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The Root System
The full E8 root system rendered, tabulated, and cross-referenced. Every root vector, every inner product, every symmetry — held at rest, reachable by coordinate.
The E8 Atlas gathers the geometry of a single extraordinary object — the E8 root system — alongside the mathematics that surround it, the projections that reveal it, and the record of what has been learned about it. The atlas is free to read. Its shape earns its place.
The full E8 root system rendered, tabulated, and cross-referenced. Every root vector, every inner product, every symmetry — held at rest, reachable by coordinate.
The four-dimensional companion to E8 — the 600-cell polytope, its H₄ symmetry group, and the icosian projection that reveals the kinship between them.
The symmetries of the root system, treated as its own object of study. Generators, relations, orbits, and the geometry those symmetries carve.
The many ways E8 has been drawn, in the plane and in higher dimensions. Each projection with its construction, its virtues, and what it hides.
The most iconic view of E8 — 240 roots projected onto a two-dimensional plane defined by the Coxeter element. Eight concentric rings of thirty vertices, rotated in a precise mandala of algebraic origin.
The four-dimensional regular polytope with 600 tetrahedral cells, 120 vertices, and H₄ symmetry — the icosahedral group's four-dimensional cousin. E8's Coxeter plane, folded, reveals it.
Seven nodes in a row, one branching off. This deceptively small graph encodes the entire structure of E8 — its roots, its Weyl group, its Lie algebra. Everything descends from it.
The full symmetry group of the E8 root system — the reflections that map the lattice to itself. A finite group of extraordinary size, whose structure encodes every isomorphism of the root system.
"An atlas is not a proof. It does not settle what is true. It gathers what is known, arranges it so a reader can find their way, and marks the places where the map ends and the territory begins."
Interactive renderings of the E8 root system, reachable directly. HELM Ph1 is the current canonical instrument; the lineage below traces how the atlas learned to draw.
Primary · HELM Ph1 The Deep Atlas — E8 root-system projection All 240 roots on canvas, Coxeter-plane projection with live lattice controls. The current canonical engine.Lineage engines are preserved as built — earlier drafts of the projection, kept for the record.