a polar zonohedron (convex) 
a spirallohedron of order 5 (non convex) 
a nice assemblage of thirteen square antiprisms 
We have seen that spherical polyhedra allow to approximate the sphere; likewise other solids can be approximated with polyhedra, or surfaces in space visualized with polyhedral surfaces.
the torus There are three types of torus: the classical ring, the closed ring and a third one, less known, where the surface intersects itself in two parts: the apple (the outside) and the lemon (the inside). Reminder: "dragright vertically" allows you to suppress faces and thus to see the inside of the torus (and especially the lemon hidden in the apple ).  


 
the sphericon and the oloid  
If we cut a bicone with hight equal to the diameter in two going through its axis (the section is a square), we get the sphericon (C. J. Roberts) by rotating one of the halves a quarter of turn around the square's axis. Ian Stewart published an article titled Cone with a Twist.
The same transformation on a cylinder with hight equal to the diameter gives the dual (the cylinder is dual of the bicone). Similarly on a cone with an apex angle of 60° we can rotate one half by 120° (visitor's suggestion). Other shapes may be used to get curious solids with this technique. 
 
 
enjoy your breakfast! A slice of watermelon and a croissant. Be aware! you can't use the Klein's bottle for your coffee; it's not a polyhedron but an unorientable surface with one side (like a Möbius strip) that has no inside or outside!  
references: 
• zonohedra by Russel Towle
• torus (MathWorld) by Eric W. Weisstein • The Sphericon by P.J. Roberts, Roger Kaufman and Steeve Mathias 
home page

convex polyhedra  non convex polyhedra  interesting polyhedra  related subjects  October 2004 updated 30092008 