MEMORANDUM RESPECTING A NEW SYSTEM OF ROOTS OF UNITY

By

William Rowan Hamilton

(Philosophical Magazine, 12 (1856), p. 446.)

Edited by David R. Wilkins Memorandum respecting a new System of Roots of Unity. By Sir William Rowan Hamilton, LL.D., M.R.I.A., F.R.A.S., &c., Andrews’ Professor of Astronomy in the University of Dublin, and Royal Astronomer of Ireland*.

[The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, 4th series, vol. xii (1856), p. 446.]

I have lately been led to the conception of a new system, or rather family of systems, of non-commutative roots of unity, which are entirely distinct from the ijk of the quaternions, though having some general analogy thereto; and which admit, even more easily than the quaternion symbols do, of geometrical interpretation. In the system which seems at present to be the most interesting one, among those included in this new family, I assume three symbols, ι, κ, λ, such that

\begin{matrix}ι^{2} = 1, κ^{3} = 1, λ^{5} = 1, \\ λ = ικ; \end{matrix} \}  (A)

where ικ must be distinguished from κι, since otherwise we should have λ^{6} = 1, λ = 1. As a very simple specimen of the symbolical conclusions deduced from these fundamental assumptions, I may mention that if we make

μ = ικ^{2} = λιλ,

we shall have also†

μ^{5} = 1, λ = μιμ;

so that μ is a new fifth root of unity, connected with the former fifth root λ by relations of perfect reciprocity. A long train of such symbolical deductions is found to follow: and every one of the results may be interpreted, as having reference to the passage from face to face (or from corner to corner) of the icosahedron (or of the dodecahedron): on which account, I am at present disposed to give the name of the “Icosian Calculus,” to this new system of symbols, and of rules for their operation. Some additional remarks on this subject may soon

* Communicated by the Author.

† In fact, by (A),

ικ = (ικ)^{−4} = (κ^{−1}ι^{−1})^{4} = (κ^{2}ι)^{4},
1 = ι . ικ . κ^{2} = ι(κ^{2}ι)^{4}κ^{2} = (ικ^{2})^{5};

also

μιμ = μκ^{2} = ικ^{4} = ικ = λ.

be offered to the Philosophical Magazine, under the title, already sanctioned by the Editors, of “Extensions of the Quaternions.”

Observatory of Trinity College, Dublin, October 29, 1856.