Syntheme illustrates ways of partitioning the 12
vertices of an icosahedron into 3 sets of 4, so that each set forms the corners
of a rectangle in the Golden Ratio. Each such rectangle is known as a **duad**.
The short sides of a duad are opposite edges of the icosahedron, and there are 30 edges,
so there are 15 duads.

Each partition of the vertices into duads is known as a **syntheme**.
There are 15 synthemes; 5 consist of duads that are mutually perpendicular, while
the other 10 consist of duads that share a common line of intersection.

Clicking on the applet causes it to alternate between a randomly chosen syntheme
(with each duad drawn in a different colour), and a randomly chosen **synthematic total**:
a set of 5 synthemes with no duads in common. The synthematic totals are drawn with all the duads
in a given syntheme sharing the same colour.

These structures play a part in a fascinating account of the symmetry group of the icosahedron, described in this article by John Baez: “Some Thoughts on the Number 6”.

Greg Egan
science fiction writer

Applets Gallery / Syntheme / created Sunday, 6 August 2000