Dual snub 24-cell

Dual snub 24-cell
Orthogonal projection of dual snub 24-cell
Type	4-polytope
Cells	96, each with three kites and nine isosceles triangles
Faces	432
Edges	480
Vertices	144
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.

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.^[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 ${\displaystyle T\oplus T'\oplus S'}$.

Cell

The cell of dual snub 24-cell

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]

See also

• Snub 24-cell honeycomb

Citations

1. Gosset 1900.
2. Coxeter 1973, pp. 151–153, §8.4. The snub {3,4,3}.
3. Koca, Al-Ajmi & Ozdes Koca 2011.
4. Koca, Al-Ajmi & Ozdes Koca 2011, pp. 986–988, 6. Dual of the snub 24-cell.
5. Koca, Al-Ajmi & Ozdes Koca 2011, p. 986–987.

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) [1948]. 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 ${\displaystyle E_{8}}$ root system". Linear Algebra and Its Applications. 434 (4): 977–989. arXiv:0906.2109. doi:10.1016/j.laa.2010.10.005. ISSN 0024-3795. S2CID 18278359.
• Koca, Mehmet; Ozdes Koca, Nazife; Al-Barwani, Muataz (2012). "Snub 24-Cell Derived from the Coxeter-Weyl Group ${\displaystyle W(D_{4})}$". Int. J. Geom. Methods Mod. Phys. 09 (8). arXiv:1106.3433. doi:10.1142/S0219887812500685. S2CID 119288632.

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	An	Bn	I2(p) / Dn	E6 / E7 / E8 / F4 / G2	Hn
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	122 • 221
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	132 • 231 • 321
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	142 • 241 • 421
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1k2 • 2k1 • k21	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds • Polytope operations