UFOs -- Unidentified Figurative Objects. A Geometric Challenge

Abstract

Choose a polynomial in three variables with not more than three or four monomials of moderate degree. Take simple coefficients as 1 and -1. Then draw a picture of the solution variety in real three space using a ray-tracing program like POV-Ray, Surf or Spicy. Go to your colleague in the next room, show her/him the picture and ask for a suitable equation. This is the theme of the article: The problem of recognizing the algebraic definition of a geometric object, in this case a real algebraic surface. We do not offer any answers or results. 24 surfaces are presented, each in two different views, but without equations. The reader may try to find them -- or consult the calendar at http://www.hh.hauser.cc, where the equations are displayed one per day together with a short animation of the surface.