can be triangulated into eight triangular regions. This triangulation has the same symmetry group as the octahedron. The triangulation basically consists of connecting the six points , , with great circles in the x-y, y-z and z-x planes. Cutting the sphere along these lines, and then laying out the eight resulting triangles in a plane can be thought of as another visualization technique for two-dimensional beings.
Similarly, can be triangulated into sixteen tetrahedral regions by connecting the eight points , , , with great spheres. This triangulation has the same symmetry as the sixteen-cell polytope. Like the triangulation of , this can be used as another method of visualization of , and it will be, later.
Notice that there are a great many more possibilities for how these sixteen tetrahedra can be arranged than there are for the triangulation of . Might there be a particularly orderly arrangement for them, that had no analogue in ?
Since an analogue of the symmetry group of the octahedron and the sixteen-cell exists in every dimension, we can generalize this process of triangulation to , decomposing it into n-cells.