`t`=?

[See also this
page for a different interactive view of E_{8}, centered
on an exploration of its Weyl group.]

The display above shows a particular projection of
the E_{8} root
system, which can be described as a remarkable polytope in 8
dimensions (also known as
the Gosset
4_{21} 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 E_{8} 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 G_{2} 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 E_{8} lattice, the
E_{8} root system being then the loop of units; G_{2}
is the (14-dimensional) subgroup of the full (28-dimensional) group
SO_{8} 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 G_{2} 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 G_{2} (i.e., a different rotation),
whereas reset to standard

sets a fixed choice of torus inside
G_{2} 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).