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>.