The Sphericon

The sphericon is a 3-dimensional geometric solid with a single continuous face and two edges. Similarly to a Möbius strip, a line traced along the centre of its side will eventually join its starting point. The sphericon rolls along a wobbly path that results in motion in a straight line.

A sphericon can be constructed from a bicone with a 90º apex. Slice it with a plane containing both apices of the bicone, giving a square section. Rotate one half by 90º along the axis that's at a tangent to the slicing plane and passes through the centre of the bicone, and rejoin the halves (see Figure 1).

The steps to construct a sphericon
Figure 1: The steps to construct a sphericon.
(a) a bicone, (b) slice, (c) rotate, (d) rejoin.

An easier way is to use a template. Glue the shaded tab to the straight edge on the left, then tape the corresponding edges. The angle a is 180 / sqrt(2)º.