The E8 root system (rotated under G2)


Mathematical background

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

The display above shows a particular projection of the E8 root system, which can be described as a remarkable polytope in 8 dimensions (also known as the Gosset 421 polytope) having 240 vertices (known, in this context, as “roots”), and 6720 edges (shown by black lines in the figure if toggle lines is clicked). Quite concretely, the roots can be described, in the coordinate system we have chosen, as 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 in question being rotated in 8-dimensional space: it is merely performing a rotation at a constant (=uniform) rate, and projected down to two dimensions, always in the same manner. 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 try warping to time 345.6).

The initial view of the polytope is chosen to illustrate an elegant 24-fold symmetry (to see it clearly, pause the animation and warp to time 0).

The rotation chosen here is special: it belongs to the exceptional Lie group G2 of automorphisms of the octonions: the octonions are a non-associative (but “alternative”) division algebra of dimension 8, whose maximal orders (integral octonions) are 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.) More precisely, we choose a torus in G2 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); every time the button reset to random is chosen, the animation is re-initialized with a different choice inside G2 (i.e., a different rotation), whereas reset to standard sets a fixed choice of torus inside G2 which is particularly symmetric relative to the chosen projection.

The pause button should be obvious, and the warp button is used to jump to a specific value of the time parameter. Finally, the toggle lines button is used to toggle display of edges of the polytope: it is far too slow on many browsers, so it is off by default (but it can be used while paused).