  We give some new explicit examples of putatively optimal projective spherical
designs. i.e., ones for which there is numerical evidence that they are of
minimal size. These form continuous families, and so have little apparent
symmetry in general, which requires the introduction of new techniques for
their construction. New examples of interest include an 11-point spherical (3,
3)-design for R 3 , and a 12-point spherical (2, 2)-design for R 4 given by
four Mercedes-Benz frames that lie on equi-isoclinic planes. We also give
results of an extensive numerical study to determine the nature of the real
algebraic variety of optimal projective real spherical designs, and in
particular when it is a single point (a unique design) or corresponds to an
infinite family of designs.
