Dual snub 24-cell

In geometry, the dual snub 24-cell is a 144 vertex convex 4-polytope composed of 96 irregular cells. Each cell has faces of two kinds: 3 kites and 6 isosceles triangles. The polytope has a total of 432 faces (144 kites and 288 isosceles triangles) and 480 edges.

Geometry
The dual snub 24-cell, first described by Koca et al. in 2011, is the dual polytope of the snub 24-cell, a semiregular polytope first described by Thorold Gosset in 1900.

Construction
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. The following describe $$T$$ and $$T'$$ 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



With quaternions $$(p,q)$$ where $$\bar p$$ is the conjugate of $$p$$ and $$[p,q]:r\rightarrow r'=prq$$ and $$[p,q]^*:r\rightarrow r''=p\bar rq$$, then the Coxeter group $$W(H_4)=\lbrace[p,\bar p] \oplus [p,\bar p]^*\rbrace $$ is the symmetry group of the 600-cell and the 120-cell of order 14400.

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


 * the snub 24-cell $$S=\sum_{i=1}^4\oplus p^i T$$
 * the 600-cell $$I=T+S=\sum_{i=0}^4\oplus p^i T$$
 * the 120-cell $$J=\sum_{i,j=0}^4\oplus p^i\bar p^{\dagger j}T'$$
 * the alternate snub 24-cell $$S'=\sum_{i=1}^4\oplus p^i\bar p^{\dagger i}T'$$

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

Dual
The dual polytope of this polytope is the Snub 24-cell.