The E_8 root system, or Gosset 4_21 polytope, is an exceptional uniform polytope in 8 dimensions, having 240 vertices and 6720 edges. This video shows a 2-dimensional projection of this polytope as it rotates in various ways.

The first 1′30″ of the video show various small rotations of the polytope to illustrate some of its highly symmetric plane projections (at 10″ we see a 30-fold symmetry known as the Petrie figure, at 20″ a 20-fold symmetry, at 30″ a 24-fold symmetry, at 50″ an 18-fold symmetry and at 1′10″ a 14-fold symmetry). The remaining 2′30″ of the video show a small sample of the 696729600 symmetries of the polytope in a different way: this time, we always return to an equivalent projection (every 10″), after some rotation which left the polytope symmetric.

Note that the polytope shown is always the same, it is merely rotated in 8-dimensional space.

It is unfortunate that compression causes the video quality to be so bad (especially in the second part of the video, where larger rotations are performed).

The E_8 polytope has 240 vertices, 6720 edges, 60480 triangular faces, 241920 tetrahedral three-cells, 483840 simplicial four-cells, 483840 simplicial five-cells, 207360 six-cells (of two different kinds, 69120 and 138240 of each, both being 6-simplices) and 19440 seven-cells (facets; 2160 being 7-simplices and 17280 being 4_11 polytopes). While it is not fully “regular” (there are only three regular solids in 8 dimensions, all boring), it is “uniform” and in many ways exceptional, being the largest of its kind. It is crystallographic in that its vertices span a lattice, the E_8 lattice, with many further remarkable properties (it is the only unimodular even lattice in dimension 8, the smallest nontrivial possible dimension).