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

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

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

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

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

@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",

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publisher = "International Union of Crystallography",

number = "6",

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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.

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JO - Acta Crystallographica Section A : Foundations of Crystallography

JF - Acta Crystallographica Section A : Foundations of Crystallography

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