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The five Platonic solids and their connection to root systems
Instantanés de recherche mathématique à Oberwolfach
The five Platonic solids and their connection to root systems
Platonic solids have fascinated humans for thousands of years. In ancient times, they were associated with the elements fire, air, water, earth, and aether. These solids are completely symmetrical three-dimensional polyhedra. In this snapshot, it is first explained that there can only be five such polyhedra in the threedimensional space. For this purpose, so-called Schläfli symbols and Coxeter graphs are introduced. More precisely, the (linear) Coxeter graphs correspond to the (linear) Schläfli symbols that, in turn, correspond exactly to the regular convex polyhedra. Through this one-to-one relationship, it is possible to classify the regular convex polytopes in any dimension by exploiting the classification of Coxeter graphs.
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Sujet mathématique
Algèbre et théorie des nombres
Géométrie et Topologie
Liens avec d'autres domaines
Informatique
Licence
DOI
10.14760/SNAP-2025-004-EN
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snapshots: overview
Sujet mathématique
Algèbre et théorie des nombres
Analyse
Pédagogie et éducation
Mathématiques discrètes et fondements des mathématiques
Géométrie et Topologie
Calcul numérique et calcul scientifique
Théorie des probabilités et statistique
Liens avec d'autres domaines
Chimie et sciences de la terre
Informatique
Ingénierie et technologie
Finances
Humanités et sciences sociales
Sciences de la vie
Physique
Pensées mathématiques