Uploaded February 2012 | Updated September 2026, 2 weeks ago
(See the description of youtube.com/watch?v=E-LC_l3gNuc for more about E_8 and its root system.)
The root system of E_8, an 8-dimensional polytope also known as the Gosset 4_21 polytope (here projected on a 2-plane that initially shows a 24-fold symmetry), is made to rotate uniformly, at a constant rate (technically: along a one-parameter subgroup). The polytope shown is always the same, it is merely rotated in 8-dimensional space, always in the same manner.
The rotation chosen here is special in that it belongs to the exceptional Lie group G_2 of automorphisms of the octonions. The E_8 root system forms the loop of units of (some copy of) the Cayley integral octonions (which are an E_8 lattice), and G_2 here refers to those rotations which preserve multiplication on the octonions (this restricts from the 28-dimensional group of all 8-dimensional rotations to a 14-dimensional subgroup). The one-parameter rotation subgroup was chosen randomly inside G_2 with the constraint that its angular velocities are in ratio of the golden ratio (so the video is never periodic, although it will come arbitrarily close to its starting position).
The vertices at the left and right which remain motionless throughout the video are the octonions −1 and 1, which are obviously fixed by G_2.
An interactive JavaScript version of this video is at madore.org/~david/math/e8rotate.html (requires a modern browser, but much prettier).
(See the description of youtube.com/watch?v=E-LC_l3gNuc for more about E_8 and its root system.)
The root system of E_8, an 8-dimensional polytope also known as the Gosset 4_21 polytope (here projected on a 2-plane that initially shows a 24-fold symmetry), is made to rotate uniformly, at a constant rate (technically: along a one-parameter subgroup). The polytope shown is always the same, it is merely rotated in 8-dimensional space, always in the same manner.
The rotation chosen here is special in that it belongs to the exceptional Lie group G_2 of automorphisms of the octonions. The E_8 root system forms the loop of units of (some copy of) the Cayley integral octonions (which are an E_8 lattice), and G_2 here refers to those rotations which preserve multiplication on the octonions (this restricts from the 28-dimensional group of all 8-dimensional rotations to a 14-dimensional subgroup). The one-parameter rotation subgroup was chosen randomly inside G_2 with the constraint that its angular velocities are in ratio of the golden ratio (so the video is never periodic, although it will come arbitrarily close to its starting position).
The vertices at the left and right which remain motionless throughout the video are the octonions −1 and 1, which are obviously fixed by G_2.
An interactive JavaScript version of this video is at madore.org/~david/math/e8rotate.html (requires a modern browser, but much prettier).
