  This paper considers the geometry of $E_8$ from a Clifford point of view in
three complementary ways. Firstly, in earlier work, I had shown how to
construct the four-dimensional exceptional root systems from the 3D root
systems using Clifford techniques, by constructing them in the 4D even
subalgebra of the 3D Clifford algebra; for instance the icosahedral root system
$H_3$ gives rise to the largest (and therefore exceptional)
non-crystallographic root system $H_4$. Arnold's trinities and the McKay
correspondence then hint that there might be an indirect connection between the
icosahedron and $E_8$. Secondly, in a related construction, I have now made
this connection explicit for the first time: in the 8D Clifford algebra of 3D
space the $120$ elements of the icosahedral group $H_3$ are doubly covered by
$240$ 8-component objects, which endowed with a `reduced inner product' are
exactly the $E_8$ root system. It was previously known that $E_8$ splits into
$H_4$-invariant subspaces, and we discuss the folding construction relating the
two pictures. This folding is a partial version of the one used for the
construction of the Coxeter plane, so thirdly we discuss the geometry of the
Coxeter plane in a Clifford algebra framework. We advocate the complete
factorisation of the Coxeter versor in the Clifford algebra into exponentials
of bivectors describing rotations in orthogonal planes with the rotation angle
giving the correct exponents, which gives much more geometric insight than the
usual approach of complexification and search for complex eigenvalues. In
particular, we explicitly find these factorisations for the 2D, 3D and 4D root
systems, $D_6$ as well as $E_8$, whose Coxeter versor factorises as
$W=\exp(\frac{\pi}{30}B_C)\exp(\frac{11\pi}{30}B_2)\exp(\frac{7\pi}{30}B_3)\exp(\frac{13\pi}{30}B_4)$.
