Robertson in 1988 suggested a model for the realization space of a convex d‐dimensional polytope and an approach via the implicit function theorem, which—in the case of a full rank Jacobian—proves that the realization space is a manifold of dimension NGfalse(Pfalse):=dfalse(f0+fd−1false)−f0,d−1. This is the natural guess for the dimension given by the number of variables minus the number of quadratic equations that are used in the definition of the realization space. While this indeed holds for many natural classes of polytopes (including simple and simplicial polytopes, as well as all polytopes of dimension at most 3), and Robertson claimed this to be true for all polytopes, Mnëv's (1986/1988) universality theorem implies that it is not true in general. Indeed, (1) the centered realization space is not a smoothly embedded manifold in general, and (2) it does not have the dimension NG(P) in general. In this paper, we develop Jacobian criteria for the analysis of realization spaces. From these we get easily that for various large and natural classes of polytopes, the realization spaces are indeed manifolds, whose dimensions are given by NG(P). However, we also identify the smallest polytopes where the dimension count NG(P) and thus Robertson's claim fails, among them the bipyramid over a triangular prism. For an explicit example with property (1), we analyze the classical 24‐cell: We show that the realization space has at least dimension NG(C4(24))=48, and it has points where it is a manifold of this dimension, but it is not smoothly embedded as a manifold everywhere.

We classify the finite groups of orthogonal transformations in 4-space, and we study these groups from the viewpoint of their geometric action, using polar orbit polytopes. For one type of groups (the toroidal groups), we develop a new classification based on their action on an invariant torus, while we rely on classic results for the remaining groups.As a tool, we develop a convenient parameterization of the oriented great circles on the 3-sphere, which leads to (oriented) Hopf fibrations in a natural way. Contents

Robertson (1988) suggested a model for the realization space of a convex d-dimensional polytope and an approach via the implicit function theorem, which -in the case of a full rank Jacobian -proves that the realization space is a manifold of dimension NG(P ) := d(f 0 + f d−1 ) − f 0,d−1 , which is the natural guess for the dimension given by the number of variables minus the number of quadratic equations that are used in the definition of the realization space. While this indeed holds for many natural classes of polytopes (including simple and simplicial polytopes, as well as all polytopes of dimension at most 3), and Robertson claimed this to be true for all polytopes, Mnëv's (1986/1988) Universality Theorem implies that it is not true in general: Indeed, (1) the centered realization space is not a smoothly embedded manifold in general, and (2) it does not have the dimension NG(P ) in general.In this paper we develop Jacobian criteria for the analysis of realization spaces. From these we get easily that for various large and natural classes of polytopes the realization spaces are indeed manifolds, whose dimensions are given by NG(P ). However, we also identify the smallest polytopes where the dimension count (2) and thus Robertson's claim fails, among them the bipyramid over a triangular prism. For the property (1), we analyze the classical 24-cell: We show that the realization space has at least the dimension NG(C (24) 4 ) = 48, and it has points where it is a manifold of this dimension, but it is not smoothly embedded as a manifold everywhere.