Uploaded April 2017 | Updated September 2026, 1 week ago
The E8 lattice (or Gosset lattice) is a very regular lattice in 8 dimensions (it is generated by the E8 root system, and realizes the optimal sphere packing in 8 dimensions). The Voronoi cell of a lattice point is the set of points closer to that lattice point than to any other: the Voronoi cells of the E8 lattice are polytopes with 240 facets and 19440 vertices (of two kinds, 2160 corresponding to the "deep" holes of E8 and 17280 to the "shallow" holes), and the Voronoi diagram of E8 is the tiling of 8-space by these polytopes.
This video shows a three-dimensional cross-section of the Voronoi diagram of E8, with two dimensions displayed as image coordinates and the third dimension as time (i.e., each frame is a random two-dimensional cross-section, and this section is translated uniformly in time, perpendicularly to its plane). Each colored region corresponds to a slice of the Voronoi cell of a lattice point. The colors are given by three additional perpendicular directions.
Unlike the companion video youtube.com/watch?v=VSAnezNAjLg (which shows a random cross-section), the plane direction in this video has been chosen to be a Coxeter plane for the lattice, which explains the 30-fold symmetry which occurs exactly around a lattice point and approximately in various places.
A lattice point (which we can call "origin") is encountered exactly in the middle of the video, at the center of the screen. So at this point, an exact 30-fold symmetry is encountered. (Of course, everything is symmetric around this point.)
The E8 lattice (or Gosset lattice) is a very regular lattice in 8 dimensions (it is generated by the E8 root system, and realizes the optimal sphere packing in 8 dimensions). The Voronoi cell of a lattice point is the set of points closer to that lattice point than to any other: the Voronoi cells of the E8 lattice are polytopes with 240 facets and 19440 vertices (of two kinds, 2160 corresponding to the "deep" holes of E8 and 17280 to the "shallow" holes), and the Voronoi diagram of E8 is the tiling of 8-space by these polytopes.
This video shows a three-dimensional cross-section of the Voronoi diagram of E8, with two dimensions displayed as image coordinates and the third dimension as time (i.e., each frame is a random two-dimensional cross-section, and this section is translated uniformly in time, perpendicularly to its plane). Each colored region corresponds to a slice of the Voronoi cell of a lattice point. The colors are given by three additional perpendicular directions.
Unlike the companion video youtube.com/watch?v=VSAnezNAjLg (which shows a random cross-section), the plane direction in this video has been chosen to be a Coxeter plane for the lattice, which explains the 30-fold symmetry which occurs exactly around a lattice point and approximately in various places.
A lattice point (which we can call "origin") is encountered exactly in the middle of the video, at the center of the screen. So at this point, an exact 30-fold symmetry is encountered. (Of course, everything is symmetric around this point.)
