The birth of E8 out of the spinors of the icosahedron

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E8 is prominent in mathematics and theoretical physics, and is generally viewed as an exceptional symmetry in an eight-dimensional (8D) space very different from the space we inhabit; for instance, the Lie group E8 features heavily in 10D superstring theory. Contrary to that point of view, here we show that the E8 root system can in fact be constructed from the icosahedron alone and can thus be viewed purely in terms of 3D geometry. The 240 roots of E8 arise in the 8D Clifford algebra of 3D space as a double cover of the 120 elements of the icosahedral group, generated by the root system H3. As a by-product, by restricting to even products of root vectors (spinors) in the 4D even subalgebra of the Clifford algebra, one can show that each 3D root system induces a root system in 4D, which turn out to also be exactly the exceptional 4D root systems. The spinorial point of view explains their existence as well as their unusual automorphism groups. This spinorial approach thus in fact allows one to construct all exceptional root systems within the geometry of three dimensions, which opens up a novel interpretation of these phenomena in terms of spinorial geometry.
Original languageEnglish
JournalProceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences
Early online date13 Jan 2016
Publication statusPublished - 2016

Bibliographical note

© Author 2015. This content is made available by the publisher under a Creative Commons CC BY Licence


  • E8
  • Icosahedral symmetry
  • Clifford algebras
  • exceptional Lie algebras
  • viruses
  • spinors

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