Scrub from E₈'s 240-root Petrie mandala, through a 3-D lift, into two concentric 600-cells inside a single 4-plane — then split one of them into five disjoint 24-cells.
The previous explainer, Exceptional Structures in Parallel, drew E₈'s 240 roots as polylines across 8 axes and as a Petrie mandala. The mandala flattens an 8D object onto a plane. E₈ is structurally two copies of H₄ glued by the golden ratio, where H₄ is the symmetry group of the 600-cell.
The scrubber runs four stages. Stage 0: the Petrie mandala. Stage 1: a 3D lift by activating a third Coxeter eigenvector. Stage 2: the fold — 240 roots become two concentric 600-cells at radii in ratio φ, in the same 4-plane. Stage 3: the inner 600-cell splits into five disjoint 24-cells (cosets of 2T inside 2I).
The Coxeter element of E₈ has eigenvalues e2πik/30 for k in {1, 7, 11, 13, 17, 19, 23, 29}. Grouping into {1, 11, 19, 29} and {7, 13, 17, 23} — images under multiplication by 7 in (ℤ/30)₊ — each set is H₄'s exponents and spans an H₄-invariant 4-plane.
Drag the scrubber; or click a stage label; or press the arrow buttons to step. The top panel shows the 3-D projection of the 240 roots at the current scrubber position. The bottom panel shows the same 240 roots as polylines across 8 Bourbaki axes — the native PC view. Colors link the two panels: every point in the scene corresponds to exactly one polyline below.
Figure 1. Stage 0: Petrie mandala. The 240 E₈ roots project to 8 concentric rings of 30 points each in the Coxeter plane.
E₈'s Coxeter element c is a 30-fold rotation of R⁸ with no real eigenvalues. Its slowest 2-plane — where c rotates by 12° — is the Petrie plane. Projecting 240 roots onto it gives 8 concentric rings of 30 points.
In parallel coordinates the rotation is less photogenic but more honest. The 240 polylines cycle through a 30-step orbit, permuting integer and half-integer families separately. Integer roots (blue) sit on 2 of 8 coordinates; half-integer roots (indigo) touch all 8.
A second 2-plane has c acting as e22πi/30, orthogonal to the Petrie plane. Together they span an H₄-invariant 4-plane.
Stage 1 adds a coordinate from the second 2-plane as depth. The mandala gains z-extent; the 8 rings pull apart vertically while remaining concentric in the original view, tracing the 600-cell's vertex structure.
Stage 2 projects onto the full 4-plane: all four coordinates a1, b1, a2, b2 of H₄'s eigenbasis. The 240 roots now live in R4. In this 4-plane they distribute over two concentric 600-cells:
Both shells are 600-cells. The complementary 4-plane (eigenplanes 7 and 13) splits the same 240 roots into inner and outer shells with the labels swapped: a root inner in one is outer in the other, norms in Galois-conjugate proportion. This is the icosian fold — E₈ viewed through the two embeddings of ℚ(√5) into R.
The shells overlay in projection. The scrubber pushes the outer shell radially outward to open a gap.
The 120 vertices of a 600-cell, identified with unit quaternions via the icosian ring, form the binary icosahedral group 2I. Inside 2I sits the binary tetrahedral group 2T of order 24, whose vertices form a 24-cell. Since |2I|/|2T| = 5, every 600-cell is the union of five disjoint 24-cells.
Both shells split independently: 5 + 5 = 10 polytopes × 24 vertices = 240 = |E₈|. Every E₈ root belongs to exactly one of the ten 24-cells.
Hover any polytope in the scene to highlight its 24 polylines in the parallel-coordinates panel below — one colored bundle of 24 lifts out of the mass, the other 216 dim away. The scrubber is paused; the toggles at the bottom control which planes the polytopes spin in and whether PC shows vertices or edge midpoints.
Why five: 2T has index 5 in 2I, so its five left cosets partition the 120 vertices into five 24-cells. This reflects the same 5-fold symmetry as the dodecahedron and φ = 2·cos(π/5). 2I can also be realized as 120 unit rotors in Cl(3), the Clifford algebra of R³ (see the PC Clifford figure and Icosahedron to E₈).
The Merge to E₈ button reverses the decomposition. The ten 24-cells re-converge into two 600-cells, the shells merge, and the 4-plane collapses to the Coxeter plane — every edge keeping its 24-cell color. The Petrie mandala from stage 0 is now traced by 960 colored edges showing which 24-cell each belongs to.
In the mandala, every ring mixes inner and outer shells and mixes cosets. The Petrie projection compresses everything that commutes with c, which happens to compress exactly the structure we want.
In parallel coordinates the structure survives: each root keeps its 8-tuple of coordinates. The scrubber is a sequence of recolorings and reorderings, not a redrawing — the polylines are identical at every stage.
The Coxeter element has 8 complex eigenvalues on R⁸ grouped into four invariant 2-planes. Stage 2 used the (1, 11) pair. The (7, 13) pair spans a second orthogonal H₄-invariant 4-plane.
This second fold is the Galois conjugate: 240 roots split 120/120 at radii √0.553 and √1.447, with labels swapped — every root inner in the first 4-plane is outer in the second. The shell radii obey |L|² + |R|² = 2 since the 4-planes are orthogonal complements.
Figure 2. The same 240 E8 roots, projected into two orthogonal H4-invariant 4-planes. Each dot is placed at its c-eigenplane angle, at radius equal to its full 4-D norm in that 4-plane. Colour is fixed by the L-shell label by default: notice that roots green-coloured in the left disc (inner shell in L, near centre) are the same green dots sitting on the outer ring of the right disc. Flip the colouring to R-shell to confirm the roles reverse. The two 4-planes are complementary slices of R8; the Galois automorphism σ: φ ↦ 1−φ swaps them.
E₈ embeds into the icosian ring I, a subring of quaternions over Q(φ). The two embeddings Q(φ) ↪ R — one sending φ to ~1.618, the other to its Galois conjugate ~−0.618 — give two inequivalent realizations as a lattice in R⁸. Neither is canonical; together they encode all of E₈'s arithmetic.
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