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. The polytope has a total of 432 faces (144 kites and 288 isosceles triangles) and 480 edges.
Geometry
The dual snub 24-cell, first described by Koca et al. in 2011, is the dual polytope of the snub 24-cell, a semiregular polytope first described by Thorold Gosset in 1900.
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. The following describe and 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 where is the conjugate of and and , then the Coxeter group is the symmetry group of the 600-cell and the 120-cell of order 14400.
Given such that and as an exchange of within where is the golden ratio, we can construct:
the snub 24-cell
the 600-cell
the 120-cell
the alternate snub 24-cell
and finally the dual snub 24-cell can then be defined as the orbits of .
Projections
Dual
The dual polytope of this polytope is the Snub 24-cell.