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600-cell
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==== Weyl orbits ==== Another construction method uses [[#Symmetries|quaternions]] and the [[Icosahedral symmetry]] of [[Weyl group]] orbits <math>O(\Lambda)=W(H_4)=I</math> of order 120.{{Sfn|Koca|Al-Ajmi|Ozdes Koca|2011|loc=6. Dual of the snub 24-cell|pp=986-988}} The following are the orbits of weights of D4 under the Weyl group W(D4): : O(0100) : T = {±1,±e1,±e2,±e3,(±1±e1±e2±e3)/2} : O(1000) : V1 : O(0010) : V2 : O(0001) : V3 [[File:120Cell-SimpleRoots-Quaternion-Tp.png|600px]] With quaternions <math>(p,q)</math> where <math>\bar p</math> is the conjugate of <math>p</math> and <math>[p,q]:r\rightarrow r'=prq</math> and <math>[p,q]^*:r\rightarrow r''=p\bar rq</math>, then the [[Coxeter group]] <math>W(H_4)=\lbrace[p,\bar p] \oplus [p,\bar p]^*\rbrace </math> is the symmetry group of the 600-cell and the [[120-cell]] of order 14400. Given <math>p \in T</math> such that <math>\bar p=\pm p^4, \bar p^2=\pm p^3, \bar p^3=\pm p^2, \bar p^4=\pm p</math> and <math>p^\dagger</math> as an exchange of <math>-1/\varphi \leftrightarrow \varphi</math> within <math>p</math>, we can construct: * the [[snub 24-cell]] <math>S=\sum_{i=1}^4\oplus p^i T</math> * the 600-cell <math>I=T+S=\sum_{i=0}^4\oplus p^i T</math> * the [[120-cell]] <math>J=\sum_{i,j=0}^4\oplus p^i\bar p^{\dagger j}T'</math>
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