miscellaneous examples of curious polyhedra

two rhombic zonohedra (Russel Towle)

Zonohedra are convex polyhedra bounded by centrally-symmetrical polygons (thus they have an even number of sides, two by two opposite, parallel and with same length). Nevertheless one may construct non convex polyhedra with such faces.
Here are two examples with rhombic faces.

a polar zonohedron (convex)

a spirallohedron of order 5 (non convex)

a near miss polyhedron (Jim McNeill)

a nice assemblage of thirteen square antiprisms

polyhedra to approximate solids (Eric W. Weisstein - MathWorld)

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: "drag-right 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.
The oloid is the convex envelope of two orthogonal disks with each center on the other's border (Paul Schatz).

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

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convex polyhedra - non convex polyhedra - interesting polyhedra - related subjects October 2004
updated 30-09-2008