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The Octonions

The Octonions 
John C. Baez 

Department of Mathematics 
University of California 
Riverside CA 92521

May 16, 2001 

Published in Bull. Amer. Math. Soc. 39 (2002), 145-205.

Errata in Bull. Amer. Math. Soc. 42 (2005), 213.

Also available in Postscript and PDF formats.



The octonions are the largest of the four normed division algebras. While somewhat neglected due to their nonassociativity, they stand at the crossroads of many interesting fields of mathematics. Here we describe them and their relation to Clifford algebras and spinors, Bott periodicity, projective and Lorentzian geometry, Jordan algebras, and the exceptional Lie groups. We also touch upon their applications in quantum logic, special relativity and supersymmetry.



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Table of Contents:
  1. Introduction
    1. Preliminaries


  2. Constructing the Octonions
    1. The Fano Plane
    2. The Cayley-Dickson Construction
    3. Clifford Algebras
    4. Spinors and Trialities


  3. Octonionic Projective Geometry
    1. Projective Lines
    2. OP1 and Bott Periodicity
    3. OP1 and Lorentzian Geometry
    4. OP2 and the Exceptional Jordan Algebra


  4. Exceptional Lie Algebras
    1. G2
    2. F4
    3. The Magic Square
    4. E6
    5. E7
    6. E8


  5. Conclusions
    1. Acknowledgements


  6. Bibliography
This website also contains some extra stuff, namely: And if all this is too hard for you, try this two-part feature in Plus Magazine: and in which Helen Joyce and I have a fun nontechnical chat about the real numbers, complex numbers, quaternions and octonions. 

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