= 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 $\sum x_i^4=3\sum x_i^2x_j^2$. 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.
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