Mathematical background

[See also David's page for a different interactive view of E₈, centered on an exploration of its Weyl group.]

Above is a particular projection of the E₈ root system, a remarkable polytope in 8 dimensions (also known as the Gosset 4₂₁ polytope) having 240 vertices (known as “roots”), and 6720 edges. The roots are the (112) points having coordinates(±1,±1,0,0,0,0,0,0) (where both signs can be chosen independently and the two non-zero coordinates can be anywhere) together with those (128) having coordinates (±½,±½,±½,±½,±½,±½,±½,±½) (where all signs can be chosen independently except that there must be an even number of minuses).

Every root (being identified with the vector leading from the origin to the vertex in question) is of the same length (i.e., all vertices are on a sphere with the origin as center; this is specific to E₈ and is not a property of all root systems); the opposite of each root is again a root, and each one is orthogonal to 126 others, while forming an angle of π/3 with 56 others (those that are connected to it by an edge): the only possible angles between two roots are 0, π/3, π/2, 2π/3 and π.

This page shows the polytope rotated in 8-dimensional space projected down to two dimensions. While in 3 dimensions a rotation of the sort is necessarily periodic (after some time the solid will have performed one full turn so ultimately return to its starting position), this is no longer the case in ≥4 dimensions (although some rotations are indeed periodic and all rotations will come back arbitrarily close to their initial position; e.g., on this page "choose time" 345.6).

The initial view of the polytope illustrates an elegant 24-fold symmetry (to see it clearly, pause the animation and "choose time" 0).

The 'standard' rotation chosen here is special, belonging to the exceptional Lie group G₂ of automorphisms of the octonions, being a non-associative (but “alternative”) division algebra of dimension 8. They have maximal orders (integral octonions) isometric to the E₈ lattice. The E₈ root system being then the loop of units; G₂ is the (14-dimensional) subgroup of the full (28-dimensional) group SO₈ of all rotations in dimension 8 which preserves this octonion product. (In particular, it must preserve 1 and −1, which explains why two points, on the left and right of the picture, remain fixed throughout the animation.) A torus in G₂ was chosen by conjugating by a uniformly chosen element of the latter, and the trajectory inside the torus is that with periods 1 and φ (the golden ratio, (1+√5)/2).
If you are enthralled by the 'everythingness' of E8, Counterpoint