Realizations of regular polytopes, IV

In earlier papers, a rich theory of geometric realizations of an abstract regular polytope has been built up. More recently, a product was described, to add to blending and scaling as a way of combining realizations. This paper introduces an inner product of cosine vectors of normalized realizations, and shows that it has certain orthogonality properties; together with induced cosine vectors, these provide powerful new tools for investigating realizations. The enhanced theory is illustrated by revisiting the realization domains of several polytopes, including the 24-cell and 600-cell.