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.Шаблон:Sfn The polytope has a total of 432 faces (144 kites and 288 isosceles triangles) and 480 edges.
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.Шаблон:Sfn The following describe <math>T</math> and <math>T'</math> 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 <math>(p,q)</math> where <math>\bar p</math> is the conjugate of <math>p</math> and <math>[p,q]:r\rightarrow r'=prq</math> and <math>[p,q]^*:r\rightarrow r=p\bar rq</math>, then the Coxeter group <math>W(H_4)=\lbrace[p,\bar p] \oplus [p,\bar p]^*\rbrace </math> is the symmetry group of the 600-cell and the 120-cell of order 14400.
Given <math>p \in T</math> such that <math>\bar p=\pm p^4, \bar p^2=\pm p^3, \bar p^3=\pm p^2, \bar p^4=\pm p</math> and <math>p^\dagger</math> as an exchange of <math>-1/\phi \leftrightarrow \phi</math> within <math>p</math> where <math>\phi=\frac{1+\sqrt{5}}{2}</math> is the golden ratio, we can construct:
the snub 24-cell <math>S=\sum_{i=1}^4\oplus p^i T</math>
the 600-cell <math>I=T+S=\sum_{i=0}^4\oplus p^i T</math>
the 120-cell <math>J=\sum_{i,j=0}^4\oplus p^i\bar p^{\dagger j}T'</math>
the alternate snub 24-cell <math>S'=\sum_{i=1}^4\oplus p^i\bar p^{\dagger i}T'</math>
and finally the dual snub 24-cell can then be defined as the orbits of <math>T \oplus T' \oplus S'</math>.
Projections
3D Orthogonal projections
Файл:DualSnub24Cell-2.png3D 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.
Файл:Dual snub 24-cell overlay with the convex hull of the 120-cell.svg3D overlay of the dual snub 24-cell with the orthogonal projection of the 120-cell which forms an outer hull of a unit circumradiuschamfered 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 <math>\pm</math>1 in the 4th dimension get projected to the origin in 3D. The 16 other vertices are the 8-cellTesseract which project to norm <math>\tfrac{\sqrt{3}}{2}=.866</math> 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.
