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 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.
[3]) Hamilton's work in this area resulted indirectly in the terms Hamiltonian circuit and Hamiltonian path in graph theory.
[4] He also invented the icosian game as a means of illustrating and popularising his discovery.
, 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, Hamilton gives one additional relation between the symbols, which is to be understood as application of
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.
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.
Each symbol corresponds to a permutation of the set of directed edges.
for the operation that produces the directed edge that results from making a left turn at the head of the directed edge to which the operation is applied.
This symbol satisfies the relations For example, the directed edge obtained by making a left turn from
Nevertheless, the group of permutations generated by these symbols is isomorphic to the rotation group of the dodecahedron, a fact that can be deduced from a specific feature of symmetric cubic graphs, of which the dodecahedron graph is an example.
The rotation group of the dodecahedron has the property that for a given directed edge there is a unique rotation that sends that directed edge to any other specified directed edge.
, a one-to-one correspondence between directed edges and rotations is established: let
be the rotation that sends the reference edge
The rotations are permutations of the set of directed edges of a different sort.
The result of applying that icosian to any other directed edge
corresponds to a sequence of right and left turns in the graph.
Specifying such a word along with an initial directed edge therefore specifies a directed path along the edges of the dodecahedron.
If the group element represented by the word equals the identity, then the path returns to the initial directed edge in the final step.
If the additional requirement is imposed that every vertex of the graph be visited exactly once—specifically that every vertex occur exactly once as the head of a directed edge in the path—then a Hamiltonian circuit is obtained.
Finding such a circuit was one of the challenges posed by Hamilton's icosian game.
[5] Any of the 60 directed edges may serve as initial edge as a consequence of the symmetry of the dodecahedron, but only 30 distinct Hamiltonian circuits are obtained in this way, up to shift in starting point, because the word consists of the same sequence of 10 left and right turns repeated twice.
interchanged has the same properties, but these give the same Hamiltonian cycles, up to shift in initial edge and reversal of direction.
[3] Hence Hamilton's word accounts for all Hamiltonian cycles in the dodecahedron, whose number is known to be 30.
The icosian calculus is one of the earliest examples of many mathematical ideas, including:
