2009/08/22 by Mehmet Koca, Nazife Ozdes Koca, Ramazan Koc +1
Materials Science · Mathematics · Physics and Astronomy · #Algebra over a field #Catalan #Dual polyhedron #Extension (predicate logic) #Finite Group Theory Research #Graph theory and applications #Quasicrystal Structures and Properties #Quaternion #Representation (politics) #Simple (philosophy) #math-ph #math.MP
paper · pdf · doi:10.1063/1.3356985
published as Journal of Mathematical Physics 51, 043501(2010) · 20 pages, 11 Figures
arxiv created 2009/08/22 · openalex publication_date 2010/04/01 · arxiv updated 2015/05/14 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
Catalan solids are the duals of the Archimedean solids, the vertices of which can be obtained from the Coxeter–Dynkin diagrams A3, B3, and H3 whose simple roots can be represented by quaternions. The respective Weyl groups W(A3), W(B3), and W(H3) acting on the highest weights generate the orbits corresponding to the solids possessing these symmetries. Vertices of the Platonic and Archimedean solids result from the orbits derived from fundamental weights. The Platonic solids are dual to each other; however, the duals of the Archimedean solids are the Catalan solids whose vertices can be written as the union of the orbits, up to some scale factors, obtained by applying the above Weyl groups on the fundamental highest weights (100), (010), and (011) for each diagram. The faces are represented by the orbits derived from the weights (010), (110), (101), (011), and (111), which correspond to the vertices of the Archimedean solids. Representations of the Weyl groups W(A3), W(B3), and W(H3) by the quaternions simplify the calculations with no reference to the computer calculations.