Conway J.H., Smith D.A. — On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry :: Электронная библиотека попечительского совета мехмата МГУ

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Conway J.H., Smith D.A. — On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry

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Название: On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry

Авторы: Conway J.H., Smith D.A.

Аннотация:

Video game programmers and spacecraft controllers use quaternions to calculate rotation, but Conway and Smith (Lafayette College) stick to the theoretical aspects of these algebras connected with special features of the geometry of complex numbers, and the symmetry of lesser-known octonian algebra

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2003

Количество страниц: 159

Добавлена в каталог: 18.11.2009

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Предметный указатель

$2^{m}$ -ions      8
$GO_{2}$       12
$GO_{3}$       23
$GO_{n}$ , general orthogonal group      6
$G_{2}$       76
$G_{2}(2)$       120
$G_{2}(p)$       133
-stabilizer      80
$PGO_{n}$ , projective general orthogonal group      6
$PSO_{8}$       97
$PSO_{n}$ , projective special orthogonal group      6
$SO_{2}$       12 89
$SO_{3}$       23
$SO_{4}$       89
$SO_{7}$       95
$SO_{8}$       89
$SO_{8}$ , and multiplication maps      92
$SO_{n}$ , special orthogonal group      6 24
$\mathbb{C}$ , set of complex numbers      5 11
$\mathbb{H}$ , set of quaternions      5
$\mathbb{M}$ , set of real numbers      3
$\mathbb{O}$ , set of octonions      5
$\mathbb{Q}$ , set of rational numbers      3
$\mathbb{Z}$ , set of rational integers      3
$\mathbf{O}= \mathbf{O}^{8}$ , set of octavian integers      10
2-dimensional space      18
2-to-1 map      24 33 42 95
3-dimensional point      18 23
3-dimensional space      49
4-dimensional space      48
Achiral      14 33
Albert, A.      77
Algebra, Clifford      9
Algebra, composition      5 10 41 67
Algebra, composition, over other fields      76
Algebra, Dickson Double      69 70 72 73 75 80
Algebra, Jordan      143
Algebra, normed division      7
Allcock, D.      145
Altmann, S.      10
Amphichiral      33
Arithmetic      5 100
Artin, E.      76 88
Aslaksen, H.      145
Associative      8
Associative, 2-generator Moups      88
Associative, di      76
Associative, power      73
Aut( $\mathbf{O}$ ) and $G_{2}(p)$       133
Aut( $\mathbf{O}$ ) is $G_{2}(2)$       120
Aut( $\mathbf{O}$ ) order of      125
Aut( $\mathbf{O}$ ), transitivity of      125
Automorphism      5
Automorphism, of octavians      119
Automorphism, of octonions      98
B $\ddot{u}$ low, R.      48
Baez, J.      5 7
Baker, A.      18
Blichfeldt, H. F.      105
Brown, H.      48
Bruck, R. H.      102
Cartan, E.      5
Catalan      59
Catalan Triangle      59
Catalan, number      116
Catalan, polynomial      59
Catalan, polynomial, truncated      59 115
Cayley number      9 99
Cayley, A.      8
Chiral      14 33
Chirality, potato      49
Chirality, shoe      49
Chirality, types of      49
Chiro      30 (see also Group)
Clifford, W. K.      9
Companions      86 88 90 96 97
Companions, for multiplication maps      97
Companions, multiplication rule for      97
complex numbers      5
Congruence, Euclidean      11 23
Conjugation      68
Conjugation, by , $T_{a}$       98
Conjugation, complex      12
Conjugation, quasi      80
Conjugation, quaternionic      42
Coordinates for quaternions and octonions      75
Coxeter, H. S. M.      10 45 46 99 102
Cross polytope      106
Degen, C. F.      9
Determinant      6
Diagonal symmetry      80
Diassociativity      76
Diassociativity, of 2-generator Moups      88
Diassociativity, of octonions      76
Dickson Doubling rule      10 69 73 80 103 126
Dickson Doubling rule, modified $\acute{a}$ la Pfister      78
Dickson Doubling rule, non-orthogonal      77
Dickson, L. E.      10 69 100 102
Diploid      33 (see also Group)
Division with small remainder      55 99 106 111
Double cover      24
Double Hurwitzian      103
Doubling      see Dickson Doubling rule
Du Val, P.      48 50
Duplex form of hexad      83
Duplex form of isotopy      84 91
Duplex form of relation      84
Eisenstein integer      16
Eisenstein, F. G.      11 16
Euclid      3
Euclidean algorithm      111
Euclidean line      3
Euler - Rodrigues parameters      9
Euler, L.      9 24
Factorization, Hurwitzian      59
Factorization, Lipschitzian      63
Factorization, modelled on another      57
Factorization, octavian      111 114
Factorization, quaternionic, of an ordinary prime      58
Factorizations, counting      59 63 114
Feaux, C.      116
Frame,       138
Frame, - and       134 138 139
Frame, even and odd      138 155
Freudenthal      143
Freudenthal, his Magic Square      143
Freudenthal, metasymplectic geometry      144
Gauss, C. F.      11 15
Gaussian prime      15
Gaussian unit      15
Generalized -gon      125
Generalized hexagon      121
Gibbs, W      8
Gosset, T.      105
Goursat, E.      44 48 50
Graves, J. T.      8 99
Group, $G_{2}$       120
Group, $G_{2}( 2)$       120
Group, $G_{2}(p)$       133
Group, $L_{3}(2)\cong L_{2}(7)$       139
Group, achiral      14 33 43 47 53
Group, achiral diploid      34
Group, achiral haploid      34
Group, automorphism of octavians      119
Group, automorphism of octonions      76 96
Group, axial rotation      30
Group, chiral      14 33-35 43
Group, chiral diploid      34
Group, chiral haploid      34
Group, chiro      36

Group, chiro-icosahedral      36
Group, chiro-octahedral      36
Group, chiro-prismatic      36
Group, chiro-pyramidal      36
Group, chiro-tetrahedral      36
Group, crystallographic      48
Group, cyclic      30 32 35
Group, cyclo-cyclic      35
Group, cyclo-dihedral      35
Group, dihedral      30 32 35
Group, dihedro-dihedral      35
Group, diplo-cyclic      35
Group, diplo-dihedral      35
Group, diplo-icosahedral      35
Group, diplo-octahedral      35
Group, diplo-tetrahedral      35
Group, diploid      33-35 43
Group, doubling a      48
Group, elliptic      34 48
Group, exceptional Lie      143
Group, general orthogonal      6 34
Group, haploid      33 43
Group, holo      36
Group, holo-antiprismatic      36
Group, holo-icosahedral      36
Group, holo-octahedral      36
Group, holo-prismatic      36
Group, holo-pyramidal      36
Group, holo-tetrahedral      36
Group, hybrid      34 35
Group, icosahedral,       30 32 35
Group, Lie      76 96 120 143
Group, Lie, exceptional      143
Group, Lie, pro-prismatic      36
Group, Lie, projective      34 43
Group, Lie, projective general orthogonal      6 34
Group, Lie, projective special orthogonal      6 34
Group, Lie, pyritohedral      36
Group, Lie, reflection      14 27 29
Group, Lie, rotation      14 27 29 30
Group, Lie, special orthogonal      6 24 34
Group, Lie, spin      24 42 91 95
Group, Lie, symplectic      143
Group, Lie, tetra-octahedral      35
Group, Lie, tetrahedral,       30 32 35
Group, Lie, unitary      143
Group, metachiral      44 49
Group, octahedral,       30 32 35
Group, of quaternions      33
Group, opposite      48
Group, orthochiral      46 49
Group, orthogonal      6 12 143
Group, parachiral      46 49
Group, permutation      30
Group, polyhedral (rotation)      30
Group, polyhedral, 4-dimensional      45
Group, pro-antiprismatic      36
Halving set      100
Halving set, -set      102
Halving set, $\infty$ -set      101 104 103
Halving set, inner and outer -sets      103
Hamilton, Sir W. R.      7 75
Haploid      33 (see also Group)
Heaviside, O.      8
Heegner, K.      18
Hexad of isotopies      84 87
Hexad of relations      83
Holo      30 (see also Group)
Hurley, A. C.      48
Hurwitz, A.      5 10 55 67 99
Hurwitzian      5 55
Hurwitzian, double      103
Hurwitzian, prime      56
Hyperhexagon      121
Ideal      4 58 99
Ideal, in $\mathbf{O}$ , triviality of      109
Ideal, principal      4 58 109
Identity, -square      77
Identity, 1-square      77
Identity, 16-square      78
Identity, 2-square      67 77
Identity, 4-square      9 77
Identity, 8-square      8 77
Imprimitive octavian factorizations of      115
Integer, -      102 103
Integer, $\infty$ -      102
Integer, double Hurwitzian      103 104
Integer, Eisenstein      5 11 16 100
Integer, Gaussian      5 11 15 55
Integer, Gravesian      100 104
Integer, Gravesian, factorization of      116
Integer, Hurwitzian      5 55 111
Integer, imprimitive      57
Integer, Kirmse      102
Integer, Kleinian      103 104 134
Integer, Lipschitzian      5 55
Integer, Lipschitzian, factorization of      61
Integer, octavian      5 10 99 105 108 109
Integer, octavian, divisor sets of      113
Integer, primitive      57
Integer, quaternion      5
Integer, rational      3
Integrality      10 55 99
Isochirality      49
Isometry      3 5 11
Isotopy      49 67 72 78 84
Isotopy, and $SO_{8}$       89
Isotopy, orthogonal      91
Isotopy, similar to automorphism      86
Jordan algebra      143
Jordan algebra, exceptional      145
Jordan product      145
Jordan, P.      143
Kaplansky, I.      77
Kelvin, Lord      8 33
Kernels, left and right      44
Kirmse, his mistake      102
Kirmse, J.      102
Korkine, A.      105
Kuipers, J. B.      xii
Lamont, P. J. C.      110 111 116
Lattice, $D^{*}_{4}$       62
Lattice, $D_{4}$       62
Lattice, $E_{8}$       99 105 107
Lattice, $E_{8}$ , mod 2      135
Lattice, cubic      62
Lattice, orthoplex      106
Lattice, simplex      106
Lattice, square      15
Lattice, triangular      16
Law, alternative      73 74
Law, associative      74 89
Law, biconjugation      68
Law, braid      68 70
Law, commutative      89
Law, composition      68
Law, composition doubling      69
Law, conjugation doubling      69
Law, exchange      68
Law, inner-product doubling      69
Law, inverse      72 74
Law, Moufang      5 73 79 95 97
Law, Moufang, and transitivity of monotopies      87
Law, Moufang, bi-multiplication      74 87 88
Law, Moufang, left-multiplication      74 88
Law, Moufang, right-multiplication      74 88
Law, product conjugation      69
Law, scaling      68
Liebeck, M.      87
Lipschitz, R. O. S.      5 99

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