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.
Click on the applet to pause or redraw. The applet alternates 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”.