Geodesic
Gosset polytopes in integral octonions
Czechoslovak Mathematical Journal, Tome 64 (2014) no. 3, pp. 683-702.

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We study the integral quaternions and the integral octonions along the combinatorics of the $24$-cell, a uniform polytope with the symmetry $D_{4}$, and the Gosset polytope $4_{21}$ with the symmetry $E_{8}$. We identify the set of the unit integral octonions or quaternions as a Gosset polytope $4_{21}$ or a $24$-cell and describe the subsets of integral numbers having small length as certain combinations of unit integral numbers according to the $E_{8}$ or $D_{4}$ actions on the $4_{21}$ 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 $2_{31}$ with the symmetry $E_{7}$, and it is a dual polytope to a Gosset polytope $3_{21}$ with the symmetry $E_{7}$ which is the set of the unit integral octonions with $\operatorname {Re}=1/2$.
DOI : 10.1007/s10587-014-0126-5
Classification : 06B99, 11Z05, 52B20
Keywords: integral octonion; 24-cell; Gosset polytope 
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Chang, Woo-Nyoung; Lee, Jae-Hyouk; Lee, Sung Hwan; Lee, Young Jun. Gosset polytopes in integral octonions. Czechoslovak Mathematical Journal, Tome 64 (2014) no. 3, pp. 683-702. doi : 10.1007/s10587-014-0126-5. http://geodesic.mathdoc.fr/articles/10.1007/s10587-014-0126-5/

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