Icosian calculus

File:Icosian_grid_small_with_labels2.svg of dodecahedron used for Hamilton's icosian game]]

The algebra is based on three symbols, $\backslash iota$, $\backslash kappa$, and $\backslash lambda$, 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,

:$$

\begin{align}

\iota^2 & = 1, \\

\kappa^3 & = 1, \\

\lambda^5 & = 1.

\end{align}

Hamilton gives one additional relation between the symbols,

:$\backslash lambda\; =\; \backslash iota\backslash kappa,$

which is to be understood as application of $\backslash kappa$ followed by application of $\backslash iota$. 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 $(A,B)$ represents a directed edge from vertex $A$ to vertex $B$. Vertex $A$ is the tail of $(A,B)$ and vertex $B$ is its head.

File:Icosian_calculus_iota2.svg

• The icosian symbol $\backslash iota$ reverses the arrow on every directed edge, that is, it interchanges the head and tail. Hence $(B,C)$ is transformed into $(C,B)$. Similarly, applying $\backslash iota$ to $(C,B)$ produces $(B,C)$, and to $(R,S)$ produces $(S,R)$.
• The icosian symbol $\backslash kappa$, applied to a directed edge $e$, produces the directed edge that (1) has the same head as $e$ and that (2) is encountered first as one moves around the head of $e$ in the anticlockwise direction. Hence applying $\backslash kappa$ to $(B,C)$ produces $(D,C)$, to $(C,B)$ produces $(Z,B)$, and to $(R,S)$ produces $(N,S)$.
• The icosian symbol $\backslash lambda$ applied to a directed edge $e$ produces the directed edge that results from making a right turn at the head of $e$. Hence applying $\backslash lambda$ to $(B,C)$ produces $(C,D)$, to $(C,B)$ produces $(B,A)$, and to $(R,S)$ produces $(S,N)$. Comparing the results of applying $\backslash kappa$ and $\backslash lambda$ to the same directed edge exhibits the rule $\backslash lambda\; =\; \backslash iota\backslash kappa$.

It is useful to define the symbol $\backslash mu$ 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

:$$

\mu = \lambda\kappa = \iota\kappa^2.

For example, the directed edge obtained by making a left turn from $(B,C)$ is $(C,P)$. Indeed, $\backslash kappa$ applied to $(B,C)$ produces $(D,C)$ and $\backslash lambda$ applied to $(D,C)$ produces $(C,P)$. Also, $\backslash kappa^2$ applied to $(B,C)$ produces $(P,C)$ and $\backslash iota$ applied to $(P,C)$ produces $(C,P)$.

These permutations are not rotations of the dodecahedron. 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. Hence by choosing a reference edge, say $(B,C)$, a one-to-one correspondence between directed edges and rotations is established: let $g\_E$ be the rotation that sends the reference edge $R$ to directed edge $E$. (Indeed, there are 60 directed edges and 60 rotations.) The rotations are permutations of the set of directed edges of a different sort. Let $g(E)$ denote the image of edge $E$ under the rotation $g$. The icosian associated to $g$ sends the reference edge $R$ to the same directed edge as does $g$, namely to $g(R)$. The result of applying that icosian to any other directed edge $E$ is $g\_Eg(R)\; =\; g\_Egg\_E^\{-1\}(E)$.{{cite journal | author-last = Biggs | author-first = Norman | title = The Icosian Calculus of Today | journal = Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences | volume = 95A | year = 1995 | pages = 23–34 | jstor = 20490184}}

A word consisting of the symbols $\backslash lambda$ and $\backslash mu$ 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. Hamilton exhibited the word $(\backslash lambda^3\backslash mu^3(\backslash lambda\backslash mu)^2)^2$ with the properties described above. 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. The word with the roles of $\backslash lambda$ and $\backslash mu$ interchanged has the same properties, but these give the same Hamiltonian cycles, up to shift in initial edge and reversal of direction. 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:

• presenting and studying a group by generators and relations;
• visualization of a group by a graph, which led to combinatorial group theory and later geometric group theory;
• Hamiltonian circuits and Hamiltonian paths in graph theory;
• dessin d'enfant{{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 = https://eudml.org/doc/121261 }}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. – see dessin d'enfant: history for details.

• Icosian

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Category:Graph theory

Category:Abstract algebra

Category:Binary operations

Category:Rotational symmetry

Category:William Rowan Hamilton