it with 24 icosahedral pyramids. Weyl orbits Another construction method uses quaternions and the Icosahedral symmetry of Weyl group orbits of order 120. 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 , we can construct the snub 24-cell as Structure The icosahedral cells fit together face-to-face leaving voids between them filled by clusters of five tetrahedral cells. Each icosahedral cell is joined to 8 other icosahedral cells at 8 triangular faces in the positions corresponding to an inscribing octahedron. The remaining triangular faces are joined to tetrahedral cells, which occur in pairs that share an edge on the icosahedral cell. The tetrahedral cells may be divided into two groups, of 96 yellow cells and 24 red cells respectively (as colored in the net illustration). Each yellow tetrahedral cell is joined via its triangular faces to 3 blue icosahedral cells and one red tetrahedral cell, while each red tetrahedral cell is joined to 4 yellow tetrahedra. Thus, the tetrahedral cells occur in clusters of five (four yellow cells face-bonded around a red central one, each red/yellow pair lying in a different hyperplane). The red central tetrahedron of the five shares each of its six edges with a different icosahedral cell, and with the pair of yellow tetrahedral cells which shares that edge on the icosahedral cell. Symmetry The snub 24-cell has three vertex-transitive colorings based on a Wythoff construction on a Coxeter group from which it is alternated: F4 defines 24 interchangeable icosahedra, while the B4 group defines two groups of icosahedra in an 8:16 counts, and finally the D4 group has 3 groups of icosahedra with 8:8:8 counts. Projections Orthographic projections Perspective projections Dual The Dual snub 24-cell has 144 identical 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. Related polytopes The snub 24-cell can be obtained as a diminishing of the 600-cell at 24 of its vertices, in fact those of a vertex inscribed 24-cell. There is