Memorandum respecting a new system of roots of unity [offprint from the Philosophical Magazine]: ...

Memorandum respecting a new system of roots of unity [offprint from the Philosophical Magazine]

HAMILTON, Sir William Rowan

Editorial: London, December 1856, 1856
Usado / Cantidad: 0
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8vo (212 x 140 mm), pp 2; in original plain wrappers (a bit creased), 'Dupl. T C D' [i.e. Trinity College Dublin] written in pencil on front wrapper. £350First edition. In this brief but important paper, Hamilton extends the class of non-commutative algebras (as they are called today) from the example of the quaternions he discovered more than a decade earlier. The new algebra, which he called the 'icosian calculus', is still generated by elements i, j and k, but whereas in the case of the quaternions we have i2 = j2 = k2 = 1, ij = k, in the algebra introduced in the present paper we have i2 = j3 = k5 = 1, ij = k. Hamilton was led to these relations by considering the symmetries of an icosahedron (or dodecahedron), and this is indeed the first determination of the group of symmetries of a regular (Platonic) solid. It is also the first time a group had been described by means of generators (i, j and k) and relations (as we say today). Previously a group had been described by its 'Cayley table', an array describing the result of multiplying any two elements of the group. This, however, is usually very cumbersome (the icosahedral group considered by Hamilton would require a 60 by 60 array), and is almost never used today. By contrast, Hamilton's description in terms of generators and relations is both economical and elegant.'? Hamilton at this early date was familiar with the properties of the groups of the regular solids, as generated by two operators or elements, and he proved that these groups may be completely defined by the orders of their two generating operators and the order of their product. His form of statement of these results coincides with that now employed with the exception of the use of certain terms. Hamilton did not use the technical term group, and he evidently was in search of things lying beyond these groups, just as his main interest as regards quaternions was beyond the quaternion group.'Notwithstanding this fact, the great importance of the groups of the regular solids as defined abstractly and the discovery of the first example of a system of numbers whose units constitute a non-commutative group as regards multiplication should secure for Hamilton an important place in the history of abstract groups ?'The commonly accepted date (1854) of the founding of the theory of abstract groups by Cayley is two years earlier than the given paper by Hamilton on the groups of the regular polyhedrons, and it is eleven years later than the discovery of the quaternions ? Cayley was assisted in his study of abstract groups by Hamilton's earlier work and the latter developed some very interesting and fundamental groups without receiving due credit by later writers' (Miller).'[Hamilton] became interested in the study of polyhedra and developed in 1856 what he called the "Icosian Calculus", a study of the properties of the icosahedron and the dodecahedron. This study resulted in an "Icosian Game" to be played on the plane projection of the dodecahedron. He sold the copyright to a Mr. Jacques of Piccadilly for twenty-five pounds. The game fascinated a mathematician like Hamilton, but it is unlikely that Mr. Jacques ever recovered his investment' (DSB).The icosian game was to find a path along the edges of the plane projection of the dodecahedron that passed through each vertex exactly once. Such a path is now known as a Hamiltonian path, and graphs which admit such a path are called Hamiltonian graphs. It is still not known, in general, how to determine whether a graph is Hamiltonian, but in the case of the dodecahedral graph Hamilton was able to construct a Hamiltonian path by means of his icosian calculus. G. A. Miller, Note on William R. Hamilton's place in the history of abstract group theory. Bibliotheca Mathematica, 3 Folge, Bd. 11, 1910-11, pp. 314-15No copy in OCLC. N° de ref. de la librería

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Título: Memorandum respecting a new system of roots ...
Editorial: London, December 1856
Año de publicación: 1856

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HAMILTON, Sir William Rowan
Editorial: London, December 1856 (1856)
Usado Cantidad: 1
Librería
WP Watson Antiquarian Books
(London, Reino Unido)
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Descripción London, December 1856, 1856. 8vo (212 x 140 mm), pp 2; in original plain wrappers (a bit creased), 'Dupl. T C D' [i.e. Trinity College Dublin] written in pencil on front wrapper.First edition. In this brief but important paper, Hamilton extends the class of non-commutative algebras (as they are called today) from the example of the quaternions he discovered more than a decade earlier. The new algebra, which he called the 'icosian calculus', is still generated by elements i, j and k, but whereas in the case of the quaternions we have i2 = j2 = k2 = 1, ij = k, in the algebra introduced in the present paper we have i2 = j3 = k5 = 1, ij = k. Hamilton was led to these relations by considering the symmetries of an icosahedron (or dodecahedron), and this is indeed the first determination of the group of symmetries of a regular (Platonic) solid. It is also the first time a group had been described by means of generators (i, j and k) and relations (as we say today). Previously a group had been described by its 'Cayley table', an array describing the result of multiplying any two elements of the group. This, however, is usually very cumbersome (the icosahedral group considered by Hamilton would require a 60 by 60 array), and is almost never used today. By contrast, Hamilton's description in terms of generators and relations is both economical and elegant.'? Hamilton at this early date was familiar with the properties of the groups of the regular solids, as generated by two operators or elements, and he proved that these groups may be completely defined by the orders of their two generating operators and the order of their product. His form of statement of these results coincides with that now employed with the exception of the use of certain terms. Hamilton did not use the technical term group, and he evidently was in search of things lying beyond these groups, just as his main interest as regards quaternions was beyond the quaternion group.'Notwithstanding this fact, the great importance of the groups of the regular solids as defined abstractly and the discovery of the first example of a system of numbers whose units constitute a non-commutative group as regards multiplication should secure for Hamilton an important place in the history of abstract groups ?'The commonly accepted date (1854) of the founding of the theory of abstract groups by Cayley is two years earlier than the given paper by Hamilton on the groups of the regular polyhedrons, and it is eleven years later than the discovery of the quaternions ? Cayley was assisted in his study of abstract groups by Hamilton's earlier work and the latter developed some very interesting and fundamental groups without receiving due credit by later writers' (Miller).'[Hamilton] became interested in the study of polyhedra and developed in 1856 what he called the "Icosian Calculus", a study of the properties of the icosahedron and the dodecahedron. This study resulted in an "Icosian Game" to be played on the plane projection of the dodecahedron. He sold the copyright to a Mr. Jacques of Piccadilly for twenty-five pounds. The game fascinated a mathematician like Hamilton, but it is unlikely that Mr. Jacques ever recovered his investment' (DSB).The icosian game was to find a path along the edges of the plane projection of the dodecahedron that passed through each vertex exactly once. Such a path is now known as a Hamiltonian path, and graphs which admit such a path are called Hamiltonian graphs. It is still not known, in general, how to determine whether a graph is Hamiltonian, but in the case of the dodecahedral graph Hamilton was able to construct a Hamiltonian path by means of his icosian calculus. G. A. Miller, Note on William R. Hamilton's place in the history of abstract group theory. Bibliotheca Mathematica, 3 Folge, Bd. 11, 1910-11, pp. 314-15No copy in OCLC. Nº de ref. de la librería 3251

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