Integer Octonions

John Baez

December 4, 2014

• Part 1: integral octonions and the Coxeter group E₁₀. Also available on the n-Category Café.
  

• Part 2 - the integral octonions, 11d supergravity, and cosmological billiards. Also available on the n-Category Café.

• Part 3 - the 240 smallest integer octonions, also known as the E₈ root polytope, and their connection to exceptional objects in mathematics. Also available on the n-Category Café.
  

• Part 4 - faces of the E₈ root polytope. Also available on the n-Category Café.
• Part 5 - Dynkin diagrams and the geometry of the E₈ root polytope. Also available on the n-Category Café.
• Part 6 - How to multiply octonions, and the Cayley integral octonions. Also available on the n-Category Café.
  

• Part 7 - Greg Egan's proof that 2 × 2 self-adjoint matrices with integral octonion entries form a copy of the E₁₀ lattice. Also available on the n-Category Café.
• Part 8 - My proof that 3 × 3 self-adjoint matrices with integral octonion entries contain a copy of K₂₇ lattice. Also available on the n-Category Café.
  

• Part 9 - Greg Egan's construction of the Leech lattice from the E₈ lattice. Also available on the n-Category Café.
• Part 10 - Fitting the Leech lattice into the exceptional Jordan algebra. Joint with Greg Egan. Also available on the n-Category Café.
• Part 11 - How to find 270 rescaled copies of the first shell of the E₈ lattice sitting in the second shell, and what you can do with these. Joint with Greg Egan. Also available on the n-Category Café.
• Part 12 - Tim Silverman on the integral octonions mod 2. Also available on the n-Category Café.

© 2014 John Baez
baez@math.removethis.ucr.andthis.edu
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