The 240 "integral octonions" of unit length can be arranged to coincide with the first shell of the E_8 lattice. These form an alternative monoid with inverses, in fact a "Moufang loop (https://en.wikipedia.org/wiki/Moufang_loop). Thorold Gosset discovered ca. 1900 what's now called the Gosset polytope, which Coxeter calls the 4_21 polytope, basically the convex hull of these 240 points in 8-space (more precisely: its boundary). —Dan ----- It has occurred to me that (vertex-transitive) spherical codes have `theta polynomials' in the same way that lattices have `theta series'. Specifically, the coefficient of q^n gives the number of points at a Euclidean distance of sqrt(n). I was somewhat surprised to see that the coefficients of the theta polynomial of the first shell of the E8 lattice -- namely 1,56,126,56,1 -- do not feature in the OEIS. When you only have three non-zero coefficients, you can define a regular graph on the vertices of the spherical code. If the spherical code has sufficiently many symmetries -- specifically, that the point stabiliser of one vertex is transitive on each of the 'shells' whose cardinalities are specified in the coefficients of the polynomial -- then this graph is strongly regular. The standard construction of the Higman-Sims graph comes from such a spherical code [living in the Leech lattice]. -----