Rotatable design

A design for a specified polynomial model is rotatable when the variance of a fitted regression mean depends on a setting only through its distance from the centre. For a full $2^d$ factorial portion plus symmetric axial points in a central composite design, fourth-order rotational symmetry requires $\sum x_i^4=3\sum x_i^2{x}_j^2$. These sums are $2^d+2{\alpha }^4$ and $2^d$, giving $\alpha =(2^d{)}^{1/4}$. Sign symmetry removes odd moments and equal second moments complete the required moment symmetry for a full quadratic model.