In 1856 Hamilton invented an `Icosian Calculus', which he used to investigate closed edge paths on a dodecahedron that visit each vertex exactly once. (Such edge paths in general graphs are today known as Hamiltonian circuits.) He published short papers describing the Icosian Calculus in the Philosophical Magazine for 1856, and in the Proceedings of the Royal Irish Academy.
Hamilton also described the Icosian Calculus, and an associated Icosian Game, at the Twenty-Seventh Meeting of the British Association for the Advancement of Science; held at Dublin in August and September 1857. His presentation was summarized in the Report of that meeting as follows:
The author stated that this calculus was entirely distinct from that of quaternions, and in it none of the roots concerned were imaginary. He then explained the leading features of the new calculus, and exemplified its use by an amusing game, which he called the Icosian, and which he had been led to invent by it, - a lithograph of which he distributed through the Section, and examples of what the game proposed to be accomplished were lithographed in the margin, the solutions being shown to be exemplifications of the calculus. The figure was the projection on a plane of the regular pentagonal dodecahedron, and at each of the angles were holes for receiving the ivory pins with which the game was played.
The paper
Memorandum respecting a new System of Roots of Unitywas published in the Philosophical Magazine, volume 12 (4th series) (1856) , p. 446.
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The paper
Account of the Icosian Calculuswas communicated on November 10th, 1856, and appeared in the Proceedings of the Royal Irish Academy, volume 6 (1858), pp. 415-416.
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D.R. Wilkins