Part 8 of 9. The familiar picture of E₈, eight rings of 30 points, is a projection of its 240 roots. This part writes them down, builds Gosset's polytope 421, and projects it to the plane and to four dimensions, where it falls onto two 600-cells whose sizes differ by the golden ratio.
Parts 3 and 4 met E₈ as the last branched tree that passes the positivity test. This part works with its 240 roots directly, as coordinates, as a polytope, and in the projections behind the familiar picture.
The first family is the 112 vectors in with two nonzero coordinates, each : , the roots of D8. The second is the vectors with all eight coordinates and an even number of minus signs, of them. Both have squared length 2, and the parity rule keeps the set closed under reflection.
Figure 1. The 240 roots in parallel coordinates, one polyline per root, read from the shared E8 library. Integer roots touch two axes at ±1 and six at zero. Half-integer roots touch every axis at ±½.
Since every root has squared length 2, the inner product of two roots fixes the angle between them. Figure 2 computes it for all pairs.
Figure 2. Inner products over all pairs of distinct roots. Only −2, −1, 0 and 1 occur. The −2 bar is the 120 antipodal pairs (α, −α). More than half of all pairs are perpendicular.
From any single root the other 239 split as 56 at inner product 1 (60°), 126 at 0, 56 at −1 and one antipode; the 126 perpendicular roots form an E7.
Join two roots by an edge when their inner product is 1. The 6,720 edges bound triangles, tetrahedra and so on up to 7-dimensional facets: the polytope 421 that Gosset described in 1900. Figure 3 counts every face as a clique, a set of roots that are all neighbours of one another.
Figure 3. The faces of 421 counted in the page, one bar per dimension on a log scale: simplices as cliques of mutually adjacent roots, cross-polytope facets by their centres. The right column counts the roots adjacent to every vertex of a face; a walk adds one at a time.
Up to dimension 5 every face has the same number of common neighbours, so each count is the last one times that number over the new vertex count: 60,480 = 6,720 × 27 / 3. The 27 around an edge have the pattern of Part 7's 27 lines. At dimension 6 the pattern breaks. 138,240 of the 6-faces close up into the 17,280 simplex facets; the other 69,120 lie only on facets that are 7-dimensional cross-polytopes, 2,160 of them. Two kinds of facet make 421 uniform but not regular.
Slicing shows it from the side. A hyperplane sliding along a coordinate axis meets vertices at five levels, splitting the roots 14 + 64 + 84 + 64 + 14; the 84 with that coordinate zero form a D7.
Slice x₁ = 0.00: 0 vertices on it, 0 edges crossing.
Figure 4. Slicing 421 along a coordinate axis. Left: the wireframe, slicing axis as depth, with red dots where the hyperplane cuts edges. Right: the cross-section head-on. Snap to next halt visits the five levels that hold vertices; every axis gives 14 + 64 + 84 + 64 + 14.
The familiar picture of E₈ is a projection onto the plane where a Coxeter element, the product of the eight simple reflections, turns slowest:
Figure 5 builds c as an integer matrix and computes its order and characteristic polynomial exactly. For E8 the order, the Coxeter number h, is 30, and the eigenvalues are e2πim/30 for the exponents m = 1, 7, 11, 13, 17, 19, 23, 29, the integers below 30 prime to it. Any order of the reflections gives the same values. Each conjugate pair of eigenvalues is a plane on which c turns by 2πm/30; the plane for m = 1 is the Coxeter plane, and projecting the roots onto it gives Figure 6.
Figure 5. Eigenvalues of the Coxeter element c on the unit circle, from its characteristic polynomial, with the h-th roots of unity as ticks and conjugate pairs joined. E6 and E7 use the first six or seven simple roots. Any order of the reflections gives the same polynomial.
Figure 6. The 240 roots projected onto the Coxeter plane, with the 6,720 edges of 421. They land on 8 rings of 30 points. The picture is unchanged by a rotation of 12°, because c acts on this plane as exactly that rotation.
Each ring is one orbit of c, 30 roots on a regular 30-gon, and 240 / 30 = 8. Only on the outer ring are consecutive roots x and cx joined by an edge, so that ring alone is a Petrie polygon of 421.
The ring radii, 0.238, 0.385, 0.474, 0.573, 0.704, 0.767, 0.927 and 1.140, pair off in the golden ratio φ: 0.385 / 0.238 = 0.767 / 0.474 = 0.927 / 0.573 = 1.140 / 0.704. The plane alone cannot explain this.
The exponents split into {1, 11, 19, 29}, those of H4, the symmetry group of the 600-cell, and {7, 13, 17, 23}, the same list times 7 modulo 30. Their eigenplanes span two orthogonal 4-dimensional subspaces. Projected onto the first, the roots fall onto two concentric 600-cells, 120 at radius √(1 − 1/√5) ≈ 0.744 and 120 at √(1 + 1/√5) ≈ 1.203, a ratio of φ; in the second the roles swap.
Read as unit quaternions, the vertices of each 600-cell form the binary icosahedral group 2I, whose subgroup 2T of order 24 has five cosets, five disjoint 24-cells. So the 240 roots split into ten 24-cells. Figure 7 computes all of this from the Coxeter element.
Figure 7. Stage 0: Petrie mandala. The 240 E₈ roots project to 8 concentric rings of 30 points each in the Coxeter plane.
Figure 8. The 240 roots in the two H4-invariant 4-spaces (exponents 1, 11 left; 7, 13 right), each dot at its angle in the exponent-1 or exponent-7 plane and at its norm in that 4-space. The green roots, inner on the left, are outer on the right.
Both subspaces are needed because E8 is built over the golden field ℚ(√5), which embeds in the reals in two ways. The icosians, the ℤ[φ]-combinations of the elements of 2I, form a rank-8 lattice that becomes E8 under a rational form using both embeddings at once (Conway and Sloane, Sphere Packings, Lattices and Groups, chapter 8).
The 600-cell can also be built from 3-dimensional symmetry. The unit normals of the icosahedron's 15 mirror planes, with both signs, are the 30 roots of H3, and in the Clifford algebra of ℝ³ the product nm of two of them is a rotor, a scalar plus a bivector, in a copy of the quaternions.
Figure 9. Products nm of the 30 roots of H3 (left) in the Clifford algebra of ℝ³, the distinct results on the right, coloured by the sign of the scalar part. The 465 products (435 pairs and 30 squares) give exactly 120 unit quaternions: the 600-cell.
The construction stops at H4. Adding 120 elements from the odd part (vectors plus trivectors) gives 240 unit vectors in the 8-dimensional algebra, but they form two orthogonal 600-cells, with inner products such as cos 72° that never occur between E8 roots. The golden-field form is what glues the two copies of H4 into E8.
Part 9 turns from the 240 roots to the infinite lattice they generate.