Uploaded March 2018 | Updated September 2026, 2 weeks ago
This video shows a three-dimensional cross-section of the Fourier transform of the E8 root system (or Gosset 4_21 polytope: see youtube.com/watch?v=E-LC_l3gNuc for a direct projection of this polytope).
The Fourier transform of the E8 root system (or more accurately, of a sum of Dirac δ distributions, one at each root of the system) is the sum of 240 complex exponentials (or 120 cosines), one for each root of E8. This function takes values between −16 and +240 (represented using an ad hoc color scheme, see below), it is periodic modulo the E8 (coroot, i.e., dual) lattice, with values +240 exactly on the points of the latter.
Here we take a three-dimensional cross-section of the (eight-dimensional) space, with two dimensions displayed as image coordinates and the third dimension as time (i.e., each frame is a two-dimensional cross-section, and this section is translated uniformly in time, perpendicularly to its plane, in a random direction). 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.
The section, axes and scale are exactly the same as in the video youtube.com/watch?v=LPVT8aDK2pc so it can be said to show a different view of the same space (the two videos correspond frame per frame and pixel per pixel).
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 color scheme is a piecewise linear gradient as follows: −16 is bright green, −8 is black, 0 is white, +16 is bright red, and +240 is bright blue (these are some of the critical values of the function).
This video shows a three-dimensional cross-section of the Fourier transform of the E8 root system (or Gosset 4_21 polytope: see youtube.com/watch?v=E-LC_l3gNuc for a direct projection of this polytope).
The Fourier transform of the E8 root system (or more accurately, of a sum of Dirac δ distributions, one at each root of the system) is the sum of 240 complex exponentials (or 120 cosines), one for each root of E8. This function takes values between −16 and +240 (represented using an ad hoc color scheme, see below), it is periodic modulo the E8 (coroot, i.e., dual) lattice, with values +240 exactly on the points of the latter.
Here we take a three-dimensional cross-section of the (eight-dimensional) space, with two dimensions displayed as image coordinates and the third dimension as time (i.e., each frame is a two-dimensional cross-section, and this section is translated uniformly in time, perpendicularly to its plane, in a random direction). 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.
The section, axes and scale are exactly the same as in the video youtube.com/watch?v=LPVT8aDK2pc so it can be said to show a different view of the same space (the two videos correspond frame per frame and pixel per pixel).
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 color scheme is a piecewise linear gradient as follows: −16 is bright green, −8 is black, 0 is white, +16 is bright red, and +240 is bright blue (these are some of the critical values of the function).
