- Icosian Calculus
The Icosian Calculus is a non-commutative
algebraic structure discovered by the Irish mathematicianWilliam Rowan Hamilton in1856 . [cite book |author=Thomas L. Hankins |title=Sir William Rowan Hamilton |publisher=The Johns Hopkins University Press |location=Baltimore |year=1980 |pages=474 |isbn=0-8018-6973-0 |oclc= |doi=] 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 adodecahedron . Hamilton’s work in this area resulted indirectly in the termsHamiltonian circuit andHamiltonian path in graph theory. [cite book |author=Norman L. Biggs, E. Keith Lloyd, Robin J. Wilson |title=Graph theory 1736-1936 |publisher=Clarendon Press |location=Oxford |year=1976 |pages=239 |isbn=0-19-853901-0 |oclc= |doi=] He also invented theIcosian 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::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 notcommutative . They generate a group of order 60, isomorphic to the group of rotations of a regularicosahedron ordodecahedron .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
External links
* [http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Icosian/NewSys.pdf Original paper on the subject] by
William Rowan Hamilton
Wikimedia Foundation. 2010.