60 research outputs found
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
Cayley-Bacharach Formulas
The Cayley-Bacharach Theorem states that all cubic curves through eight given
points in the plane also pass through a unique ninth point. We write that point
as an explicit rational function in the other eight.Comment: 13 pages, 4 figure
The Complexity of Finding Small Triangulations of Convex 3-Polytopes
The problem of finding a triangulation of a convex three-dimensional polytope
with few tetrahedra is proved to be NP-hard. We discuss other related
complexity results.Comment: 37 pages. An earlier version containing the sketch of the proof
appeared at the proceedings of SODA 200
Every non-Euclidean oriented matroid admits a biquadratic final polynomial
Richter-Gebert proved that every non-Euclidean uniform oriented matroid admits a biquadratic final polynomial. We extend this result to the non-uniform cas
Emergence of complex patterns in a higher-dimensional phyllotactic system
A hypothesis commonly known as Hofmeister’s rule states that primordia appearing at the apical ring of a plant shoot in periodic time steps are formed in the position where the most space is available with respect to the space occupation of already-formed primordia. A corresponding two-dimensional dynamical model has been extensively studied by Douady and Couder, and shown to generate a variety of observable phyllotactic patterns indeed. In this study, motivated by mathematical interest in a theoretical phyllotaxis-inspired system rather than by a concrete biological problem, we generalize this model to three dimensions and present the dynamics observed in simulations, thereby illustrating the range of complex structures that phyllotactic mechanisms can give rise to. The patterns feature unexpected additional properties compared to the two-dimensional case, such as periodicity and chaotic behavior of the divergence angle
Extremal properties for dissections of convex 3-polytopes
A dissection of a convex d-polytope is a partition of the polytope into
d-simplices whose vertices are among the vertices of the polytope.
Triangulations are dissections that have the additional property that the set
of all its simplices forms a simplicial complex. The size of a dissection is
the number of d-simplices it contains. This paper compares triangulations of
maximal size with dissections of maximal size. We also exhibit lower and upper
bounds for the size of dissections of a 3-polytope and analyze extremal size
triangulations for specific non-simplicial polytopes: prisms, antiprisms,
Archimedean solids, and combinatorial d-cubes.Comment: 19 page
Konzeptuelles Verständnis von Brüchen mit Visualisierungen auf iPads fördern: Eine empirische Studie
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