Epistemology m a t h 4 w i s d o m - g m a i l +370 607 27 665 My work is in the Public Domain for all to share freely. 读物 书 影片 维基百科 Introduction E9F5FC Questions FFFFC0 Software Understand the significance of the octonions. Rotations Complex numbers describes rotations in two-dimensions, and quaternions can be used to describe rotations in three dimensions. Is there a connection between octonions and rotations in four dimensions? Clifford algebras Relate the octonions to Clifford algebras. Compare the (associative) Clifford algebra construction with the (nonassociative) Cayley-Dickson construction. Give combinatorial interpretations of both and see how they differ. Is the Cayley-Dickson construction associative up to plus or minus signs (up to reflection)? Modeling cognition How are the (nonassociative) octonions relate to the (associative) split-biquaternions? Compare the learning three-cycle (for the quaternions) and Fano's plane (eightfold way) for the octonions. Is there a way that the octonions get identified with the reals? The eightsome = nullsome gets understood as a onesome? And the identification of nullsome with onesome is related to the field with one element. And we are left with exceptional Lie structures. Readings Facts about octonions John Baez, based on Dixon: The group of symmetries (or technically, "automorphisms") of the octonions is the exceptional group {$G_2$}, which contains {$SU(3)$}. To get {$SU(3)$}, we can take the subgroup of {$G_2$} that preserves a given unit imaginary octonion... say {$e_1$}. Modeling with octonions The octonions can model the nonassociativity of perspectives. The Clifford algebra {$\textrm{Cl}_{0,7}$} with seven generators (squaring to {$-1$}) and {$2^7$} basis elements models the sevensome. The Clifford algebra {$\textrm{Cl}_{0,3}$}, the split-biquaternions, with three generators (squaring to {$-1$}) and {$2^3$} basis elements models the threesome. The octonions have three generators and eight basis elements. Notes
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