The Boy’s Surface is a minimal, non-orientable surface. A non-orientable surface is one for which it is impossible to make a consistent choice of “front” and “back” everywhere on the surface. For a surface that is embedded in three-dimensional space, there is just one direction at each non-singular point that is perpendicular the surface, and there are two unit normals at that point, pointing in opposite directions. (Palais.)

An implicit surface surfaces is a two-dimensional, geometric shapes that exist in three dimensional space and defined as a level surface of a function. This surface is topologically equivalent to the Cross-cap and the Boy’s surface, which is basically a semi-sphere with the rims sewed together. (Xhalee.) Boys’ Apery surface is one without such singularity.

Both objects have a very intriguing and inspiring shape that contribute to the dynamic of two-dimensional designs.  A 16 image portfolio of  implicit and Boy’s surface visualizations are available at hermay.org