Designs for fitting first-order response surfaces over spherical regions are often constructed from the vertices of regular geometric figures. The well-known method of generating a simplex design for any number of factors gives the factors unequal, asymmetric ranges. The unequal, asymmetric ranges of the coded design factors create a problem in applying the designs: the geometric and statistical properties of the design must be distorted to give the experimental factors the desired ranges. Symmetric orientations are given for the simplex designs for 4–6, 8–10, 12, and 13 factors. In the new orientations, the simplex designs have up to six levels of the factors.