regular n-simplex in Rn which can then be translated, rotated, and scaled as desired. One way to do this is as follows. Denote the basis vectors of Rn by e1 through en. Begin with the standard -simplex which is the convex hull of the basis vectors. By adding an additional vertex, these become a face of a regular -simplex. The additional vertex must lie on the line perpendicular to the barycenter of the standard simplex, so it has the form for some real number α. Since the squared distance between two basis vectors is 2, in order for the additional vertex to form a regular n-simplex, the squared distance between it and any of the basis vectors must also be 2. This yields a quadratic equation for α. Solving this equation shows that there are two choices for the additional vertex: Either of these, together with the standard basis vectors, yields a regular n-simplex. The above regular n-simplex is not centered on the origin. It can be translated to the origin by subtracting the mean of its vertices. By rescaling, it can be given unit side length. This results in the simplex whose vertices are: for , and Note that there are two sets of vertices described here. One set uses in each calculation. The other set uses in each calculation. This simplex is inscribed in a hypersphere of radius . A different rescaling produces a simplex that is inscribed in a unit hypersphere. When this is done, its vertices are where , and The side length of this simplex is . A highly symmetric way to construct a regular -simplex is to use a representation of the cyclic group by orthogonal matrices. This is an orthogonal matrix such that is the identity matrix, but no lower power of is. Applying powers of this matrix to an appropriate vector will produce the vertices of a regular -simplex. To carry this out, first observe that for any orthogonal matrix , there is a choice of basis in which is a block diagonal matrix where each is orthogonal and either or . In order for to have order , all of these matrices must have order dividing . Therefore each is either a matrix whose only entry is or, if is odd, ; or it is a matrix of the form where each is an integer between zero and inclusive. A sufficient