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.

Dual snub 24-cell
Orthogonal projection
Type	4-polytope
Cells	96
Faces	432	144 kites 288 Isosceles triangle
Edges	480
Vertices	144
Dual	Snub 24-cell
Properties	convex

Geometry

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]

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

Projections

3D Orthogonal projections

3D Visualization of the hull of the dual snub 24-cell, with vertices colored by overlap count: The (42) yellow have no overlaps. The (51) orange have 2 overlaps. The (18) sets of tetrahedral surfaces are uniquely colored.

3D overlay of the dual snub 24-cell with the orthogonal projection of the 120-cell which forms an outer hull of a unit circumradius chamfered dodecahedron. Of the 600 vertices in the 120-cell (J), 120 of the dual snub 24-cell (T'+S') are a subset of J and 24 (the T 24-cell) are not. Some of those 24 can be seen projecting outside the convex 3D hull of the 120-cell. As itemized in the hull data of this diagram, the 8 16-cell vertices of T have 6 with unit norm and can be seen projecting outside the center of 6 hexagon faces, while 2 with a ${\displaystyle \pm }$1 in the 4th dimension get projected to the origin in 3D. The 16 other vertices are the 8-cell Tesseract which project to norm ${\displaystyle {\tfrac {\sqrt {3}}{2}}=.866}$ inside the 120-cell 3D hull. Please note: the face and cell count data, along with the area and volume, within this image are from Mathematica automated tetrahedral cell analysis and are not based on the 96 kite cells of the dual snub 24-cell.

2D Orthogonal projections

2D projection of the dual snub 24-cell with color coded vertex overlaps 2D Projections to selected Coxeter Planes

Dual

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

See also

• Snub 24-cell honeycomb