Icosian Calculus - Informal Definition

Informal Definition

The algebra is based on three symbols that are each roots of unity, in that repeated application of any of them yields the value 1 after a particular number of steps. They are:

Hamilton also gives one other relation between the symbols:

(In modern terms this is the (2,3,5) triangle group.)

The operation is associative but not commutative. They generate a group of order 60, isomorphic to the group of rotations of a regular icosahedron or dodecahedron, and therefore to the alternating group of degree five.

Although the algebra exists as a purely abstract construction, it can be most easily visualised in terms of operations on the edges and vertices of a dodecahedron. Hamilton himself used a flattened dodecahedron as the basis for his instructional game.

Imagine an insect crawling along a particular edge of Hamilton's labelled dodecahedron in a certain direction, say from to . We can represent this directed edge by .

• The Icosian symbol equates to changing direction on any edge, so the insect crawls from to (following the directed edge ).

• The Icosian symbol equates to rotating the insect's current travel anti-clockwise around the end point. In our example this would mean changing the initial direction to become .

• The Icosian symbol equates to making a right-turn at the end point, moving from to .