page_title: E8 (mathematics)
section_title: E8 root system
hierarchical_section_title: E8 (mathematics) / E8 root system
caption_attribution_description: English: The E8 4_21 polytope is projected to 3D using 3 rows of the E8 to H4 folding matrix giving H3 symmetry. The vertices are sorted and tallied by their 3D norm. Generating the increasingly transparent hull of each set of tallied norms shows: 1) 4 points at the origin 2) 2 icosahedrons 3) 2 dodecahedrons 4) 4 icosahedrons 5) 1 icosadodecahedron 6) 2 dodecahedrons 7) 2 icosahedrons 8) 1 icosadodecahedron for a total of 240 vertices. This is, of course, 2 concentric sets of hulls from the H4 symmetry of the w:600-cell scaled by the golden ratio. For more information, see my blog post here. [1]
context_page_description: In mathematics, E₈ is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8. The designation E₈ comes from the Cartan–Killing classification of the complex simple Lie algebras, which fall into four infinite series labeled Aₙ, Bₙ, Cₙ, Dₙ, and five exceptional cases labeled E₆, E₇, E₈, F₄, and G₂. The E₈ algebra is the largest and most complicated of these exceptional cases.
context_section_description: A root system of rank r is a particular finite configuration of vectors, called roots, which span an r-dimensional Euclidean space and satisfy certain geometrical properties. In particular, the root system must be invariant under reflection through the hyperplane perpendicular to any root.
  The E₈ root system is a rank 8 root system containing 240 root vectors spanning R⁸. It is irreducible in the sense that it cannot be built from root systems of smaller rank. All the root vectors in E₈ have the same length. It is convenient for a number of purposes to normalize them to have length √2.  These 240 vectors are the vertices of a semi-regular polytope discovered by Thorold Gosset in 1900, sometimes known as the 4₂₁ polytope.
caption: Shown in 3D projection using the basis vectors [u,v,w] giving H3 symmetry: u = (1, φ, 0, −1, φ, 0,0,0)v = (φ, 0, 1, φ, 0, −1,0,0)w = (0, 1, φ, 0, −1, φ,0,0)  The projected 421 polytope vertices are sorted and tallied by their 3D norm generating the increasingly transparent hulls of each set of tallied norms. These show: 4 points at the origin2 icosahedrons2 dodecahedrons4 icosahedrons1 icosadodecahedron2 dodecahedrons2 icosahedrons1 icosadodecahedron  for a total of 240 vertices. This is, of course, 2 concentric sets of hulls from the H4 symmetry of the 600-cell scaled by the golden ratio.[2]
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