## Abstract

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 Re = 1/2.

Original language | English |
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Pages (from-to) | 683-702 |

Number of pages | 20 |

Journal | Czechoslovak Mathematical Journal |

Volume | 64 |

Issue number | 3 |

DOIs | |

State | Published - Sep 2014 |

### Bibliographical note

Publisher Copyright:© 2014, Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic.

## Keywords

- 24-cell
- Gosset polytope
- integral octonion