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:

$iota^2\; =\; 1,!$

$kappa^3\; =\; 1,!$

$lambda^5\; =\; 1,!$

Hamilton also gives one other relation between the symbols:

$lambda\; =\; iotakappa,!$

These symbols can only be multiplied (not added) and although they are all associative they are not commutative. They generate a group of order 60, isomorphic to the group of rotations of a regular icosahedron or dodecahedron.

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 $B$ to $C$. We can represent this directed edge by $BC$.

• The Icosian symbol $iota$ equates to changing direction on any edge, so the insect crawls from $C$ to $B$ (following the directed edge $CB$).
• The Icosian symbol $kappa$ equates to rotating the insect's current travel anti-clockwise around the end point. In our example this would mean changing the initial direction $BC$ to become $PC$.
• The Icosian symbol $lambda$ equates to making a right-turn at the end point, moving from $BC$ to $CD$.

References