The icosian calculus is a non-commutative algebraic structure discovered by the Irish mathematician William Rowan Hamilton in 1856. 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 correspond to moves between vertices on a dodecahedron. (Hamilton originally thought in terms of moves between the faces of an icosahedron, which is equivalent by duality. This is the origin of the name "icosian".) Hamilton's work in this area resulted indirectly in the terms Hamiltonian circuit and Hamiltonian path in graph theory. He also invented the icosian game as a means of illustrating and popularising his discovery.
Informal definition
The algebra is based on three symbols, ι {\displaystyle \iota } , κ {\displaystyle \kappa } , and λ {\displaystyle \lambda } , that Hamilton described as "roots of unity", by which he meant that repeated application of any of them a particular number of times yields the identity, which he denoted by 1. Specifically, they satisfy the relations
ι 2 = 1 , κ 3 = 1 , λ 5 = 1. {\displaystyle {\begin{aligned}\iota ^{2}&=1,\\\kappa ^{3}&=1,\\\lambda ^{5}&=1.\end{aligned}}}
Hamilton gives one additional relation between the symbols,
λ = ι κ , {\displaystyle \lambda =\iota \kappa ,}
which is to be understood as application of κ {\displaystyle \kappa } followed by application of ι {\displaystyle \iota } . Hamilton points out that application in the reverse order produces a different result, implying that composition or multiplication of symbols is not generally commutative, although it is associative. The symbols 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. This, however, is not how Hamilton described them. Hamilton drew comparisons between the icosians and his system of quaternions, but noted that, unlike quaternions, which can be added and multiplied, obeying a distributive law, the icosians could only, as far as he knew, be multiplied. Hamilton understood his symbols by reference to the dodecahedron, which he represented in flattened form as a graph in the plane. The dodecahedron has 30 edges, and if arrows are placed on edges, there are two possible arrow directions for each edge, resulting in 60 directed edges. Each symbol corresponds to a permutation of the set of directed edges. The definitions below refer to the labeled diagram above. The notation ( A , B ) {\displaystyle (A,B)} represents a directed edge from vertex A {\displaystyle A} to vertex B {\displaystyle B} . Vertex A {\displaystyle A} is the tail of ( A , B ) {\displaystyle (A,B)} and vertex B {\displaystyle B} is its head.
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