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.^[1] The polytope has a total of 432 faces (144 kites and 288 isosceles triangles) and 480 edges.

The dual snub 24-cell, first described by Koca et al. in 2011,^[2] is the dual polytope of the snub 24-cell, a semiregular polytope first described by Thorold Gosset in 1900.^[3]

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.^[4] The following describe ${\displaystyle T}$ and ${\displaystyle T'}$ 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 ${\displaystyle (p,q)}$ where ${\displaystyle {\bar {p}}}$ is the conjugate of ${\displaystyle p}$ and ${\displaystyle [p,q]:r\rightarrow r'=prq}$ and ${\displaystyle [p,q]^{*}:r\rightarrow r''=p{\bar {r}}q}$, then the Coxeter group ${\displaystyle W(H_{4})=\lbrace [p,{\bar {p}}]\oplus [p,{\bar {p}}]^{*}\rbrace }$ is the symmetry group of the 600-cell and the 120-cell of order 14400.

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

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

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

The dual polytope of this polytope is the Snub 24-cell.^[5]

• Snub 24-cell honeycomb