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.
Part 7 ended at E₈, whose smallest representation is its own 248-dimensional adjoint. Parts 3 and 4 met E₈ as a Dynkin diagram, the last branched tree that passes the positivity test. This part works with its 240 roots directly: as coordinates, as the vertices of a polytope, and in the projections that give the familiar picture of E₈.
The first family is the 112 integer vectors in with exactly two nonzero coordinates, each : choose two positions and two signs, . These are the roots of D8, and each has squared length .
The second family turns D8 into E8: the vectors whose eight coordinates are all with an even number of minus signs. Of the sign patterns, half qualify, , and each has squared length . The parity rule keeps the set closed under reflection, since reflecting a half-integer root in an integer root flips two signs. Together,
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.
Every root has squared length 2, so the inner product of two roots decides the angle between them. Figure 2 computes 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.
Seen from any single root, the other 239 split the same way: 56 at inner product 1 (60°), 126 at 0, 56 at −1 and one antipode. The 126 perpendicular roots are the roots of an E7.
Join two roots by an edge when their inner product is exactly 1, the closest possible, at angle
Every root then has 56 neighbours, so there are
These edges bound triangles, tetrahedra and so on up to 7-dimensional facets: the uniform polytope Gosset described in 1900.
| Face dimension | Count | Face type |
|---|---|---|
| 0 (vertices) | 240 | points |
| 1 (edges) | 6,720 | segments |
| 2 | 60,480 | triangles |
| 3 | 241,920 | tetrahedra |
| 4 | 483,840 | 4-simplices |
| 5 | 483,840 | 5-simplices |
| 6 | 207,360 | 6-simplices |
| 7 (facets) | 19,440 | 17,280 7-simplices and 2,160 7-orthoplexes |
Figure 3. Face counts of 421. Every face below the top dimension is a simplex. The facets come in two kinds, which is why 421 is uniform but not regular.
The polytope can also be sliced. Slide a hyperplane along a coordinate axis: most positions cut edges and miss every vertex, but at five positions a whole sheet of vertices lies on it. They split the 240 roots as 14 + 64 + 84 + 64 + 14: the 84 roots with that coordinate zero (a D7), the 14 integer roots with that coordinate ±1 on each side, and the 64 half-integer roots with that coordinate ±½.
Figure 4. Slicing 421 along a coordinate axis. Left: the wireframe, with the slicing axis as depth and the Coxeter plane as the face; the pink sheet is the hyperplane and red dots are where it cuts edges. Right: the same cross-section head-on. Snap to next halt jumps to the five levels where the hyperplane passes through vertices. Every axis gives the same 14 + 64 + 84 + 64 + 14 profile.
Deleting a node of the Dynkin diagram picks out a smaller root system inside the 240, as Figure 6 of Part 3 computes for all eight nodes. Deleting α8 leaves the 126 roots of E7, the same 126 that are perpendicular to the highest root.
The familiar picture of E₈ is a projection onto one particular plane in ℝ⁸, the plane where a Coxeter element acts as its slowest rotation. A Coxeter element is the product of all eight simple reflections, each used once:
Different orders give conjugate elements, so any order will do. Its order is the Coxeter number:
As a real 8 × 8 matrix, c has eight eigenvalues, all 30th roots of unity:
The exponents {1, 7, 11, 13, 17, 19, 23, 29} are the integers below 30 that share no factor with it, and they also give the order of the Weyl group, . Each conjugate pair of eigenvalues is a real 2-plane on which c rotates by 2πm/30. The plane for m = 1 turns 12° per step, the slowest, and that is the Coxeter plane.
| pair | m | angle | dim |
|---|
Figure 5. The eight eigenvalues of c, in four conjugate pairs. Each pair is a real plane on which c rotates by . The pair is the slowest rotation, the Coxeter plane.
With an orthonormal basis (u, v) of that plane, each root x lands at
The shared library finds (u, v) as the null space of c2 − 2 cos(2π/30) c + I, which isolates the eigenvalue pair e±2πi/30 without complex arithmetic.
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. Every orbit has exactly 30 roots, so there are 240 / 30 = 8 orbits; Coxeter number times rank is the number of roots in every irreducible root system. Since c rotates the plane by 12°, each orbit lands on a regular 30-gon, and the eight orbits sit at eight different radii. Only on the outer ring are neighbouring roots x and cx joined by an edge of 421 (inner product 1), so that ring alone is a Petrie polygon of 421.
In the library's normalisation the eight ring radii are 0.238, 0.385, 0.474, 0.573, 0.704, 0.767, 0.927 and 1.140, and they pair off in the golden ratio: 0.385 / 0.238, 0.767 / 0.474, 0.927 / 0.573 and 1.140 / 0.704 all equal φ = (1 + √5)/2. The plane alone cannot explain the pairing.
The exponents split into two sets, {1, 11, 19, 29} and {7, 13, 17, 23}. The first is the list of exponents of H4, the symmetry group of the 600-cell; the second is its image under multiplication by 7 modulo 30, the same list seen through the other square root of 5. Their eigenplanes span two orthogonal 4-dimensional subspaces of ℝ⁸, each preserved by c. Project the 240 roots onto the first one and they fall onto two concentric 600-cells: 120 roots at radius √(1 − 1/√5) ≈ 0.744 and 120 at radius √(1 + 1/√5) ≈ 1.203, a ratio of exactly φ. Project onto the second subspace and the same thing happens with the roles exchanged: a root on the inner shell in one is on the outer shell in the other. Since the two subspaces are orthogonal complements, the squared radii of each root add to 2.
Each 600-cell carries a finer structure. Its 120 vertices, read as unit quaternions, form the binary icosahedral group 2I, which contains the binary tetrahedral group 2T of order 24 as a subgroup of index 5. The five cosets of 2T are five disjoint 24-cells, so the 240 roots split into ten 24-cells, five per shell. Figure 7 computes all of this from the Coxeter element and lets you move between the stages.
Figure 7. Stage 0: Petrie mandala. The 240 E₈ roots project to 8 concentric rings of 30 points each in the Coxeter plane.
The parallel-coordinates panel under the scene checks the pictures above it. Every stage is a projection or a recolouring, never a change to the roots, and the panel's eight coordinates per root stay fixed while the scene moves.
Figure 8. The same 240 E8 roots, projected into two orthogonal H4-invariant 4-planes. Each dot sits at its c-eigenplane angle, at radius equal to its norm in that 4-plane. Coloured by L-shell, the green roots on the inner shell of the left disc are on the outer ring of the right disc; colour by R-shell to see the reverse. The Galois automorphism σ: φ ↦ 1−φ swaps the two 4-planes.
Both subspaces are needed because E8 is built from the golden field ℚ(√5), which has two embeddings into the reals, sending √5 to +√5 or to −√5. The icosians, the ℤ[φ]-combinations of the 120 elements of 2I, form a ring that is a free abelian group of rank 8. Measured with a suitable rational quadratic form that uses both embeddings of the golden field at once (Conway and Sloane give it in chapter 8 of Sphere Packings, Lattices and Groups), that rank-8 group is the E8 lattice. The two 4-dimensional projections are the two embeddings, and φ ↦ −1/φ is what exchanges inner and outer shells.
The 600-cell itself can be built from 3-dimensional symmetry. The icosahedron has 15 mirror planes, and their unit normals, with both signs, are the 30 roots of H3. In the Clifford algebra of ℝ³, the product nm of two unit vectors is a scalar plus a bivector, and the even part of the algebra, spanned by 1 and the three bivectors, is a copy of the quaternions. Composing two reflections gives a rotation, and nm is the corresponding rotor.
Figure 9. Multiplying H3 roots in pairs. Left: the 30 roots of H3; the current pair (n, m) is highlighted. Right: the distinct products nm found so far, coloured by the sign of the scalar part. The 465 unordered pairs produce exactly 120 distinct unit quaternions. When the run finishes, the page reports the inner products among them, which are the cosines of the multiples of 36° together with 60° and 120° (0, ±1/2, ±φ/2, ±1/(2φ), ±1): these are the 120 vertices of the 600-cell, the binary icosahedral group 2I.
The construction stops at H4. Adding the 120 odd elements of the Clifford algebra (vectors and trivectors) gives 240 unit vectors in the 8-dimensional algebra, but under its Euclidean inner product they form two orthogonal copies of the 600-cell. Their inner products include cos 72° ≈ 0.309, which never occurs between unit E8 roots, where only 0, ±½ and ±1 are possible. The Clifford algebra supplies the two copies of H4; the golden-field quadratic form above is what glues them into E8.
Part 9 turns from the 240 roots to the infinite lattice they generate.