The Octonions
John C. Baez

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.

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.

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 a lot of extra material on the octonions:

• Octonions online
  links to other websites containing material about the octonions.

• Brougham bridge
  pictures of the bridge where Hamilton carved his definition of the quaternions.

• Integral octonions
  integral octonions and their connections to geometry and physics.
  

• Octonions and the Standard Model
  the octonions and exceptional Jordan algebra in particle physics.
  

• On Quaternions and Octonions: Their Geometry, Arithmetic and Symmetry
  my review of John Conway and Derek Smith's book.
  
  

For a lighter approach, try this two-part feature in Plus Magazine:

• Curious quaternions

and

• Ubiquitous octonions

in which Helen Joyce and I have a fun nontechnical chat about the real numbers, complex numbers, quaternions and octonions. Also try this article John Huerta and I wrote for Scientific American:

• The strangest numbers in string theory.

John Huerta and I have written a number of technical papers about the octonions and other normed division algebras in physics:

• John Baez, Division algebras and quantum theory.
• John Baez and John Huerta, Division algebras and supersymmetry I.
• John Baez and John Huerta, Division algebras and supersymmetry II.
• John Huerta, Division algebras and supersymmetry III.
• John Huerta, Division algebras and supersymmetry IV.
• John Huerta, Division algebras, supersymmetry and higher gauge theory.

Huerta and I have also written about a curious connection between octonions, the exceptional group G₂ and a ball rolling on another ball 3 times as big:

• John Baez and John Huerta, G₂ and the rolling ball.

We also have some blog articles explaining this paper:

• John Baez, Rolling circles and balls.
• John Huerta, G₂ and the rolling ball.

In the broad light of day mathematicians check their equations and their proofs, leaving no stone unturned in their search for rigour. But, at night, under the full moon, they dream, they float among the stars and wonder at the miracle of the heavens. They are inspired. Without dreams there is no art, no mathematics, no life. - Michael Atiyah

© 2023 John Baez
baez@math.removethis.ucr.andthis.edu

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