By John Horton Conway

This publication investigates the geometry of quaternion and octonion algebras. Following a entire ancient creation, the booklet illuminates the specific houses of three- and four-dimensional Euclidean areas utilizing quaternions, resulting in enumerations of the corresponding finite teams of symmetries. the second one half the ebook discusses the fewer regularly occurring octonion algebra, targeting its amazing "triality symmetry" after a suitable research of Moufang loops. The authors additionally describe the arithmetics of the quaternions and octonions. The e-book concludes with a brand new thought of octonion factorization. subject matters coated comprise the geometry of complicated numbers, quaternions and third-dimensional teams, quaternions and four-dimensional teams, Hurwitz critical quaternions, composition algebras, Moufang loops, octonions and 8-dimensional geometry, vital octonions, and the octonion projective plane.