  $E_8$ is prominent in mathematics and theoretical physics, and is generally
viewed as an exceptional symmetry in an eight-dimensional space very different
from the space we inhabit; for instance the Lie group $E_8$ features heavily in
ten-dimensional superstring theory. Contrary to that point of view, here we
show that the $E_8$ root system can in fact be constructed from the icosahedron
alone and can thus be viewed purely in terms of three-dimensional geometry. The
$240$ roots of $E_8$ 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 $H_3$. 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.
