dual snub 24-cell

The dual snub 24-cell, first described by Koca et al. in 2011,{{Sfn|Koca|Al-Ajmi|Ozdes Koca|2011}} is the dual polytope of the snub 24-cell, a semiregular polytope first described by Thorold Gosset in 1900.{{Sfn|Gosset|1900}}

The vertices of a dual snub 24-cell are obtained using quaternion simple roots (T') in the generation of the 600 vertices of the 120-cell.{{Sfn|Koca|Al-Ajmi|Ozdes Koca|2011|loc=6. Dual of the snub 24-cell|pp=986-988}} The following describe $T$ and $T\text{'}$ 24-cells as quaternion orbit 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

600px

With quaternions $(p,q)$ where $\backslash bar\; p$ is the conjugate of $p$ and $[p,q]:r\backslash rightarrow\; r\text{'}=prq$ and $[p,q]^*:r\backslash rightarrow\; r\text{'}\text{'}=p\backslash bar\; rq$, then the Coxeter group $W(H\_4)=\backslash lbrace[p,\backslash bar\; p]\; \backslash oplus\; [p,\backslash bar\; p]^*\backslash rbrace$ is the symmetry group of the 600-cell and the 120-cell of order 14400.

Given $p\; \backslash in\; T$ such that $\backslash bar\; p=\backslash pm\; p^4,\; \backslash bar\; p^2=\backslash pm\; p^3,\; \backslash bar\; p^3=\backslash pm\; p^2,\; \backslash bar\; p^4=\backslash pm\; p$ and $p^\backslash dagger$ as an exchange of $-1/\backslash phi\; \backslash leftrightarrow\; \backslash phi$ within $p$ where $\backslash phi=\backslash frac\{1+\backslash sqrt\{5\}\}\{2\}$ is the golden ratio, we can construct:

• the snub 24-cell $S=\backslash sum\_\{i=1\}^4\backslash oplus\; p^i\; T$
• the 600-cell $I=T+S=\backslash sum\_\{i=0\}^4\backslash oplus\; p^i\; T$
• the 120-cell $J=\backslash sum\_\{i,j=0\}^4\backslash oplus\; p^i\backslash bar\; p^\{\backslash dagger\; j\}T\text{'}$
• the alternate snub 24-cell $S\text{'}=\backslash sum\_\{i=1\}^4\backslash oplus\; p^i\backslash bar\; p^\{\backslash dagger\; i\}T\text{'}$

and finally the dual snub 24-cell can then be defined as the orbits of $T\; \backslash oplus\; T\text{'}\; \backslash oplus\; S\text{'}$.

The dual polytope of this polytope is the Snub 24-cell.{{Sfn|Coxeter|1973|pp=151-153|loc=§8.4. The snub {3,4,3} }}

• Snub 24-cell honeycomb

{{Reflist}}

{{Refbegin}}

• {{Cite journal|first=Thorold|last=Gosset|author-link=Thorold Gosset|title=On the Regular and Semi-Regular Figures in Space of n Dimensions|journal=Messenger of Mathematics|publisher=Macmillan|year=1900}}
• {{Cite book | last=Coxeter | first=H.S.M. | author-link=Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=Regular Polytopes (book) }}
• {{Cite book|author-link=John Horton Conway|first1=John|last1=Conway|first2=Heidi|last2=Burgiel|first3=Chaim|last3=Goodman-Strauss|title=The Symmetries of Things|year=2008|isbn=978-1-56881-220-5}}
• {{Cite journal|url=http://arxiv-web3.library.cornell.edu/abs/1106.3433|title=Snub 24-Cell Derived from the Coxeter-Weyl Group W(D4)|first1=Mehmet|last1=Koca|first2=Nazife|last2=Ozdes Koca|first3=Muataz|last3=Al-Barwani|year=2012|journal=Int. J. Geom. Methods Mod. Phys.|volume=09|issue=8 |doi=10.1142/S0219887812500685 |arxiv=1106.3433 |s2cid=119288632 }}
• {{Cite journal|title=Quaternionic representation of snub 24-cell and its dual polytope derived from E8 root system|first1=Mehmet|last1=Koca|first2= Mudhahir|last2=Al-Ajmi|first3=Nazife|last3=Ozdes Koca|journal=Linear Algebra and Its Applications|volume=434|issue=4|year=2011|pages=977–989|doi=10.1016/j.laa.2010.10.005 |s2cid=18278359 |issn=0024-3795|doi-access=free|arxiv=0906.2109}}

{{Refend}}

{{Polytopes}}

{{DEFAULTSORT:Dual snub 24-Cell}}

Category:4-polytopes