# Sphericon

Sphericon animation
The sphericon as a ruled surface.The two identical bicone halves are marked in different colors.

The sphericon is a solid that has a continuous developable surface with two congruent semi circular edges, and four vertices that define a square. It is a member of a special family of rollers that, while being rolled on a flat surface, bring all the points of their surface to contact with the surface they are rolling on.[1] It was first introduced by the Israeli game and toy inventor David Haran Hirsch who patented it in Israel in 1980.[2] It was given its name by Colin Roberts, who also explored it.

It may be constructed from a bicone (a double cone) with an apex angel of 90 degrees, by splitting the bicone along a plane through both apexes, rotating one of the two halves by 90 degrees, and reattaching the two halves.[3]

Alternatively, the surface of a sphericon can be formed by cutting and gluing a paper template in the form of four circular sectors (with central angles π/√2) joined edge-to-edge.[4]

Ian Stewart of the University of Warwick and Tony Phillips of Stony Brook University have also investigated the sphericon, and it has helped the latter develop theories about mazes.[citation needed]

## Geometric properties

The surface area of a sphericon with radius r is given by:

${\displaystyle \!S=2{\sqrt {2}}\pi r^{2}}$

The volume is given by:

${\displaystyle \!V={\frac {2}{3}}\pi r^{3}}$

exactly half the volume of a sphere with the same radius.

Comparison of an oloid (left) and sphericon (right) —
in the SVG image, move over the image to rotate the shapes

## References

1. ^ "Mathematical Recreations: Cone with a Twist", Scientific American, October 1999
2. ^ David Haran Hirsch (1980): "Patent no. 59720: A device for generating a meander motion; Patent drawings; Patent application form; Patent claims
3. ^ Paul J. Roberts (2010). "The Sphericon". Archived from the original on 2012-07-23.
4. ^ A mesh at www.pjroberts.com/sphericon, archived by web.archive.org