243 research outputs found
A unique representation of polyhedral types
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
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
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 vertices. We show that, modulo
planar rigid motions, this number is at most . We also exhibit several families which realize lower bounds of the order
of , and .
For the upper bound we use techniques from complex algebraic geometry, based
on the (projective) Cayley-Menger variety over the complex numbers . In this context, point configurations
are represented by coordinates given by squared distances between all pairs of
points. Sectioning the variety with hyperplanes yields at most
zero-dimensional components, and one finds this degree to be
. 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 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
Let be a -dimensional polytope. The {\em realization space}
of~ is the space of all polytopes that are combinatorially
equivalent to~, 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~ defined over~, there is a -polytope whose realization
space is ``stably equivalent'' to~. This implies that the realization space
of a -polytope can have the homotopy type of an arbitrary finite simplicial
complex, and that all algebraic numbers are needed to realize all -
polytopes. The proof is constructive. These results sharply contrast the
-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
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
The topic of the paper are developments of -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 -dimensional domain.Comment: 20pages, 15 figure
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