We study the integral quaternions and the integral octonions along the combinatorics of the 24-cell, a uniform polytope with the symmetry D4, and the Gosset polytope 421 with the symmetry E8.
We identify the set of the unit integral octonions or quaternions as a Gosset polytope 421 or a 24-cell and describe the subsets of integral numbers having small length as certain combinations of unit integral numbers according to the E8 or D4 actions on the 421 or the 24-cell, respectively.
Moreover, we show that each level set in the unit integral numbers forms a uniform polytope, and we explain the dualities between them. In particular, the set of the pure unit integral octonions is identified as a uniform polytope 231 with the symmetry E7, and it is a dual polytope to a Gosset polytope 321 with the symmetry E7 which is the set of the unit integral octonions with Re = 1/2.
|Number of pages||20|
|Journal||Czechoslovak Mathematical Journal|
|State||Published - Sep 2014|
- Gosset polytope
- integral octonion