Dual snub 24-cell

Template:Infobox 4-polytope 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.

Geometry

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. Its dual is a semiregular, first described by (Koca Al-Ajmi).

The vertices of a dual snub 24-cell are obtained using quaternion simple roots $T^{\prime }$ in the generation of the 600 vertices of the 120-cell. The following describe $T$ and $T^{\prime }$ 24-cells as quaternion orbit weights of $D_4$ under the Weyl group $W(D_4)$:$$\begin{array}{rl}O(0100)& :T=\{\pm 1,\pm e_1,\pm e_2,\pm e_3,\frac{\pm 1\pm e_1\pm e_2\pm e_3}{2}\}\\ O(1000)& :V_1\\ O(0010)& :V_2\\ O(0001)& :V_3\\ T^{\prime }& =\sqrt{2}(V_1\oplus V_2\oplus V_3)=\left[\begin{array}{cccccccc}\frac{-1-e_1}{\sqrt{2}}& \frac{1-e_1}{\sqrt{2}}& \frac{-1+e_1}{\sqrt{2}}& \frac{1+e_1}{\sqrt{2}}& \frac{-e_2-e_3}{\sqrt{2}}& \frac{e_2-e_3}{\sqrt{2}}& \frac{-e_2+e_3}{\sqrt{2}}& \frac{e_2+e_3}{\sqrt{2}}\\ \frac{-1-e_2}{\sqrt{2}}& \frac{1-e_2}{\sqrt{2}}& \frac{-1+e_2}{\sqrt{2}}& \frac{1+e_2}{\sqrt{2}}& \frac{-e_1-e_3}{\sqrt{2}}& \frac{e_1-e_3}{\sqrt{2}}& \frac{-e_1+e_3}{\sqrt{2}}& \frac{e_1+e_3}{\sqrt{2}}\\ \frac{-e_1-e_2}{\sqrt{2}}& \frac{e_1-e_2}{\sqrt{2}}& \frac{-e_1+e_2}{\sqrt{2}}& \frac{e_2+e_3}{\sqrt{2}}& \frac{-1-e_3}{\sqrt{2}}& \frac{1-e_3}{\sqrt{2}}& \frac{-1+e_3}{\sqrt{2}}& \frac{1+e_3}{\sqrt{2}}\end{array}\right].\end{array}$$

With quaternions $(p,q)$ where $\overline{p}$ is the conjugate of $p$ and $[p,q]:r\to r^{\prime }=prq$ and $[p,q{]}^{*}:r\to r^{″}=p\overline{r}q$, then the Coxeter group $W(H_4)=\{[p,\overline{p}]\oplus [p,\overline{p}{]}^{*}\}$ is the symmetry group of the 600-cell and the 120-cell of order 14400.

Given $p\in T$ such that $\overline{p}=\pm p^4$, ${\overline{p}}^2=\pm p^3$, ${\overline{p}}^3=\pm p^2$, ${\overline{p}}^4=\pm p$ and $p^{†}$ as an exchange of $-1/\varphi \leftrightarrow \varphi$ within $p$, where $\varphi =\frac{1+\sqrt{5}}{2}$ is the golden ratio, one can construct the snub 24-cell $S$, 600-cell $I$, 120-cell $J$, and alternate snub 24-cell $S^{\prime }$ in the following, respectively:$$\begin{array}{rl}S=\sum _{i=1}^4\oplus p^{i}T,& I=T+S=\sum _{i=0}^4\oplus p^{i}T,\\ J=\sum _{i,j=0}^4\oplus p^i{\overline{p}}^{†j}{T}^{\prime },& S^{\prime }=\sum _{i=1}^4\oplus p^i{\overline{p}}^{†i}{T}^{\prime }.\end{array}$$This finally can define the dual snub 24-cell as the orbits of $T\oplus T^{\prime }\oplus S^{\prime }$.

Cell

The cell of dual snub 24-cell

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

See also

• Snub 24-cell honeycomb

References

• Gosset, Thorold (1900). "On the Regular and Semi-Regular Figures in Space of n Dimensions". Messenger of Mathematics (Macmillan). 
• Coxeter, H.S.M. (1973). Regular Polytopes (3rd ed.). New York: Dover. 
• Conway, John; Burgiel, Heidi; Goodman-Strauss, Chaim (2008). The Symmetries of Things. ISBN 978-1-56881-220-5. 
• Koca, Mehmet; Al-Ajmi, Mudhahir; Ozdes Koca, Nazife (2011). "Quaternionic representation of snub 24-cell and its dual polytope derived from $E_8$ root system". Linear Algebra and Its Applications 434 (4): 977–989. doi:10.1016/j.laa.2010.10.005. ISSN 0024-3795. https://www.sciencedirect.com/science/article/pii/S0024379510005173. 
• Koca, Mehmet; Ozdes Koca, Nazife; Al-Barwani, Muataz (2012). "Snub 24-Cell Derived from the Coxeter-Weyl Group $W(D_4)$". Int. J. Geom. Methods Mod. Phys. 09 (8). doi:10.1142/S0219887812500685. https://arxiv-web3.library.cornell.edu/abs/1106.3433.