The birth of E8 out of the spinors of the icosahedron

TY - JOUR

T1 - The birth of E8 out of the spinors of the icosahedron

AU - Dechant, Pierre-Philippe

N1 - © Author 2015. This content is made available by the publisher under a Creative Commons CC BY Licence

PY - 2016

Y1 - 2016

N2 - 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.

AB - 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.

KW - E8

KW - Icosahedral symmetry

KW - Clifford algebras

KW - exceptional Lie algebras

KW - viruses

KW - spinors

UR - https://arxiv.org/abs/1712.06436

UR - https://cameroncounts.wordpress.com/2017/03/15/the-ade-affair-6/

UR - https://www.lms.ac.uk/sites/lms.ac.uk/files/files/NLMS_476_for%20web.pdf

U2 - 10.1098/rspa.2015.0504

DO - 10.1098/rspa.2015.0504

M3 - Article

SN - 1364-5021

JO - Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences

JF - Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences

ER -