents at the triple points are orthogonal two by two (and therefore so are the three tangent planes of the surface at this triple point). See also quadrifolium.  The above parametrization of Morin-Apery makes it an algebraic surface of degree 6, and it has been proved that this degree could not be lowered without cuspidal points appearing (the Roman surface and the cross-cap are quartics).

 Here is a model of Boy's surface in the form of a reunion of linked rectangles. The self-intersection curve is composed of 3 squares located in 3 planes two by two perpendicular to one another. Since some adjacent rectangles are coplanar, this model should be deformed to truly become a polyhedron, but there exists a true (generalized) polyhedron that is a model for Boy's surface: Brehm's polyhedron.  If Boy's surface is painted, the layer of paint obtained (which is in one piece since this surface is one-sided) is an immersion of the sphere (because the two-sheeted covering space of the projective plane is the sphere); it is the reason why Boy's surface was used as the central step in the process of turning a sphere inside out: see for example this text in Pour la Science. 