The Sphericon
In the 1960's C.J. Roberts invented the "Sphericon". The first Sphericon model he made was of Mahogany which was given to his sister and the topic was forgotten. In May of 1999, Roberts wrote to Ian Stewart, a writer for Scientific American and in October 1999, Stewart published an article titled "Cone with a Twist". This was the first time most people learned of the Sphericon's existance.
Links to other pages for further reading:
A smooth Sphericon is made from a bicone. As such, the base model being two cones joined at the base, it is not a polyhedron.
A type of Sphericon can be made from a dipyramid. In the Nicon Study, higher order Nicons are built programmatically from polygons and are thus, all faceted. If the faceting is ignored they are still valid models for the study of their conic counterparts.


A Sphericon can be made of a bicone or any dipyramid with a 90° apex has been divided in two such that its cross section is a square. Here is a split dipyramid displayed in LiveGraphics3D. (see this page for how to control a Live3D graphic with your mouse). 



One of the two parts is rotated 90° then the two parts are reassembled. In this LiveGraphics3D image, the half on the right side has been rotated. 




The assembled Sphericon. In this form the Sphericon would be what Roberts calls a "Faceted Sphericon". The Faceted Sphericon lends itself to be easily modeled since most computer modeling is done with polygons. A dipyramid will be made of isosceles triangles and the more facets the dipyramid has, the smoother the Faceted Sphericon model will be visually.
The Sphericon has one continuous surface and two discontinuous edges. 



A nearly Smooth Sphericon can be modeled by increasing the number of facets to a very high number. This "Smooth" Sphericon is made of a total of 576 isosceles triangles. 



When a square is swept around the midpoint of opposite edges, a cylinder results such that its width and length are equal. This is known as the side cut. One half is turned 90° and the halves are rejoined just as they were in building the Sphericon.
The Side Cut of the Sphericon has two discontinous surfaces and one continuous edge.
The Sphercon Side Cut is the dual of the smooth Sphericon. In the faceted form the Sphericon Side Cut is not exactly dual but resembles the dual to the faceted Sphericon. But even in faceted form, the circuitry of the surface and edge paths of the two are codual. 

Question or comments about the web page should be directed to polyhedra@bigfoot.com.
The generation of OFF, VRML, and Live3D files was done with
antiprism. The
Hedron application by
Jim McNeill was used to generate VRML Switch files.
History:
20090307 Revised commentary on duals
20070906 Initial Release