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Platonic solids generate their four-dimensional analogues

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Platonic solids generate their four-dimensional analogues. / Dechant, Pierre-Philippe.

In: Acta Crystallographica Section A : Foundations of Crystallography, Vol. 69, No. 6, 26.11.2013, p. 592-602.

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Harvard

Dechant, P-P 2013, 'Platonic solids generate their four-dimensional analogues', Acta Crystallographica Section A : Foundations of Crystallography, vol. 69, no. 6, pp. 592-602. https://doi.org/10.1107/S0108767313021442

APA

Dechant, P-P. (2013). Platonic solids generate their four-dimensional analogues. Acta Crystallographica Section A : Foundations of Crystallography, 69(6), 592-602. https://doi.org/10.1107/S0108767313021442

Vancouver

Dechant P-P. Platonic solids generate their four-dimensional analogues. Acta Crystallographica Section A : Foundations of Crystallography. 2013 Nov 26;69(6):592-602. https://doi.org/10.1107/S0108767313021442

Author

Dechant, Pierre-Philippe. / Platonic solids generate their four-dimensional analogues. In: Acta Crystallographica Section A : Foundations of Crystallography. 2013 ; Vol. 69, No. 6. pp. 592-602.

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@article{85404abf046648f88f0f7a065d51df5c,
title = "Platonic solids generate their four-dimensional analogues",
abstract = "This paper shows how regular convex 4-polytopes – the analogues of the Platonic solids in four dimensions – can be constructed from three-dimensional considerations concerning the Platonic solids alone. Via the Cartan–Dieudonn{\'e} theorem, the reflective symmetries of the Platonic solids generate rotations. In a Clifford algebra framework, the space of spinors generating such three-dimensional rotations has a natural four-dimensional Euclidean structure. The spinors arising from the Platonic solids can thus in turn be interpreted as vertices in four-dimensional space, giving a simple construction of the four-dimensional polytopes 16-cell, 24-cell, the F4 root system and the 600-cell. In particular, these polytopes have `mysterious' symmetries, that are almost trivial when seen from the three-dimensional spinorial point of view. In fact, all these induced polytopes are also known to be root systems and thus generate rank-4 Coxeter groups, which can be shown to be a general property of the spinor construction. These considerations thus also apply to other root systems such as A_{1}\oplus I_{2}(n) which induces I_{2}(n)\oplus I_{2}(n), explaining the existence of the grand antiprism and the snub 24-cell, as well as their symmetries. These results are discussed in the wider mathematical context of Arnold's trinities and the McKay correspondence. These results are thus a novel link between the geometries of three and four dimensions, with interesting potential applications on both sides of the correspondence, to real three-dimensional systems with polyhedral symmetries such as (quasi)crystals and viruses, as well as four-dimensional geometries arising for instance in Grand Unified Theories and string and M-theory.",
author = "Pierre-Philippe Dechant",
note = "This is an author produced version of a paper published in Acta Crystallographica Section A. Uploaded in accordance with the publisher's self-archiving policy.",
year = "2013",
month = nov,
day = "26",
doi = "10.1107/S0108767313021442",
language = "English",
volume = "69",
pages = "592--602",
journal = "Acta Crystallographica Section A : Foundations of Crystallography",
issn = "0108-7673",
publisher = "International Union of Crystallography",
number = "6",

}

RIS (suitable for import to EndNote) - Download

TY - JOUR

T1 - Platonic solids generate their four-dimensional analogues

AU - Dechant, Pierre-Philippe

N1 - This is an author produced version of a paper published in Acta Crystallographica Section A. Uploaded in accordance with the publisher's self-archiving policy.

PY - 2013/11/26

Y1 - 2013/11/26

N2 - This paper shows how regular convex 4-polytopes – the analogues of the Platonic solids in four dimensions – can be constructed from three-dimensional considerations concerning the Platonic solids alone. Via the Cartan–Dieudonné theorem, the reflective symmetries of the Platonic solids generate rotations. In a Clifford algebra framework, the space of spinors generating such three-dimensional rotations has a natural four-dimensional Euclidean structure. The spinors arising from the Platonic solids can thus in turn be interpreted as vertices in four-dimensional space, giving a simple construction of the four-dimensional polytopes 16-cell, 24-cell, the F4 root system and the 600-cell. In particular, these polytopes have `mysterious' symmetries, that are almost trivial when seen from the three-dimensional spinorial point of view. In fact, all these induced polytopes are also known to be root systems and thus generate rank-4 Coxeter groups, which can be shown to be a general property of the spinor construction. These considerations thus also apply to other root systems such as A_{1}\oplus I_{2}(n) which induces I_{2}(n)\oplus I_{2}(n), explaining the existence of the grand antiprism and the snub 24-cell, as well as their symmetries. These results are discussed in the wider mathematical context of Arnold's trinities and the McKay correspondence. These results are thus a novel link between the geometries of three and four dimensions, with interesting potential applications on both sides of the correspondence, to real three-dimensional systems with polyhedral symmetries such as (quasi)crystals and viruses, as well as four-dimensional geometries arising for instance in Grand Unified Theories and string and M-theory.

AB - This paper shows how regular convex 4-polytopes – the analogues of the Platonic solids in four dimensions – can be constructed from three-dimensional considerations concerning the Platonic solids alone. Via the Cartan–Dieudonné theorem, the reflective symmetries of the Platonic solids generate rotations. In a Clifford algebra framework, the space of spinors generating such three-dimensional rotations has a natural four-dimensional Euclidean structure. The spinors arising from the Platonic solids can thus in turn be interpreted as vertices in four-dimensional space, giving a simple construction of the four-dimensional polytopes 16-cell, 24-cell, the F4 root system and the 600-cell. In particular, these polytopes have `mysterious' symmetries, that are almost trivial when seen from the three-dimensional spinorial point of view. In fact, all these induced polytopes are also known to be root systems and thus generate rank-4 Coxeter groups, which can be shown to be a general property of the spinor construction. These considerations thus also apply to other root systems such as A_{1}\oplus I_{2}(n) which induces I_{2}(n)\oplus I_{2}(n), explaining the existence of the grand antiprism and the snub 24-cell, as well as their symmetries. These results are discussed in the wider mathematical context of Arnold's trinities and the McKay correspondence. These results are thus a novel link between the geometries of three and four dimensions, with interesting potential applications on both sides of the correspondence, to real three-dimensional systems with polyhedral symmetries such as (quasi)crystals and viruses, as well as four-dimensional geometries arising for instance in Grand Unified Theories and string and M-theory.

U2 - 10.1107/S0108767313021442

DO - 10.1107/S0108767313021442

M3 - Article

VL - 69

SP - 592

EP - 602

JO - Acta Crystallographica Section A : Foundations of Crystallography

JF - Acta Crystallographica Section A : Foundations of Crystallography

SN - 0108-7673

IS - 6

ER -