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 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. Its dual is a semiregular, first described by.
The vertices of a dual snub 24-cell are obtained using quaternion simple roots in the generation of the 600 vertices of the 120-cell. The following describe and 24-cells as quaternion orbit weights of under the Weyl group :
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, one can construct the snub 24-cell, 600-cell, 120-cell, and alternate snub 24-cell in the following, respectively:This finally can define the dual snub 24-cell as the orbits of.

Cell

The dual snub 24-cell has 96 identical cells. The cell can be constructed by multiplying to the eight Cartesian coordinates:
where and. These vertices form six isosceles triangles and three kites, where the legs and the base of an isosceles triangle are and, and the two pairs of adjacent equal-length sides of a kite are and.