An investigation of the geometry of quaternion and octonion algebras, this book is intended for mathematicians, physicists, and crystallographers at any level--from undergraduate to professional--who are interested in the symmetries of low-dimensional space. The book can also be used as a text for graduate courses in many mathematical fields, including geometry, group theory, algebra, and number theory. A Selection of Topics Covered: The geometry of complex numbers Quaternions and 3-dimensional groups Quaternions and 4-dimensional groups The Hurwitz integral quaternions Moufang loops Octonions and 8-dimensional geometry Integral octonions The octonion projective plane