Gosset polytopes in integral octonions

Woo Nyoung Chang; Jae Hyouk Lee; Sung Hwan Lee; Young Jun Lee

doi:10.1007/s10587-014-0126-5

Gosset polytopes in integral octonions

Woo Nyoung Chang, Jae Hyouk Lee, Sung Hwan Lee, Young Jun Lee

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

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.

Original language	English
Pages (from-to)	683-702
Number of pages	20
Journal	Czechoslovak Mathematical Journal
Volume	64
Issue number	3
DOIs	https://doi.org/10.1007/s10587-014-0126-5
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

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10.1007/s10587-014-0126-5

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title = "Gosset polytopes in integral octonions",

abstract = "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.",

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T1 - Gosset polytopes in integral octonions

AU - Chang, Woo Nyoung

AU - Lee, Jae Hyouk

AU - Lee, Sung Hwan

AU - Lee, Young Jun

PY - 2014/9

Y1 - 2014/9

N2 - 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.

AB - 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.

KW - 24-cell

KW - Gosset polytope

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JO - Czechoslovak Mathematical Journal

JF - Czechoslovak Mathematical Journal

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ER -

Gosset polytopes in integral octonions

Abstract

Bibliographical note

Keywords

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