On quaternions and octonions : their geometry, arithmetic, and symmetry / John H. Conway, Derek A. Smith.

Format:

Book

Author/Creator:

Conway, John H. (John Horton)

Contributor:

Smith, Derek Alan, 1970-

Language:

English

Subjects (All):

Quaternions.

Cayley numbers (Algebra).

Physical Description:

xii, 159 pages : illustrations ; 24 cm

Place of Publication:

Natick, Mass. : AK Peters, [2003]

Summary:

This book investigates the geometry of quaternion and octonion algebras. Following a comprehensive historical introduction, the book illuminates the special properties of 3- and 4-dimensional Euclidean spaces using quaternions, leading to enumerations of the corresponding finite groups of symmetries. The second half of the book discusses the less familiar octonion algebra, concentrating on its remarkable "triality symmetry" after an appropriate study of Moufang loops. The authors also describe the arithmetics of the quaternions and octonions. The book concludes with a new theory of octonion factorization. Topics covered include the geometry of complex numbers, quaternions and 3-dimensional groups, quaternions and 4-dimensional groups, Hurwitz integral quaternions, composition algebras, Moufang loops, octonions and 8-dimensional geometry, integral octonions, and the octonion projective plane.

Contents:

I The Complex Numbers 1

1.1 The Algebra R of Real Numbers 3

1.2 Higher Dimensions 5

1.3 The Orthogonal Groups 6

1.4 The History of Quaternions and Octonions 6

2 Complex Numbers and 2-Dimensional Geometry 11

2.1 Rotations and Reflections 11

2.2 Finite Subgroups of GO[subscript 2] and SO[subscript 2] 14

2.3 The Gaussian Integers 15

2.4 The Kleinian Integers 18

2.5 The 2-Dimensional Space Groups 18

II The Quaternions 21

3 Quaternions and 3-Dimensional Groups 23

3.1 The Quaternions and 3-Dimensional Rotations 23

3.2 Some Spherical Geometry 26

3.3 The Enumeration of Rotation Groups 29

3.4 Discussion of the Groups 30

3.5 The Finite Groups of Quaternions 33

3.6 Chiral and Achiral, Diploid and Haploid 33

3.7 The Projective or Elliptic Groups 34

3.8 The Projective Groups Tell Us All 35

3.9 Geometric Description of the Groups 36

Appendix v to vqv Is a Simple Rotation 40

4 Quaternions and 4-Dimensional Groups 41

4.2 Two 2-to-1 Maps 42

4.3 Naming the Groups 43

4.4 Coxeter's Notations for the Polyhedral Groups 45

4.5 Previous Enumerations 48

4.6 A Note on Chirality 49

Appendix Completeness of the Tables 50

5 The Hurwitz Integral Quaternions 55

5.1 The Hurwitz Integral Quaternions 55

5.2 Primes and Units 56

5.3 Quaternionic Factorization of Ordinary Primes 58

5.4 The Metacommutation Problem 61

5.5 Factoring the Lipschitz Integers 61

III The Octonions 65

6 The Composition Algebras 67

6.1 The Multiplication Laws 68

6.2 The Conjugation Laws 68

6.3 The Doubling Laws 69

6.4 Completing Hurwitz's Theorem 70

6.5 Other Properties of the Algebras 72

6.6 The Maps L[subscript x], R[subscript x], and B[subscript x] 73

6.7 Coordinates for the Quaternions and Octonions 75

6.8 Symmetries of the Octonions: Diassociativity 76

6.9 The Algebras over Other Fields 76

6.10 The 1-, 2-, 4-, and 8-Square Identities 77

6.11 Higher Square Identities: Pfister Theory 78

Appendix What Fixes a Quaternion Subalgebra? 80

7 Moufang Loops 83

7.1 Inverse Loops 83

7.2 Isotopies 84

7.3 Monotopies and Their Companions 86

7.4 Different Forms of the Moufang Laws 88

8 Octonions and 8-Dimensional Geometry 89

8.1 Isotopies and SO[subscript 8] 89

8.2 Orthogonal Isotopies and the Spin Group 91

8.3 Triality 92

8.4 Seven Rights Can Make a Left 92

8.5 Other Multiplication Theorems 94

8.6 Three 7-Dimensional Groups in an 8-Dimensional One 95

8.7 On Companions 97

9 The Octavian Integers O 99

9.1 Defining Integrality 99

9.2 Toward the Octavian Integers 100

9.3 The E[superscript 8] Lattice of Korkine, Zolotarev, and Gosset 105

9.4 Division with Remainder, and Ideals 109

9.5 Factorization in O[superscript 8] 111

9.6 The Number of Prime Factorizations 114

9.7 "Meta-Problems" for Octavian Factorization 116

10 Automorphisms and Subrings of O 119

10.1 The 240 Octavian Units 119

10.2 Two Kinds of Orthogonality 120

10.3 The Automorphism Group of O 121

10.4 The Octavian Unit Rings 125

10.5 Stabilizing the Unit Subrings 128

Appendix Proof of Theorem 5 131

11 Reading O Mod 2 133

11.1 Why Read Mod 2? 133

11.2 The E[subscript 8] Lattice, Mod 2 135

11.3 What Fixes [left angle bracket] lambda [right angle bracket]? 138

11.4 The Remaining Subrings Modulo 2 140

12 The Octonion Projective Plane OP[superscript 2] 143

12.1 The Exceptional Lie Groups and Freudenthal's "Magic Square" 143

12.2 The Octonion Projective Plane 144

12.3 Coordinates for OP[superscript 2] 145.

Notes:

Includes bibliographical references (pages 149-152) and index.

ISBN:

1568811349

OCLC:

50761334