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The '''icosian calculus''' is a non-commutative [[algebraic structure]] discovered by the Irish mathematician [[William Rowan Hamilton]] in 1856.<ref>{{Cite journal
|title=Memorandum respecting a new System of Roots of Unity
|author=Sir William Rowan Hamilton
|author-link=William Rowan Hamilton
|url=http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Icosian/NewSys.pdf
|journal=[[Philosophical Magazine]]
|volume=12
|year=1856
|page=446
}}</ref><ref>{{cite book |author=Thomas L. Hankins |title=Sir William Rowan Hamilton |publisher=The Johns Hopkins University Press |location=Baltimore |year=1980 |page=474 |isbn=0-8018-6973-0 |oclc= |doi=}}</ref>
In modern terms, he gave a [[group presentation]] of the [[icosahedral group|icosahedral rotation group]] by [[Generating set of a group|generators]] and relations.


Hamilton’s discovery derived from his attempts to find an algebra of [[tuple|"triplets" or 3-tuples]] that he believed would reflect the three [[Cartesian coordinate system#Cartesian coordinates in three dimensions|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.<ref name="biggs">{{cite book |author=Norman L. Biggs, E. Keith Lloyd, Robin J. Wilson |title=Graph theory 1736–1936 |publisher=Clarendon Press |location=Oxford |year=1976 |page=239 |isbn=0-19-853901-0 |oclc= |doi=}}</ref> He also invented the [[icosian game]] as a means of illustrating and popularising his discovery.


== Informal definition ==
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[[File:Icosian_grid_small_with_labels2.svg|thumb|right|250px|[[Stereographic projection]] of dodecahedron used for Hamilton's [[icosian game]]]]
 
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:
 
:<math>
\begin{align}
\iota^2 & = 1, \\
\kappa^3 & = 1, \\
\lambda^5 & = 1.
\end{align}
</math>
 
Hamilton also gives one other relation between the symbols:
 
:<math>\lambda = \iota\kappa.\,\!</math>
(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 (mathematics)|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 <math>B</math> to <math>C</math>. We can represent this [[Directed graph|directed edge]] by <math>BC</math>.
 
[[File:Icosian_calculus_iota2.svg|thumb|right|400px|Geometrical illustration of operation iota in icosian calculus]]
 
*The icosian symbol <math>\iota</math> equates to changing direction on any edge, so the insect crawls from <math>C</math> to <math>B</math> (following the directed edge <math>CB</math>).
 
*The icosian symbol <math>\kappa</math> equates to rotating the insect's current travel anti-clockwise around the end point. In our example this would mean changing the initial direction <math>BC</math> to become <math>DC</math>.
 
*The icosian symbol <math>\lambda</math> equates to making a right-turn at the end point, moving from <math>BC</math> to <math>CD</math>.
 
== Legacy ==
The icosian calculus is one of the earliest examples of many mathematical ideas, including:
* presenting and studying a group by [[Presentation of a group|generators and relations]];
* a [[triangle group]], later generalized to [[Coxeter group]]s;
* visualization of a group by a graph, which led to [[combinatorial group theory]] and later [[geometric group theory]];
* [[Hamiltonian circuit]]s and [[Hamiltonian path]]s in graph theory;<ref name="biggs" />
* [[dessin d'enfant]]<ref>{{Cite journal | title = Dessins d'enfants: bipartite maps and Galois groups | first = Gareth | last = Jones | journal = [[Séminaire Lotharingien de Combinatoire]] | volume = B35d | year = 1995 | pages = 4 | url = http://radon.mat.univie.ac.at/~slc/s/s35jones.html | postscript =, [http://www.emis.de/journals/SLC/wpapers/s35jones.pdf PDF] }}</ref><ref>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.</ref> – see [[dessin d'enfant#History|dessin d'enfant: history]] for details.
 
== References ==
{{reflist}}
 
[[Category:Graph theory]]
[[Category:Abstract algebra]]
[[Category:Binary operations]]
[[Category:Rotational symmetry]]

Latest revision as of 03:54, 20 November 2014


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