In hyperbolic space exist four regular platonic solids whose vertices lie in infinity. Further details are available in the following article . Cover shows sample scenes from Touching Soap Films . It belongs to a rare set of new compact CMC surfaces found by numerical experiments. It belongs to a family of surfaces whose fundamental domains are given as solution of free-boundary value problems in hyperbolic orthoschemes.
Details are available in the following paper .
Compact CMC surfaces have a long history in geometry, more details are given in Numerical Examples of Compact Constant Mean Curvature Surfaces . The picture is part of an animation of hyperbolic space shown in the video Touching Soap Films . The picture was created for an article of Ruth Kellerhals on the volume of hyperbolic polyhedra. The animation uses a specific arrangement of reflectors to obtain the caleidoskopic effect. Touching Soap Films shows the Bidenoid minimal surface discovered by H. Karcher.