The Birth of $E_8$ out of the Spinors of the Icosahedron

摘要: $E_8$ is prominent in mathematics and theoretical physics, generally viewed as an exceptional symmetry eight-dimensional space very different from the we inhabit; for instance Lie group features heavily ten-dimensional superstring theory. Contrary to that point of view, here show root system can fact be constructed icosahedron alone thus purely terms three-dimensional geometry. The $240$ roots arise 8D Clifford algebra 3D a double cover $120$ elements icosahedral group, generated by $H_3$. As by-product, restricting even products vectors (spinors) 4D subalgebra algebra, one each induces 4D, which turn out also exactly systems. spinorial view explains their existence well unusual automorphism groups. This approach allows construct all systems within geometry three dimensions, opens up novel interpretation these phenomena