Sphericon

The sphericon is a 3D shape with one side and two edges, discovered by Colin Roberts, of Hertfordshire, England.

To construct, take a bicone (a double cone) with an apex of 90 degrees, and split it from apex to apex. Rotate one half by 90 degrees, and stick the two halves back together again.

An easier method of making a sphericon is to use a [http://homepage.ntlworld.com/paul.roberts99/How_to_make.htm template] . Cut it out, and form a ring by gluing the tab to the respective edge. Then just tape the remaining curved edges together. The four angles are each $180\; /\; sqrt\{2\}$ degrees.

After making this shape, Roberts realised that these could be put into many different lattices which could all revolve around each other when a single sphericon was turned.

If you picture a square swept into three dimensions about a line that goes through two diagonally opposite apexes, you have the double cone from which the sphericon originates. If you use literally any other regular 2D polygon, (such as a triangle or even a 126-agon), you can create another new shape by cutting and rotating. Some sweeps can create many shapes, by rotating different amounts. All initial shapes with an even number of sides can be swept along the line from mid-side to mid-side instead of from apex to apex, to give another infinite number of new shapes, each with their own lattices and geometric secrets.

Professors Ian Stewart (University of Warwick) and Tony Phillips (State University of New York at Stony Brook) have investigated the sphericon in different ways, and it has helped the latter develop theories about mazes.

References

*"Mathematical Recreations" - "Cone with a Twist", "Scientific American", October 1999
* [http://www.pjroberts.com/sphericon Sphericon Homepage.] It contains images, text and animation that give a fuller view of the sphericon family.
* [http://www.korthalsaltes.com/sphericon.htm Paper model of a sphericon] Make a sphericon
* [http://www.softcom.net/users/sbmathias/series.htm Sphericon variations] using regular polygons with different numbers of sides