Second Order Rotatable Designs in Three Dimensions

0. Summary. The technique of fitting a response surface is one widely used (especially in the chemical industry) to aid in the statistical analysis of experimental work in which the "yield" of a product depends, in some unknown fashion, on one or more controllable variables. Before the details of such an analysis can be carried out, experiments must be performed at predetermined levels of the conitrollable factors, i.e., an experimental design must be selected prior to experimentation. Box and Hunter [3] suggested designs of a certain type, which they called rotatable, as being suitable for such experimentation. Very few of these designs were then known. Since that time the work of R. L. Carter [6] has provided many new second order rotatable designs in two factors. However, additional methods were needed wvhich would provide both second and third order designs in three and more factors. The present work represents an attempt to meet, in part, this need. New construction methods for obtaining rotatable designs of second order in three dimensions are here presented. By use of these methods various infinite classes of designs are obtained, and it may be shown that all the rotatable designs previously known can be derived as special cases of these infinite classes. Also derived is an infinite class of second order rotatable designs which contain only 16 points; only two particular designs contain fewer points.