Icosian Calculus
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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 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.
[edit] 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:
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.
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 ι equates to changing direction on any edge, so the insect crawls from C to B (following the directed edge CB).
- The Icosian symbol κ 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 λ equates to making a right-turn at the end point, moving from BC to CD.
[edit] References
- ^ Sir William Rowan Hamilton (1856), "Memorandum respecting a new System of Roots of Unity", Philosophical Magazine 12: 446, http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Icosian/NewSys.pdf
- ^ Thomas L. Hankins (1980). Sir William Rowan Hamilton. Baltimore: The Johns Hopkins University Press. pp. 474. ISBN 0-8018-6973-0.
- ^ Norman L. Biggs, E. Keith Lloyd, Robin J. Wilson (1976). Graph theory 1736-1936. Oxford: Clarendon Press. pp. 239. ISBN 0-19-853901-0.
