Sphericon. Credit: Steve Mathias.
A sphericon is a curious and mathematically delightful three-dimensional object made from a right double-cone – two identical, 90-degree cones joined base to base – and an added twist. To create a sphericon, a right double-cone is sliced along a plane that includes both vertices. The resulting cross-section is a square, which enables one of the halves to be rotated through a right angle and the two halves to be glued back together without any overlap. This final twist enables the sphericon to roll – but in an unusual way.
An ordinary cone placed on a flat surface rolls around in circles. A double-cone can roll in a clockwise circle or a counterclockwise one. A sphericon, in contrast, performs a controlled wiggle, with first one conical sector in contact with the flat surface, then the other. Two sphericons placed next to each other can roll on each other's surfaces. Four sphericons arranged in a square block can all roll around one another simultaneously. And eight sphericons can fit on the surface of one sphericon so that any one of the outer solids can roll on the surface of the central one.
The sphericon was first found by the Englishman Colin Roberts in 1969, while he was still in school. In 1999 he brought his discovery to the attention of Ian Stewart who subsequently wrote about the new object in his "Mathematical Recreations" column in Scientific American.1
1. Stewart, Ian. "Cone with a Twist." Scientific American, 281: 116-117 (Oct. 1999).