243 research outputs found

    A unique representation of polyhedral types

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    It is known that for each combinatorial type of convex 3-dimensional polyhedra, there is a representative with edges tangent to the unit sphere. This representative is unique up to projective transformations that fix the unit sphere. We show that there is a unique representative (up to congruence) with edges tangent to the unit sphere such that the origin is the barycenter of the points where the edges touch the sphere.Comment: 4 pages, 2 figures. v2: belated upload of final version (of March 2004

    Many projectively unique polytopes

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    We construct an infinite family of 4-polytopes whose realization spaces have dimension smaller or equal to 96. This in particular settles a problem going back to Legendre and Steinitz: whether and how the dimension of the realization space of a polytope is determined/bounded by its f-vector. From this, we derive an infinite family of combinatorially distinct 69-dimensional polytopes whose realization is unique up to projective transformation. This answers a problem posed by Perles and Shephard in the sixties. Moreover, our methods naturally lead to several interesting classes of projectively unique polytopes, among them projectively unique polytopes inscribed to the sphere. The proofs rely on a novel construction technique for polytopes based on solving Cauchy problems for discrete conjugate nets in S^d, a new Alexandrov--van Heijenoort Theorem for manifolds with boundary and a generalization of Lawrence's extension technique for point configurations.Comment: 44 pages, 18 figures; to appear in Invent. mat

    On the Number of Embeddings of Minimally Rigid Graphs

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    Rigid frameworks in some Euclidian space are embedded graphs having a unique local realization (up to Euclidian motions) for the given edge lengths, although globally they may have several. We study the number of distinct planar embeddings of minimally rigid graphs with nn vertices. We show that, modulo planar rigid motions, this number is at most (2n−4n−2)≈4n{{2n-4}\choose {n-2}} \approx 4^n. We also exhibit several families which realize lower bounds of the order of 2n2^n, 2.21n2.21^n and 2.88n2.88^n. For the upper bound we use techniques from complex algebraic geometry, based on the (projective) Cayley-Menger variety CM2,n(C)⊂P(n2)−1(C)CM^{2,n}(C)\subset P_{{{n}\choose {2}}-1}(C) over the complex numbers CC. In this context, point configurations are represented by coordinates given by squared distances between all pairs of points. Sectioning the variety with 2n−42n-4 hyperplanes yields at most deg(CM2,n)deg(CM^{2,n}) zero-dimensional components, and one finds this degree to be D2,n=1/2(2n−4n−2)D^{2,n}={1/2}{{2n-4}\choose {n-2}}. The lower bounds are related to inductive constructions of minimally rigid graphs via Henneberg sequences. The same approach works in higher dimensions. In particular we show that it leads to an upper bound of 2D3,n=2n−3n−2(n−6n−3)2 D^{3,n}= {\frac{2^{n-3}}{n-2}}{{n-6}\choose{n-3}} for the number of spatial embeddings with generic edge lengths of the 1-skeleton of a simplicial polyhedron, up to rigid motions

    Realization spaces of 4-polytopes are universal

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    Let P⊂RdP\subset\R^d be a dd-dimensional polytope. The {\em realization space} of~PP is the space of all polytopes P′⊂RdP'\subset\R^d that are combinatorially equivalent to~PP, modulo affine transformations. We report on work by the first author, which shows that realization spaces of \mbox{4-dimensional} polytopes can be ``arbitrarily bad'': namely, for every primary semialgebraic set~VV defined over~Z\Z, there is a 44-polytope P(V)P(V) whose realization space is ``stably equivalent'' to~VV. This implies that the realization space of a 44-polytope can have the homotopy type of an arbitrary finite simplicial complex, and that all algebraic numbers are needed to realize all 44- polytopes. The proof is constructive. These results sharply contrast the 33-dimensional case, where realization spaces are contractible and all polytopes are realizable with integral coordinates (Steinitz's Theorem). No similar universality result was previously known in any fixed dimension.Comment: 10 page

    Polytopality and Cartesian products of graphs

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    We study the question of polytopality of graphs: when is a given graph the graph of a polytope? We first review the known necessary conditions for a graph to be polytopal, and we provide several families of graphs which satisfy all these conditions, but which nonetheless are not graphs of polytopes. Our main contribution concerns the polytopality of Cartesian products of non-polytopal graphs. On the one hand, we show that products of simple polytopes are the only simple polytopes whose graph is a product. On the other hand, we provide a general method to construct (non-simple) polytopal products whose factors are not polytopal.Comment: 21 pages, 10 figure

    Rolling of Coxeter polyhedra along mirrors

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    The topic of the paper are developments of nn-dimensional Coxeter polyhedra. We show that the surface of such polyhedron admits a canonical cutting such that each piece can be covered by a Coxeter (n−1)(n-1)-dimensional domain.Comment: 20pages, 15 figure
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