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: three kites and six isosceles triangles. The polytope has a total of 432 faces (144 kites and 288 isosceles triangles) and 480 edges.

The snub 24-cell is a convex uniform 4-polytope that consists of 120 regular tetrahedra and 96 icosahedra as its cell, firstly described by Thorold Gosset in 1900.^[1] Its dual is a semiregular,^[2] first described by Koca, Al-Ajmi & Ozdes Koca (2011).^[3]

The vertices of a dual snub 24-cell are obtained using quaternion simple roots ${\displaystyle T'}$ in the generation of the 600 vertices of the 120-cell. The following describe ${\displaystyle T}$ and ${\displaystyle T'}$ 24-cells as quaternion orbit weights of ${\displaystyle D_{4}}$ under the Weyl group ${\displaystyle W(D_{4})}$:^[4]$${\displaystyle {\begin{aligned}O(0100)&:T=\left\{\pm 1,\pm e_{1},\pm e_{2},\pm e_{3},{\frac {\pm 1\pm e_{1}\pm e_{2}\pm e_{3}}{2}}\right\}\\O(1000)&:V_{1}\\O(0010)&:V_{2}\\O(0001)&:V_{3}\\T'&={\sqrt {2}}(V_{1}\oplus V_{2}\oplus V_{3})={\begin{bmatrix}{\frac {-1-e_{1}}{\sqrt {2}}}&{\frac {1-e_{1}}{\sqrt {2}}}&{\frac {-1+e_{1}}{\sqrt {2}}}&{\frac {1+e_{1}}{\sqrt {2}}}&{\frac {-e_{2}-e_{3}}{\sqrt {2}}}&{\frac {e_{2}-e_{3}}{\sqrt {2}}}&{\frac {-e_{2}+e_{3}}{\sqrt {2}}}&{\frac {e_{2}+e_{3}}{\sqrt {2}}}\\{\frac {-1-e_{2}}{\sqrt {2}}}&{\frac {1-e_{2}}{\sqrt {2}}}&{\frac {-1+e_{2}}{\sqrt {2}}}&{\frac {1+e_{2}}{\sqrt {2}}}&{\frac {-e_{1}-e_{3}}{\sqrt {2}}}&{\frac {e_{1}-e_{3}}{\sqrt {2}}}&{\frac {-e_{1}+e_{3}}{\sqrt {2}}}&{\frac {e_{1}+e_{3}}{\sqrt {2}}}\\{\frac {-e_{1}-e_{2}}{\sqrt {2}}}&{\frac {e_{1}-e_{2}}{\sqrt {2}}}&{\frac {-e_{1}+e_{2}}{\sqrt {2}}}&{\frac {e_{2}+e_{3}}{\sqrt {2}}}&{\frac {-1-e_{3}}{\sqrt {2}}}&{\frac {1-e_{3}}{\sqrt {2}}}&{\frac {-1+e_{3}}{\sqrt {2}}}&{\frac {1+e_{3}}{\sqrt {2}}}\end{bmatrix}}.\end{aligned}}}$$

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 ${\displaystyle {\bar {p}}=\pm p^{4}}$, ${\displaystyle {\bar {p}}^{2}=\pm p^{3}}$, ${\displaystyle {\bar {p}}^{3}=\pm p^{2}}$, ${\displaystyle {\bar {p}}^{4}=\pm p}$ and ${\displaystyle p^{\dagger }}$ as an exchange of ${\displaystyle -1/\phi \leftrightarrow \phi }$ within ${\displaystyle p}$, where ${\textstyle \phi ={\frac {1+{\sqrt {5}}}{2}}}$ is the golden ratio, one can construct the snub 24-cell ${\displaystyle S}$, 600-cell ${\displaystyle I}$, 120-cell ${\displaystyle J}$, and alternate snub 24-cell ${\displaystyle S'}$ in the following, respectively:$${\displaystyle {\begin{aligned}S=\sum _{i=1}^{4}\oplus p^{i}T,&\qquad I=T+S=\sum _{i=0}^{4}\oplus p^{i}T,\\J=\sum _{i,j=0}^{4}\oplus p^{i}{\bar {p}}^{\dagger j}T',&\qquad S'=\sum _{i=1}^{4}\oplus p^{i}{\bar {p}}^{\dagger i}T'.\end{aligned}}}$$This finally can define the dual snub 24-cell as the orbits of ${\displaystyle T\oplus T'\oplus S'}$.

The dual snub 24-cell has 96 identical cells. The cell can be constructed by multiplying ${\textstyle {\frac {1}{2{\sqrt {2}}}}}$ to the eight Cartesian coordinates: $${\displaystyle {\begin{matrix}(-\phi ,0,1),&\qquad (0,-1,-\phi ),&\qquad (1,\phi ,0),\\(-\varphi ,\varphi ,-\varphi ),&\qquad (\varphi ,-\varphi ,\varphi ),&\qquad (\varphi ^{2},0,1),\\(1,-\varphi ^{2},0),&\qquad (0,-1,\varphi ^{2}),\end{matrix}}}$$ where ${\textstyle \phi ={\frac {1+{\sqrt {5}}}{2}}}$ and ${\textstyle \varphi ={\frac {1-{\sqrt {5}}}{2}}}$. These vertices form six isosceles triangles and three kites, where the legs and the base of an isosceles triangle are ${\textstyle {\frac {1}{\sqrt {2}}}}$ and ${\textstyle {\frac {\phi }{\sqrt {2}}}}$, and the two pairs of adjacent equal-length sides of a kite are ${\textstyle {\frac {1}{\sqrt {2}}}}$ and ${\textstyle {\frac {\varphi ^{2}}{\sqrt {2}}}}$.^[5]

• Snub 24-cell honeycomb