Catalan solids derived from three-dimensional-root systems and quaternions

Mehmet Koca, Nazife Ozdes Koca, Ramazan Koç

Research output: Contribution to journalArticle

4 Citations (Scopus)

Abstract

Catalan solids are the duals of the Archimedean solids, the vertices of which can be obtained from the Coxeter-Dynkin diagrams A 3, B 3, and H 3 whose simple roots can be represented by quaternions. The respective Weyl groups W(A 3), W(B 3), and W(H 3) 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(A 3), W(B 3), and W(H 3) by the quaternions simplify the calculations with no reference to the computer calculations.

Original languageEnglish
Article number051003JMP
JournalJournal of Mathematical Physics
Volume51
Issue number4
DOIs
Publication statusPublished - Apr 2010

Fingerprint

Catalan Solid
Archimedean solid
quaternions
Root System
Quaternion
Weyl Group
Orbit
Platonic solid
Three-dimensional
apexes
orbits
Dynkin Diagram
Scale factor
Simplify
Union
Diagram
diagrams
Roots
Face
Symmetry

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Catalan solids derived from three-dimensional-root systems and quaternions. / Koca, Mehmet; Ozdes Koca, Nazife; Koç, Ramazan.

In: Journal of Mathematical Physics, Vol. 51, No. 4, 051003JMP, 04.2010.

Research output: Contribution to journalArticle

@article{0e913ca4992740f589332ccd02b115af,
title = "Catalan solids derived from three-dimensional-root systems and quaternions",
abstract = "Catalan solids are the duals of the Archimedean solids, the vertices of which can be obtained from the Coxeter-Dynkin diagrams A 3, B 3, and H 3 whose simple roots can be represented by quaternions. The respective Weyl groups W(A 3), W(B 3), and W(H 3) 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(A 3), W(B 3), and W(H 3) by the quaternions simplify the calculations with no reference to the computer calculations.",
author = "Mehmet Koca and {Ozdes Koca}, Nazife and Ramazan Ko{\cc}",
year = "2010",
month = "4",
doi = "10.1063/1.3356985",
language = "English",
volume = "51",
journal = "Journal of Mathematical Physics",
issn = "0022-2488",
publisher = "American Institute of Physics Publising LLC",
number = "4",

}

TY - JOUR

T1 - Catalan solids derived from three-dimensional-root systems and quaternions

AU - Koca, Mehmet

AU - Ozdes Koca, Nazife

AU - Koç, Ramazan

PY - 2010/4

Y1 - 2010/4

N2 - Catalan solids are the duals of the Archimedean solids, the vertices of which can be obtained from the Coxeter-Dynkin diagrams A 3, B 3, and H 3 whose simple roots can be represented by quaternions. The respective Weyl groups W(A 3), W(B 3), and W(H 3) 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(A 3), W(B 3), and W(H 3) by the quaternions simplify the calculations with no reference to the computer calculations.

AB - Catalan solids are the duals of the Archimedean solids, the vertices of which can be obtained from the Coxeter-Dynkin diagrams A 3, B 3, and H 3 whose simple roots can be represented by quaternions. The respective Weyl groups W(A 3), W(B 3), and W(H 3) 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(A 3), W(B 3), and W(H 3) by the quaternions simplify the calculations with no reference to the computer calculations.

UR - http://www.scopus.com/inward/record.url?scp=77953266804&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77953266804&partnerID=8YFLogxK

U2 - 10.1063/1.3356985

DO - 10.1063/1.3356985

M3 - Article

AN - SCOPUS:77953266804

VL - 51

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 4

M1 - 051003JMP

ER -