E₈'s 240 roots, the Leech lattice's 196,560 minimal vectors, the 4,096 Golay codewords, the Monster group's representation ladder — seen through the PC lens.
Alongside the five exceptional Lie algebras G₂, F₄, E₆, E₇, E₈ sit three moonshine-adjacent isolated objects: the Leech lattice in 24D, the binary Golay code, and the Monster group. E₈ is the densest 8D sphere packing; Leech is the densest 24D packing; Golay is Leech's combinatorial skeleton; the Monster's smallest non-trivial representation matches the first j-invariant coefficient to a constant.
These objects are covered fully in E₈ & Lie algebras and the Moonshine act of Modular Forms. This page is the PC lens: each structure as a polyline bundle, with axis value vocabulary, symmetry as bundle invariance, sparsity as flat segments, codeword density as visual texture.
G₂ has 12 roots in 3 zero-sum coordinates. F₄ has 48 roots on 4 axes (the root system of the 24-cell). E₆, E₇, E₈ live in 8 axes with 72, 126, and 240 polylines respectively. Viewed at a common visual scale, the escalation from sparse to dense is immediate.
Figure 1. Small multiples: parallel coordinates for all five exceptional root systems at the same visual scale. Hover any polyline to isolate one root. The density of crossings and the value vocabulary per axis change abruptly at each step. E₈ is the saturation point of the exceptional root-system list — no larger exceptional root system exists, though the classical families continue.
Petrie/Coxeter projections flatten 8 dimensions onto 2 as radial mandalas. Parallel coordinates keep the full coordinate record. The value alphabet per axis is readable: G₂ visits −2, −1, 0, 1, 2; F₄ and E₆–E₈ mix integer roots (flat at 0 on most axes) with half-integer roots (±½ on every axis). Each root's sparsity is legible at a glance.
The Coxeter element c is a product of the 8 simple reflections. It has order 30: apply it 30 times and every root returns home. Each application rotates the Petrie mandala by 12° and permutes the 240 polylines cyclically.
Figure 2. The Coxeter element acting on E₈'s 240 roots. Left: 2D Coxeter-plane projection (the Petrie mandala). Right: 8-axis parallel coordinates. Each step applies c once. After 30 steps every root and every polyline returns to its starting position. Blue polylines are Type 1 (integer) roots; indigo polylines are Type 2 (half-integer) roots.
The Petrie view shows rotation as a rigid rotating mandala. The PC view shows the same rotation as a braid: polylines slide along their axes in lockstep. The half-integer family stays in [−½, ½]; the integer family visits −1, 0, 1.
Dechant (2015) showed E₈ can be built deterministically from the 3D icosahedron. H₃ in R³ has 8-dimensional Clifford algebra Cl(3). Inside Cl(3) sit 120 unit rotors — the binary icosahedral group 2I — each a spinor double cover of an icosahedral rotation. Unioning those 120 rotors with a φ-scaled copy in R⁸ via the Cl(3) inner product gives E₈'s 240 roots.
The figure below pulls out the middle step: 2I as 120 unit icosians in 4 quaternion axes. These are the spinors produced by Cl(3)'s even subalgebra acting on itself. Each polyline is one rotor. The 120 polylines split into 5 cosets of the binary tetrahedral subgroup 2T (the 24 vertices of a 24-cell); the five colours mark the cosets.
Figure 2b. The Clifford construction in three steps. The icosahedron (left) has 12 vertices and 20 triangular faces; the 15 mirror planes through its symmetry axes generate H3. Inside Cl(3), H3 lifts to a spin double cover of order 120 — the binary icosahedral group 2I — whose elements are shown as parallel-coordinate polylines in 4 quaternion axes (right). The 120 split naturally into three sub-families: 8 basis units ±1, ±i, ±j, ±k + 16 half-integer units ½(±1,±1,±1,±1) = the 24 vertices of a 24-cell (2T); and 96 “golden rotors” obtained by cyclic permutation of ½(0,±1,±φ¹,±φ⁻¹). Stacking these 120 with their φ-multiples in R8 produces the 240 roots of E8.
Every E₈ root factors as rotor × scale, with one of 120 Cl(3) spinors and a scale of 1 or φ. The 240 PC polylines of E₈ are two overlaid copies of the 120 icosian polylines: one unit-norm, one φ-scaled. See E₈'s 600-cells and Icosahedron to E₈.
The eight simple roots of E8 form a tree. Add a ninth simple root — the negative of the highest root, α0 = −θ — and the tree becomes a closed loop: the affine Dynkin diagram Ẽ8. The Borel–de Siebenthal theorem says that any node deletion from Ẽ8 gives the Dynkin diagram of a maximal rank-8 subalgebra of E8.
Deleting the affine node α0 recovers E8 itself — the trivial case. The other eight deletions produce eight proper maximal subalgebras, ranging from D8 (112 roots) down to A5 ⊕ A2 ⊕ A1 (just 38 roots). Each subalgebra's root system sits inside E8's 240 roots as a specific sub-polytope. Click a node to highlight its deletion result below.
Figure 2c. Affine Dynkin diagram Ẽ8 (top, 9 nodes) with Kac marks (1, 2, 3, 4, 5, 6, 3, 4, 2). The deleted node is crossed out; the surviving rank-8 simple roots generate a maximal subalgebra. The parallel-coordinate panel below shows all 240 E8 roots, with the selected subalgebra's roots drawn solid and the rest faded. The eight non-trivial deletions yield eight subalgebras: E7⊕A1, E6⊕A2, D5⊕A3, A4⊕A4, A5⊕A2⊕A1, A8, A7⊕A1, D8.
Both lattices are even unimodular, and both achieve densest sphere packing in their dimension (Viazovska, 2016; Cohn-Kumar-Miller-Radchenko-Viazovska, 2017). But the jump from 8 to 24 dimensions is a phase transition in complexity. Place all 240 E₈ roots next to 240 randomly sampled Leech minimal vectors: same polyline count, three times the axes, a far wider coordinate vocabulary.
E₈ roots: 240 vectors across 8 axes
Leech sample: 240 vectors across 24 axes
Figure 3. E₈'s 240 roots (left) against 240 randomly sampled Leech minimal vectors (right). E₈ coordinates live in {−1, −½, 0, ½, 1}; Leech integer coordinates span {−4, …, 4}. Polyline colors mark vector types.
E₈ is quiet and structured: five possible values per axis, visible symmetry between root types. Leech is loud at the same polyline count: 24 axes compress horizontally, the three Leech types (Type A: two ±4 entries; Type B: eight ±2 on a Golay octad; Type C: one ±3 plus twenty-three ±1) overlap in a wilder braid. Visual density tracks mathematical density.
There are 196,560 shortest nonzero vectors in the Leech lattice. They decompose into three shape classes:
Figure 4. The 196,560 minimal vectors, viewed statistically. Polyline view samples 500 vectors colored by type. The heatmap counts coordinate values per axis across the full population.
Each Leech type has a distinct PC signature. Type A: flat at 0 with two ±4 spikes. Type B: flat on 16 axes, ±2 on an octad. Type C: ±1 everywhere except one ±3 spike. Visual sparsity equals mathematical support.
The extended binary Golay code is a 12-dimensional subspace of F₂²⁴. All 4,096 codewords fit in one figure. Each is a 24-bit vector; drawn as a polyline across 24 parallel axes (values 0 or 1), the population self-organizes by weight. Only weights 0, 8, 12, 16, 24 occur: the code is doubly even and self-dual, and these two constraints collapse 25 possible weights to 5.
Figure 5. Every codeword of the extended binary Golay code [24,12,8], drawn as a binary polyline across 24 axes. Color encodes Hamming weight. Filter buttons isolate each weight class.
Weight is the count of axis-top visits — readable visually. Octads (weight 8, blue) and their complements (weight 16, orange) are literal mirror images. Dodecads (weight 12, green) are the largest, densest class.
The Monster group's 194 irreducible representations have dimensions spanning from 1 to about 2.6×10²&sup6;. McKay and Thompson's observation was that every Fourier coefficient of the j-function (minus 744) is a small non-negative integer combination of these dimensions. Each coefficient cn becomes a polyline in the space of multiplicities.
Figure 6. Parallel-coordinates decomposition of the first seven j-function coefficients c₁…c₇. Each polyline is one coefficient; axes d₁…d₈ are the first eight Monster irreducible representations (dimensions 1, 196,883, 21,296,876, …). The y-value on each axis is the multiplicity of that representation in the coefficient's decomposition. Hover a line to see the full numerical decomposition.
The McKay–Thompson pattern is geometric: polylines stair-step downward, with small multiplicities on large-dimensional reps and large multiplicities on small-dimensional reps. Successive coefficients sit above previous ones on small-d axes and extend right as higher representations enter.
For construction theorems, character tables, genus-zero arguments, sporadic-group chains, and the Leech-to-Monster path, see E₈ & Lie algebras and the Moonshine act of Modular Forms.