Realizations of regular polytopes. IV (Q2248823)

A comprehensive theory of geometric realizations of a finite abstract regular polytope \(\mathcal P\) in Euclidean space has been elaborated by the author of the paper under review, partly in co-authorship with B. Monson. The paper under review is Part IV in a series of papers. For Parts I, II, and III see the author [ibid. 37, No. 1, 38--56 (1989; Zbl 0676.51008)], the author and \textit{B. Monson} [ibid. 65, No. 1--2, 102--112 (2003; Zbl 1022.51019)], and the author [ibid. 82, No. 1-2, 35-63 (2011; Zbl 1226.51005)], respectively. Part I introduces a realization \(P\) of a (finite) abstract regular polytope \(\mathcal P\) in Euclidean space, which can be thought of as a a geometric realization of \(\mathcal P\) so that each combinatorial cellular automorphism of \(\mathcal P\) extends to a geometric symmetry of \(P\). Such a realization is called by the reviewer a realization without hidden symmetries [\textit{S. Lawrencenko}, Geombinatorics 24, No. 1, 11--20 (2014; Zbl 1312.51007)]. One vertex is distinguished as the base vertex to apply Wythoff's construction. The realizations of \(\mathcal P\) are parametrized by diagonal vectors and are represented as vectors in Euclidean space of certain dimension. The space of all realizations of \(\mathcal P\) forms a compact convex cone under the operations of scaling and blending which combine different realizations. In Parts I and II, the properties of the space of realizations of \(P\) are thoroughly studied. In Part III, the tensor product of realizations, which can be described in terms of cosine vectors, is introduced in addition to blending and scaling as a way of combining realizations. The normalized realizations (in which the vertices of \(P\) lie on the unit sphere) form a compact convex set. The paper under review (Part IV) develops further the theory by introducing one more operation on normalized realizations. This is called the inner product of cosine vectors and reveals certain orthogonality properties. The author also provides powerful new tools for investigating realizations in addition to induced cosine vectors. One of those tools is a cosine matrix which enables to study realizations on the base of combinatorial properties of \(\mathcal P\). To illustrate the application of the enhanced theory, the author considers the realization domains of several polytopes including the 24-cell, the 600-cell, and the self-Petrie map \(\{ 5, 5\}_5\).