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Platonic solids generate their four-dimensional analogues
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| Zeitschriftentitel: | Acta Crystallographica Section A Foundations of Crystallography |
|---|---|
| Personen und Körperschaften: | |
| In: | Acta Crystallographica Section A Foundations of Crystallography, 69, 2013, 6, S. 592-602 |
| Format: | E-Article |
| Sprache: | Unbestimmt |
| veröffentlicht: |
International Union of Crystallography (IUCr)
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| Schlagwörter: |
| author_facet |
Dechant, Pierre-Philippe Dechant, Pierre-Philippe |
|---|---|
| author |
Dechant, Pierre-Philippe |
| spellingShingle |
Dechant, Pierre-Philippe Acta Crystallographica Section A Foundations of Crystallography Platonic solids generate their four-dimensional analogues Structural Biology |
| author_sort |
dechant, pierre-philippe |
| spelling |
Dechant, Pierre-Philippe 0108-7673 International Union of Crystallography (IUCr) Structural Biology http://dx.doi.org/10.1107/s0108767313021442 <jats:p>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.<jats:italic>Via</jats:italic>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<jats:italic>F</jats:italic><jats:sub>4</jats:sub>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.</jats:p> Platonic solids generate their four-dimensional analogues Acta Crystallographica Section A Foundations of Crystallography |
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2013 |
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International Union of Crystallography (IUCr) |
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Acta Crystallographica Section A Foundations of Crystallography |
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| title |
Platonic solids generate their four-dimensional analogues |
| title_unstemmed |
Platonic solids generate their four-dimensional analogues |
| title_full |
Platonic solids generate their four-dimensional analogues |
| title_fullStr |
Platonic solids generate their four-dimensional analogues |
| title_full_unstemmed |
Platonic solids generate their four-dimensional analogues |
| title_short |
Platonic solids generate their four-dimensional analogues |
| title_sort |
platonic solids generate their four-dimensional analogues |
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Structural Biology |
| url |
http://dx.doi.org/10.1107/s0108767313021442 |
| publishDate |
2013 |
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592-602 |
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<jats:p>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.<jats:italic>Via</jats:italic>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<jats:italic>F</jats:italic><jats:sub>4</jats:sub>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.</jats:p> |
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| description | <jats:p>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.<jats:italic>Via</jats:italic>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<jats:italic>F</jats:italic><jats:sub>4</jats:sub>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.</jats:p> |
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| spelling | Dechant, Pierre-Philippe 0108-7673 International Union of Crystallography (IUCr) Structural Biology http://dx.doi.org/10.1107/s0108767313021442 <jats:p>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.<jats:italic>Via</jats:italic>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<jats:italic>F</jats:italic><jats:sub>4</jats:sub>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.</jats:p> Platonic solids generate their four-dimensional analogues Acta Crystallographica Section A Foundations of Crystallography |
| spellingShingle | Dechant, Pierre-Philippe, Acta Crystallographica Section A Foundations of Crystallography, Platonic solids generate their four-dimensional analogues, Structural Biology |
| title | Platonic solids generate their four-dimensional analogues |
| title_full | Platonic solids generate their four-dimensional analogues |
| title_fullStr | Platonic solids generate their four-dimensional analogues |
| title_full_unstemmed | Platonic solids generate their four-dimensional analogues |
| title_short | Platonic solids generate their four-dimensional analogues |
| title_sort | platonic solids generate their four-dimensional analogues |
| title_unstemmed | Platonic solids generate their four-dimensional analogues |
| topic | Structural Biology |
| url | http://dx.doi.org/10.1107/s0108767313021442 |