Part 10½ of 15. Clifford spinors promote 3D symmetry to 4D, then to 8D. The icosahedron's 60 rotations lift to H₄'s 120 spinors, and the full 120-element reflection group lifts to 240 pinors, matching the root count of E₈.
The rest of this atlas treats E₈ as a 248-dimensional Lie algebra with 240 roots in ℝ⁸. That is the Dynkin story. A second story starts from the regular icosahedron.
The claim: E₈ is the shadow of the icosahedron's symmetry group through the Clifford algebra of 3-space. Start with a solid in ℝ³, ask Clifford algebra for its reflection group, and the algebra returns a chain of reflection groups ending in E₈.
The chain is due to Dechant (2012–2016), building on Arnold's trinity. It gives a different reason for E₈'s existence: the icosahedral rotation group lifts to 120 spinors in Cl⁺(3), while the full reflection group lifts to 240 pinors in Cl(3); under the Clifford inner product, those 240 pinors satisfy the E₈ root-system axioms.
Figure 1. The chain of induction. Each arrow is a Clifford-algebra construction: 30 H₃ roots pair-multiply to 120 even multivectors (spinors), which are the 120 vertices of the 600-cell and the 120 roots of H₄. Extending to all 240 multivectors (even + odd) reproduces E₈'s 240 roots under the Euclidean norm Cl(3) inherits from ℝ³.
Fix the regular icosahedron with 12 vertices, 30 edges, 20 triangular faces. Its full symmetry group Ih (including reflections) has order 120. Sixty of those symmetries are rotations: the subgroup I of order 60, isomorphic to the alternating group A5. The remaining 60 involve a reflection or improper rotation.
As a reflection group, Ih is generated by 15 reflections, one through each of 15 mirror planes. Those 15 mirrors are the Coxeter group H3. The 15 reflection normals and their 15 negatives give a 30-vector root system, also called H3, whose Weyl group is exactly Ih.
Figure 2. The icosahedron with its 12 vertices at cyclic permutations of (0, ±1, ±φ). Drag to rotate. Switch modes to highlight a single vertex, a single mirror plane, or the full set of 15 mirrors (all pass through the origin). The 30 root-vector arrows are the pairs of unit normals to those 15 mirrors.
Clifford algebra puts every geometric transformation of ℝ³ in a single multiplicative arena. Start with three orthonormal vectors e1, e2, e3 satisfying
No other relations. The geometric product of two unit vectors is a bivector, the oriented plane they span. The product e1e2e3 is the pseudoscalar, an oriented volume.
One scalar, three vectors, three bivectors, one pseudoscalar gives Cl(3), 8-dimensional. That 8 is the same 8 as E₈'s rank — not a coincidence.
Figure 3. The full 8×8 multiplication table of Cl(3). Each cell shows a·b for basis multivectors a (row) and b (column). The table is a 4×4 block structure: scalar+vector block, bivector+pseudoscalar block, and the two mixed blocks. The pseudoscalar I commutes with everything and satisfies I² = −1, so I behaves as an imaginary unit.
In Clifford algebra, reflecting a vector x in a mirror with unit normal n has a clean formula: . And composing two reflections, first in m then in n, is another clean formula: . The object R := nm is a rotor, a product of two 1-vectors. By the geometric product, R lives in the even subalgebra Cl⁺(3): scalars + bivectors, which is 1 + 3 = 4-dimensional.
The even subalgebra Cl⁺(3) is the space of spinors. Geometrically, each rotor R encodes a rotation of ℝ³ (composition of two reflections = rotation about their intersection line by twice the angle between them). Algebraically, R is a scalar plus a bivector: it looks like a quaternion, and indeed Cl⁺(3) ≅ ℍ.
Figure 4. Two reflections compose into a rotation. Slide to change the angle θ between mirror normals m and n. The vector x first reflects across m (green trace), then across n (orange trace). The composite is a rotation by 2θ about the axis m × n — encoded as the rotor R = nm = cos(θ) + sin(θ) · B, a scalar plus a bivector, in Cl⁺(3).
The 120 spinors. The icosahedral rotation group I ≅ A5 has 60 elements. Inside Cl⁺(3), rotations are represented by rotors R = nm (products of two unit 1-vectors), and each rotation lifts to exactly two rotors, R and −R. So the rotation group I has a double cover inside Cl⁺(3) of order 2 × 60 = 120 elements. That double cover is the binary icosahedral group 2I ⊂ Spin(3), and its 120 elements are exactly the 120 distinct even multivectors we'll generate in the next figure, equivalently the 120 roots of H4, the 120 vertices of the 600-cell. Reflections (odd products of 1-vectors) live in Cl⁻(3) and extend the story to Pin(3) for the full 240-element cover of the reflection group Ih.
Here is the mechanical claim. Take the 30 unit root vectors of H3, one for each oriented mirror. Multiply every pair, including each with itself. Every such product nm is an element of Cl⁺(3). Collect the distinct results. You find exactly 120 of them, and they form a 4-dimensional root system, identified with Cl⁺(3) ≅ ℝ⁴ under the Euclidean norm. That root system is H4, the root system of the regular 600-cell.
Figure 5. Clifford induction H3 → H4. Press Play to run through every pair of H3 roots (left panel) and deposit the distinct products (right panel, colour-coded by scalar-part sign: green > 0, red < 0, grey 0). The count on the right climbs to exactly 120 — the order of the binary icosahedral group 2I and the vertex count of the 600-cell. No other dimension of root system gives this shape.
The 120 even multivectors generated by the previous figure are not just a root system on paper. Identified with unit quaternions, they are the binary icosahedral group 2I, the double cover of the rotation group I ≅ A5. And their 120 positions in ℝ⁴ are the 120 vertices of the regular 600-cell, the most symmetric regular 4-polytope.
Explicit coordinates for these 120 vertices (after normalising):
The φ here is the golden ratio, . The golden ratio is the telltale that we came from icosahedral symmetry. An, Bn, Dn, F4, and G2 have root systems with only rational coordinates; H3, H4, and E8 (in certain coordinate systems) are the ones that smuggle in √5.
Figure 6. The 600-cell, stereographically projected from ℝ⁴ into ℝ³. The 120 vertices carry four natural partitions. Inscribed 24-cell: the 24 of form ±ei and ½(±1,±1,±1,±1) — exactly the binary tetrahedral group 2T ⊂ 2I. Snub 24-cell: the remaining 96 vertices, the even-permutation set ½(0, ±1, ±φ, ±1/φ); their convex hull is a diminished 600-cell with tridiminished-icosahedron vertex figure. 5 × 24-cells compound: the left cosets of 2T inside 2I partition the full 120 vertices into five interlocking 24-cells — the icosian 5 = |2I|/|2T|. Icosahedral shells: the 120 vertices organise into parallel cross-sections of 1 + 12 + 20 + 12 + ... icosahedra stacked along the w-axis. Drag to rotate through the fourth dimension.
We have used only the even half of Cl(3), the four-dimensional space of scalars plus bivectors. What about the odd half? That is Cl⁻(3) = vectors plus pseudoscalar, also four-dimensional. Together Cl(3) = Cl⁺(3) ⊕ Cl⁻(3), total 4 + 4 = 8 dimensions.
A single reflection is represented by a single 1-vector, an element of Cl⁻(3). In general, the full pin group Pin(3) consists of products of any number of unit vectors, even or odd. The spinor subgroup Spin(3) = Pin(3) ∩ Cl⁺(3) has 120 elements for icosahedral symmetry. Pin(3), the full double cover of Ih, has 240 elements, twice 120.
And 240, of course, is the root count of E8. The claim this explainer is building toward:
Clifford induction H3 → E8. Take the 120 even multivectors (Cl⁺(3)) and the 120 odd multivectors (Cl⁻(3)) generated from H3's 30 roots. The 240 multivectors, taken as vectors in ℝ8 with the Euclidean norm Cl(3) inherits from ℝ³, satisfy the root-system axioms and form the root system of E8.
We do not reprove that here; it is the main theorem of Dechant's work. What we can do is verify it visually: take the 240 multivectors, project them through the Coxeter element onto E8's Coxeter plane, and check they land on the 8-ring 30-gon mandala that atlas #13 derived by a completely different route.
Figure 7. E8's Coxeter-plane mandala, split into two shells of 120 roots each. The 8 concentric rings carry 30 roots apiece; the inner four rings (120 roots, green) make up one copy of H4, and the outer four rings (120 roots, red) are its golden-ratio image φH4. This is the golden fold E8 = H4 ⊕ φH4; in the Clifford picture the two copies carry the even and odd multivectors. For the direct derivation of this mandala see Part 13.
The chain H3 → H4 → E8 is not alone. Vladimir Arnold observed many trinities of exceptional objects in mathematics, and they often align into a 3×3 grid. One such grid places our chain as the third row:
Figure 8. Arnold's trinity of reflection groups. The three columns are parallel constructions: simply-laced Dynkin ↔ folded Dynkin ↔ icosahedral. The Clifford induction operates on the rows: given a rank-3 reflection group, build its even multivector algebra to get the rank-4 group; extend to the full Clifford algebra to get the rank-8 group. The H column is the only one where the last step produces an exceptional rather than a classical algebra — which is why E8 is exceptional and D4, F4 are relatively tame.
The Dynkin classification proves E8 exists by listing allowed trees, but does not say why eight dimensions, or why the exceptional tower stops with E6, E7, E8.
The Clifford induction reads it differently: 8D exceptional Lie phenomena are the algebraic shadow of the icosahedron's exceptional 3D symmetry. The icosahedron is exceptional in 3D as the only non-crystallographic Platonic solid, and that drives the cascade H3 → H4 → E8.
The trinity's other columns run the same Clifford machine on the tetrahedron and cube, landing on D4 and F4. Only the icosahedral column produces an exceptional terminal. The octonionic story and the Freudenthal magic square give a complementary picture.
The induction adds a reason: the icosahedron's 60 rotations double-cover to 120 spinors, and the full 120-element reflection group double-covers to 240 pinors — the root count of E8.