On Quaternions and Octonions
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 f
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3-dimensional angle associative composition algebra automorphism group Chapter chiral chiral group complex numbers composition algebra congruences conjugation contains coordinates corresponding define Dickson double dimensions divisor E8 Lattice Eisenstein Eisenstein integers element enumerate equivalent Euclidean evenly orthogonal Figure finite group finite subgroups Gaussian integers geometry halving-set Hamilton hexad Hurwitz integers Hurwitz units Hurwitzian identity isomorphism isotopy k-frame Lemma Lipschitzian modulo monotopy Moufang laws Moufang loops n-integers n-sets namely nonzero norm normed division algebra notation obtain octavian integers octavian units octonions orthogonal group orthoplex pair prime factorization projective plane Proof prove quaternion triplet Quaternions and Octonions reflections right multiplications rotation group shows simple rotation special orthogonal group spin group subrings symmetry Taylor & Francis Theorem triality triangle unique factorization unit orthogonal unit quaternions unit rings unit-migration vectors