by two chiral five-click rotations of any one tetrahedron. In each dodecahedral cell, one tetrahedral cell comes from each of the ten 600-cells inscribed in the 120-cell. Therefore the whole 120-cell, with all ten inscribed 600-cells, can be generated from just one 600-cell by rotating its cells. Augmentation Another consequence of the 120-cell containing inscribed 600-cells is that it is possible to construct it by placing 4-pyramids of some kind on the cells of the 600-cell. These tetrahedral pyramids must be quite irregular in this case (with the apex blunted into several 'apexes'), but we can discern their shape in the way a tetrahedron lies inscribed in a dodecahedron. Only 120 tetrahedral cells of each 600-cell can be inscribed in the 120-cell's dodecahedra; its other 480 tetrahedra span dodecahedral cells. Each dodecahedron-inscribed tetrahedron is the center cell of a cluster of five tetrahedra, with the four others face-bonded around it lying only partially within the dodecahedron. The central tetrahedron is edge-bonded to an additional 12 tetrahedral cells, also lying only partially within the dodecahedron. The central cell is vertex-bonded to 40 other tetrahedral cells which lie entirely outside the dodecahedron. 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 the 600-cell the 120-cell the alternate snub 24-cell the dual snub 24-cell = . As a configuration This configuration matrix represents the 120-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 120-cell. The nondiagonal numbers say how many of the column's element occur in or at the row's element. Here is the configuration expanded with k-face elements and k-figures. The diagonal element counts are the ratio of the full Coxeter group order, 14400, divided by the order of the subgroup with mirror removal. Visualization The 120-cell consists of 120 dodecahedral cells. For visualization purposes, it is convenient that the dodecahedron has opposing parallel