Third Order Rotatable Designs in Three Dimensions

Abstract

Two recent papers by Bose and Draper [1] and Draper [3] showed how it was possible, by combining certain sets of points, to construct infinite classes of second order rotatable designs in three and more dimensions. In this paper, third order rotatable designs in three dimensions are discussed. First, a general theorem is proved that provides the conditions under which a third order rotatable arrangement of points in $k$ dimensions is non-singular. The four previously known third order designs in three dimensions are stated; it is then shown how some of the second order design classes constructed earlier [1] may be combined in pairs to give infinite classes of sequential third order rotatable designs in three dimensions. One example of such a combination is worked out in full and it is shown that two of the four known designs are special cases of this class. A summary of further third order rotatable design classes that have been shown to exist, and that have been tabulated by the author, concludes the paper.