Method and credits

The model

The explorer constructs all 240 roots directly in eight dimensions. It joins two roots when their inner product is 1. This produces 6,720 edges and 56 neighbours at each root.

Every screen position is an orthogonal projection onto two vectors from an orthonormal 8D frame. Rotations update that frame. The root coordinates and their distances remain fixed.

Dragging a root

A screen movement gives two constraints on an 8D rotation. The explorer solves for the smallest skew-symmetric rotation generator that satisfies those constraints. It then applies the resulting plane rotations to the complete frame.

This minimum-change rule resolves the hidden directions consistently. A projected root can resist an outward drag when it reaches its full 8D radius.

Sources

David A. Madore's Weyl-group explorer supplies the coordinate convention, the order-30 Weyl word and its matching Petrie projection. Madore placed that code and text in the public domain and requested recognition.

Madore's separate continuous rotation under G₂ demonstrated a smooth E8 projection in 2012.

Samim's 2025 page embedded Madore's first explorer and prompted this direct-manipulation version.

The four Coxeter-plane views use the conjugate exponent pairs 1, 7, 11 and 13. The Coxeter element rotates them by 12°, 84°, 132° and 156°. Each view has eight rings of 30 roots.

The specialist menu also preserves Madore's alternate, 24-fold, 20-fold, blurred 18-fold and blurred 14-fold projections.

This implementation

David Ryan commissioned this explorer in 2026. OpenAI Codex wrote its original implementation. It uses plain JavaScript and the browser's 2D canvas, with no external library or downloaded model.