# Icosian calculus

The **icosian calculus** is a non-commutative algebraic structure discovered by the Irish mathematician William Rowan Hamilton in 1856.[1][2]
In modern terms, he gave a group presentation of the icosahedral rotation group by generators and relations.

Hamilton’s discovery derived from his attempts to find an algebra of "triplets" or 3-tuples that he believed would reflect the three Cartesian axes. The symbols of the icosian calculus can be equated to moves between vertices on a dodecahedron. Hamilton’s work in this area resulted indirectly in the terms Hamiltonian circuit and Hamiltonian path in graph theory.[3] He also invented the icosian game as a means of illustrating and popularising his discovery.

## 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 .

## Legacy

The icosian calculus is one of the earliest examples of many mathematical ideas, including:

- presenting and studying a group by generators and relations;
- a triangle group, later generalized to Coxeter groups;
- visualization of a group by a graph, which led to combinatorial group theory and later geometric group theory;
- Hamiltonian circuits and Hamiltonian paths in graph theory;[3]
- dessin d'enfant[4][5] – see dessin d'enfant: history for details.

## References

- William Rowan Hamilton (1856). "Memorandum respecting a new System of Roots of Unity" (PDF).
*Philosophical Magazine*.**12**: 446. - Thomas L. Hankins (1980).
*Sir William Rowan Hamilton*. Baltimore: The Johns Hopkins University Press. p. 474. ISBN 0-8018-6973-0. - Norman L. Biggs; E. Keith Lloyd; Robin J. Wilson (1976).
*Graph theory 1736–1936*. Oxford: Clarendon Press. p. 239. ISBN 0-19-853901-0. - Jones, Gareth (1995). "Dessins d'enfants: bipartite maps and Galois groups".
*Séminaire Lotharingien de Combinatoire*.**B35d**: 4, PDF - W. R. Hamilton, Letter to John T. Graves "On the Icosian" (17 October 1856),
*Mathematical papers, Vol. III, Algebra,*eds. H. Halberstam and R. E. Ingram, Cambridge University Press, Cambridge, 1967, pp. 612–625.