Shapes of the future are here!

A general framework and methodology for the creation of shape is outlined. The introduction of several new classes of objects, shape operators, and the development of a generalized shape construction is presented. A family of figures and bodies associated with a planar convex polygon and spatial convex polytope are constructed, and include a number of historically interesting and contemporary geometric objects such as the centroid, cross-section, crystal, difference, Petty, polar, podal, projection, intersection, and symmetry figures. Shape operators defined in terms of geometric and mixed functionals are computed and the formulation of natural extremal problems associated with the geometric structures are solved using numerical optimization techniques.