Hecke Groups, Dessins d'Enfants and the Archimedean Solids
Grothendieck's dessins d'enfants arise with ever-increasing frequency in many areas of 21st century mathematical physics. In this paper, we review the connections between dessins and the theory of Hecke groups. Focussing on the restricted class of highly symmetric dessins corresponding to...
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| Format: | Article |
| Language: | English |
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Frontiers Media S.A.
2015-12-01
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| Series: | Frontiers in Physics |
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| Online Access: | http://journal.frontiersin.org/Journal/10.3389/fphy.2015.00091/full |
| _version_ | 1818995929404932096 |
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| author | Yang-Hui eHe Yang-Hui eHe James eRead |
| author_facet | Yang-Hui eHe Yang-Hui eHe James eRead |
| author_sort | Yang-Hui eHe |
| collection | DOAJ |
| description | Grothendieck's dessins d'enfants arise with ever-increasing frequency in many areas of 21st century mathematical physics. In this paper, we review the connections between dessins and the theory of Hecke groups. Focussing on the restricted class of highly symmetric dessins corresponding to the so-called Archimedean solids, we apply this theory in order to provide a means of computing representatives of the associated conjugacy classes of Hecke subgroups in each case. The aim of this paper is to demonstrate that dessins arising in mathematical physics can point to new and hitherto unexpected directions for further research. In addition, given the particular ubiquity of many of the dessins corresponding to the Archimedean solids, the hope is that the computational results of this paper will prove useful in the further study of these objects in mathematical physics contexts. |
| first_indexed | 2024-12-20T21:21:39Z |
| format | Article |
| id | doaj.art-5845224afd0c4b8e8b1eaa988d9e78ab |
| institution | Directory Open Access Journal |
| issn | 2296-424X |
| language | English |
| last_indexed | 2024-12-20T21:21:39Z |
| publishDate | 2015-12-01 |
| publisher | Frontiers Media S.A. |
| record_format | Article |
| series | Frontiers in Physics |
| spelling | doaj.art-5845224afd0c4b8e8b1eaa988d9e78ab2022-12-21T19:26:15ZengFrontiers Media S.A.Frontiers in Physics2296-424X2015-12-01310.3389/fphy.2015.00091159335Hecke Groups, Dessins d'Enfants and the Archimedean SolidsYang-Hui eHe0Yang-Hui eHe1James eRead2City University, London; Nankai University, China and University of OxfordUniversity of OxfordUniversity of OxfordGrothendieck's dessins d'enfants arise with ever-increasing frequency in many areas of 21st century mathematical physics. In this paper, we review the connections between dessins and the theory of Hecke groups. Focussing on the restricted class of highly symmetric dessins corresponding to the so-called Archimedean solids, we apply this theory in order to provide a means of computing representatives of the associated conjugacy classes of Hecke subgroups in each case. The aim of this paper is to demonstrate that dessins arising in mathematical physics can point to new and hitherto unexpected directions for further research. In addition, given the particular ubiquity of many of the dessins corresponding to the Archimedean solids, the hope is that the computational results of this paper will prove useful in the further study of these objects in mathematical physics contexts.http://journal.frontiersin.org/Journal/10.3389/fphy.2015.00091/fullArchimedean solidsHecke groupsPlatonic solidsDessins d'enfantsBelyi maps |
| spellingShingle | Yang-Hui eHe Yang-Hui eHe James eRead Hecke Groups, Dessins d'Enfants and the Archimedean Solids Frontiers in Physics Archimedean solids Hecke groups Platonic solids Dessins d'enfants Belyi maps |
| title | Hecke Groups, Dessins d'Enfants and the Archimedean Solids |
| title_full | Hecke Groups, Dessins d'Enfants and the Archimedean Solids |
| title_fullStr | Hecke Groups, Dessins d'Enfants and the Archimedean Solids |
| title_full_unstemmed | Hecke Groups, Dessins d'Enfants and the Archimedean Solids |
| title_short | Hecke Groups, Dessins d'Enfants and the Archimedean Solids |
| title_sort | hecke groups dessins d 39 enfants and the archimedean solids |
| topic | Archimedean solids Hecke groups Platonic solids Dessins d'enfants Belyi maps |
| url | http://journal.frontiersin.org/Journal/10.3389/fphy.2015.00091/full |
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