Robert M. Erdahl

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Georges Voronoi (1908-09) introduced two important reduction methods for positive quadratic forms: the reduction with perfect forms, and the reduction with L-type domains. A form is perfect if it can be reconstructed from all representations of its arithmetic minimum. Two forms have the same L-type if Delaunay tilings of their lattices are affinely(More)
A polytope D, whose vertices belong to a lattice of rank d, is Delaunay if it can be circumscribed by an ellipsoid E with interior free of lattice points, and so that the vertices of D are the only lattice points on the quadratic surface E. If in addition E is uniquely determined by D, we call D a perfect Delaunay polytope. Thus, in the perfect case, the(More)
We show how a d-stress on a piecewise-linear realization of an oriented (non-simplicial, in general) d-manifold in Rd naturally induces stresses of lower dimensions on this manifold, and discuss implications of this construction to the analysis of self-stresses in spatial frameworks. The mappings we construct are not linear, but polynomial. In the 1860–70s(More)
I will consider two outstanding problems in the theory of Delaunay and Voronoi tilings for lattices. There is a new classification problem that has arisen from a new structure theorem. Sergei Ryshkov proved that: The Minkowski sum of two Voronoi polytopes is a Voronoi polytope, if and only if the corresponding Delaunay tilings are commensurate. This result(More)