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### Summary

### 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.

### Reviews 1

### Choice Review

William Hamilton invented the quaternion algebra in 1843 and precipitated an intellectual fad for quaternions as a tool for mechanics that prevailed until matrix algebra pushed quaternions aside early in the 20th century. Remarkable structures, the quaternion algebra and also its eight-dimensional analog, octonian algebra, remain on the scene; each generation finds new virtues in them. Conway and his student Smith show how quaternions help classify the finite groups of isometries in three and four dimensions; they cleanly prove Hurwitz's famous classification of finite-dimensional composition algebras and develop higher-dimensional analogs of unique factorization in these settings. They also touch on Pfister theory (part of quadratic form theory). Readers will find here Conway's celebrated style: fastidious verbal economy married to ingenious graphics. ^BSumming Up: Recommended. General readers; lower-division undergraduates through professionals. D. V. Feldman University of New Hampshire

### Table of Contents

Preface | p. xi |

I The Complex Numbers | p. 1 |

1 Introduction | p. 3 |

1.1 The Algebra R of Real Numbers | p. 3 |

1.2 Higher Dimensions | p. 5 |

1.3 The Orthogonal Groups | p. 6 |

1.4 The History of Quaternions and Octonions | p. 6 |

2 Complex Numbers and 2-Dimensional Geometry | p. 11 |

2.1 Rotations and Reflections | p. 11 |

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

2.3 The Gaussian Integers | p. 15 |

2.4 The Kleinian Integers | p. 18 |

2.5 The 2-Dimensional Space Groups | p. 18 |

II The Quaternions | p. 21 |

3 Quaternions and 3-Dimensional Groups | p. 23 |

3.1 The Quaternions and 3-Dimensional Rotations | p. 23 |

3.2 Some Spherical Geometry | p. 26 |

3.3 The Enumeration of Rotation Groups | p. 29 |

3.4 Discussion of the Groups | p. 30 |

3.5 The Finite Groups of Quaternions | p. 33 |

3.6 Chiral and Achiral, Diploid and Haploid | p. 33 |

3.7 The Projective or Elliptic Groups | p. 34 |

3.8 The Projective Groups Tell Us All | p. 35 |

3.9 Geometric Description of the Groups | p. 36 |

Appendix v to vqv Is a Simple Rotation | p. 40 |

4 Quaternions and 4-Dimensional Groups | p. 41 |

4.1 Introduction | p. 41 |

4.2 Two 2-to-1 Maps | p. 42 |

4.3 Naming the Groups | p. 43 |

4.4 Coxeter's Notations for the Polyhedral Groups | p. 45 |

4.5 Previous Enumerations | p. 48 |

4.6 A Note on Chirality | p. 49 |

Appendix Completeness of the Tables | p. 50 |

5 The Hurwitz Integral Quaternions | p. 55 |

5.1 The Hurwitz Integral Quaternions | p. 55 |

5.2 Primes and Units | p. 56 |

5.3 Quaternionic Factorization of Ordinary Primes | p. 58 |

5.4 The Metacommutation Problem | p. 61 |

5.5 Factoring the Lipschitz Integers | p. 61 |

III The Octonions | p. 65 |

6 The Composition Algebras | p. 67 |

6.1 The Multiplication Laws | p. 68 |

6.2 The Conjugation Laws | p. 68 |

6.3 The Doubling Laws | p. 69 |

6.4 Completing Hurwitz's Theorem | p. 70 |

6.5 Other Properties of the Algebras | p. 72 |

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

6.7 Coordinates for the Quaternions and Octonions | p. 75 |

6.8 Symmetries of the Octonions: Diassociativity | p. 76 |

6.9 The Algebras over Other Fields | p. 76 |

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

6.11 Higher Square Identities: Pfister Theory | p. 78 |

Appendix What Fixes a Quaternion Subalgebra? | p. 80 |

7 Moufang Loops | p. 83 |

7.1 Inverse Loops | p. 83 |

7.2 Isotopies | p. 84 |

7.3 Monotopies and Their Companions | p. 86 |

7.4 Different Forms of the Moufang Laws | p. 88 |

8 Octonions and 8-Dimensional Geometry | p. 89 |

8.1 Isotopies and SO[subscript 8] | p. 89 |

8.2 Orthogonal Isotopies and the Spin Group | p. 91 |

8.3 Triality | p. 92 |

8.4 Seven Rights Can Make a Left | p. 92 |

8.5 Other Multiplication Theorems | p. 94 |

8.6 Three 7-Dimensional Groups in an 8-Dimensional One | p. 95 |

8.7 On Companions | p. 97 |

9 The Octavian Integers O | p. 99 |

9.1 Defining Integrality | p. 99 |

9.2 Toward the Octavian Integers | p. 100 |

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

9.4 Division with Remainder, and Ideals | p. 109 |

9.5 Factorization in O[superscript 8] | p. 111 |

9.6 The Number of Prime Factorizations | p. 114 |

9.7 "Meta-Problems" for Octavian Factorization | p. 116 |

10 Automorphisms and Subrings of O | p. 119 |

10.1 The 240 Octavian Units | p. 119 |

10.2 Two Kinds of Orthogonality | p. 120 |

10.3 The Automorphism Group of O | p. 121 |

10.4 The Octavian Unit Rings | p. 125 |

10.5 Stabilizing the Unit Subrings | p. 128 |

Appendix Proof of Theorem 5 | p. 131 |

11 Reading O Mod 2 | p. 133 |

11.1 Why Read Mod 2? | p. 133 |

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

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

11.4 The Remaining Subrings Modulo 2 | p. 140 |

12 The Octonion Projective Plane OP[superscript 2] | p. 143 |

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

12.2 The Octonion Projective Plane | p. 144 |

12.3 Coordinates for OP[superscript 2] | p. 145 |

Bibliography | p. 149 |

Index | p. 153 |