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The icosian calculus of today. (English) Zbl 1280.05058

Summary: Hamilton’s icosian calculus is less well known than his algebra of quaternions, but it is nevertheless an interesting system of non-commutative algebra. He used it to study complete cycles on the dodecahedron, a subject which so fascinated him that he attempted to popularise a game based on it. The game was not a success in commercial terms, but it resulted in the term ‘Hamiltonian cycle’ being used for a complete cycle on any graph. This was not only inappropriate, because Hamilton was not the first to study such things, but also inauspicious, because Hamilton’s methods have little relevance to the study of ‘Hamiltonian’ cycles in general.
A modern development which can be linked more positively with the icosian calculus is the use of generators and relations to study graphs which have certain symmetry properties. In particular, there is a remarkable theorem of W. T. Tutte, concerning graphs of degree three, the proof of which involves calculations like those of Hamilton. Similar ideas were also used by J. H. Conway in unpublished, but seminal, investigations on the same subject. This is the legacy of the icosian calculus which is discussed in the paper.

MSC:

05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
05-03 History of combinatorics
01A55 History of mathematics in the 19th century
05C45 Eulerian and Hamiltonian graphs
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