# On Quaternions and Octonions

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## Book Description

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.

## Table of Contents

Preface, I The Complex Numbers, 1 Introduction, 1.1 The Algebra ℝ of Real Numbers, 1.2 Higher Dimensions, 1.3 The Orthogonal Groups, 1.4 The History of Quaternions and Octonions, 2 Complex Numbers and 2-Dimensional Geometry, 2.1 Rotations and Reflections, 2.2 Finite Subgroups of GO2 and SO2, 2.3 The Gaussian Integers, 2.4 The Kleinian Integers, 2.5 The 2-Dimensional Space Groups, II The Quaternions, 3 Quaternions and 3-Dimensional Groups, 3.1 The Quaternions and 3-Dimensional Rotations, 3.2 Some Spherical Geometry, 3.3 The Enumeration of Rotation Groups, 3.4 Discussion of the Groups, 3.5 The Finite Groups of Quaternions, 3.6 Chiral and Achiral,Diploid and Haploid, 3.7 The Projective or Elliptic Groups, 3.8 The Projective Groups Tell Us All, 3.9 Geometric Description of the Groups, Appendix: v → v̄qv Is a Simple Rotation, 4 Quaternions and 4-Dimensional Groups, 4.1 Introduction, 4.2 Two 2-to-1Maps, 4.3 Naming the Groups, 4.4 Coxeter’s Notations for the Polyhedral Groups, 4.5 Previous Enumerations, 4.6 A Note on Chirality, Appendix: Completeness of the Tables, 5 The Hurwitz Integral Quaternions, 5.1 The Hurwitz Integral Quaternions, 5.2 Primes and Unit, 5.3 Quaternionic Factorization of Ordinary Primes, 5.4 The Metacommutation Problem, 5.5 Factoring the Lipschitz Integers, III The Octonions, 6 The Composition Algebras, 6.1 TheMultiplication Laws, 6.2 The Conjugation Laws, 6.3 The Doubling Laws, 6.4 Completing Hurwitz’s Theorem, 6.5 Other Properties of the Algebras, 6.6 The Maps Lx,Rx,and Bx, 6.7 Coordinates for the Quaternions and Octonions, 6.8 Symmetries of the Octonions: Diassociativity, 6.9 The Algebras over Other Fields, 6.10 The 1-,2-,4-,and 8-Square Identities, 6.11 Higher Square Identities: Pfister Theory, Appendix: What Fixes a Quaternion Subalgebra?, 7 Moufang Loops, 7.1 Inverse Loops, 7.2 Isotopies, 7.3 Monotopies and Their Companions, 7.4 Different Forms of the Moufang Laws, 8 Octonions and 8-Dimensional Geometry, 8.1 Isotopies and SO8, 8.2 Orthogonal Isotopies and the Spin Group, 8.3 Triality, 8.4 Seven Rights Can Make a Left, 8.5 Other Multiplication Theorems, 8.6 Three 7-Dimensional Groups in an 8-Dimensional One, 8.7 On Companions, 9 The Octavian Integers O, 9.1 Defining Integrality, 9.2 Toward the Octavian Integers, 9.3 The E8 Lattice of Korkine,Zolotarev,and Gosset, 9.4 Division with Remainder,and Ideals, 9.5 Factorization in O8, 9.6 The Number of Prime Factorizations, 9.7 “Meta-Problems” for Octavian Factorization, 10 Automorphisms and Subrings of O, 10.1 The 240Octavian Units, 10.2 Two Kinds of Orthogonality, 10.3 The Automorphism Group of O, 10.4 The Octavian Unit Rings, 10.5 Stabilizing the Unit Subrings, Appendix: Proof of Theorem5, 11 Reading O Mod 2, 11.1 Why Read Mod 2?, 11.2 The E8 Lattice,Mod 2, 11.3 What Fixes (λ)?, 11.4 The Remaining Subrings Modulo 2, 12 The Octonion Projective Plane OP2, 12.1 The Exceptional Lie Groups and Freudenthal’s “Magic Square”, 12.2 The Octonion Projective Plane, 12.3 Coordinates for OP2, Bibliography, Index

## Author(s)

### Biography

John H. Conway, Derek A. Smith

## Reviews

"Conway and Smith’s book is a wonderful introduction to the normed division algebras … They develop these number systems from scratch, explore their connections to geometry, and even study

number theoryin quaternionic and octonionic versions of the integers. … a lucid and elegant introduction. … remarkably self-contained. It assumes no knowledge of number theory, string theory, Lie theory, or lower-case Gothic letters."

—John C. Baez,Bulletin of the American Mathematical Society, January 2005"A resonant spike above background noise in one parameter as another parameter is varied is a frequent indicator…"

—Geoffrey Dixon,Mathematical Intelligencer, May 2004"Those readers who are fascinated by the links between geometry and groups will find that this book gives them new insights."

—Hugh Williams,The Mathematical Gazette, July 2004"This is a beautiful and fascinating book on the geometry and arithmetic of the quaternion algebra and the octonion algebra. … most intriguing to read: it is an excellent exposition of very attractive topics, and it contains several new and significant results."

—Theo Grundhöfer,Mathematical Reviews, 2003