On Quaternions and Octonions

by John H. Conway, Derek Smith

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.

Genres: Mathematics, Nonfiction, Science, Reference

159 pages, Hardcover

First published January 1, 2003

About the author

John Horton Conway, often credited as John H. Conway, was a Professor of Mathematics at Princeton University, known for inventing the "Game of Life."

Reviews

[Alex Lee · Rating 5 out of 5]

The beginning of the book was really exciting. The end was really interesting. Unfortunately I do not have a developed mathematical reasoning to follow in the middle. This book is meant to be an introduction about how these amazing extra-dimensional figures can be manipulated and interrelate as a kind of extra-symmetry that amounts to a series of spatial arrangements and rotations. I will have to read it again in a few years.

[Michael Weaver · Rating 4 out of 5]

Conway does a great job at exploring three and four dimensional euclidean spaces - detailing finite group of symmetries and dissecting the calculations of quaternions and octonions.