Polycons
In 2005, Mason Green invented the concept of taking conic sections and attaching them to polyhedra. These were termed "Polycons".
If no constraints are made on how conic sections are attached to polyhedra, there are an infinte number of ways it could be done. I've only included two models but many other interesting ones are possible. This is an area of research for further study.


Cube with 6 Half Sphericons 


One way to do it is to allow nonconvexity. Pictured is a cube with six half Sphericons attached. Adding the half Sphericons gives the model tetrahedral symmetry.
If one could stand on a cube face, and could pass through the faces, from each cube face there would be a smooth path to four other of the cube's faces.
The transparent VRML model is rather interesting as it gives the illusion the internal cube is displayed orthogonally but it is not. 




Some of the ones Mason described are convex. This one is called a "Pyricon" for it pyrite symmetry. Place specifically shaped conic sections on twelve faces of an icosahedron and the result is this model. 

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:
20070906 Initial Release