Volumes of Polyhedra in Non-Euclidean Spaces of Constant Curvature

Computation of the volumes of polyhedra is a classical geometry problem known since ancient mathematics and preserving its importance until present time. Deriving volume formulas for 3-dimensional non-Euclidean polyhedra of a given combinatorial type is a very difficult problem. Nowadays, it is fully solved for a tetrahedron, the most simple polyhedron in the combinatorial sense. However, it is well known that for a polyhedron of a special type its volume formula becomes much simpler. This fact was noted by Lobachevsky who found the volume of the so-called ideal tetrahedron in hyperbolic space (all vertices of this tetrahedron are on the absolute).In this survey, we present main results on volumes of arbitrary non-Euclidean tetrahedra and polyhedra of special types (both tetrahedra and polyhedra of more complex combinatorial structure) in 3-dimensional spherical and hyperbolic spaces of constant curvature K = 1 and K = -1, respectively. Moreover, we consider the new method by Sabitov for computation of volumes in hyperbolic space (described by the Poincare model in upper half-space). This method allows one to derive explicit volume formulas for polyhedra of arbitrary dimension in terms of coordinates of vertices. Considering main volume formulas for non-Euclidean polyhedra, we will give proofs (or sketches of proofs) for them. This will help the reader to get an idea of basic methods for computation of volumes of bodies in non-Euclidean spaces of constant curvature.

A Randomized Approach to Volume Constrained Polyhedronization Problem

Given a finite set of points in R3, polyhedronization deals with constructing a simple polyhedron such that the vertices of the polyhedron are precisely the given points. In this paper, we present randomized approximation algorithms for minimal volume polyhedronization (MINVP) and maximal volume polyhedronization (MAXVP) of three dimensional point sets in general position. Both, MINVP and MAXVP, problems have been shown to be NP-hard and to the best of our knowledge, no practical algorithms exist to solve these problems. It has been shown that for any point set S in R3, there always exists a tetrahedralizable polyhedronization of S. We exploit this fact to develop a greedy heuristic for MINVP and MAXVP constructions. Further, we present an empirical analysis on the quality of the approximation results of some well defined point sets. The algorithms have been validated by comparing the results with the optimal results generated by an exhaustive searching (brute force) method for MINVP and MAXVP for some well chosen point sets of smaller sizes. Finally, potential applications of minimum and maximum volume polyhedra in 4D printing and surface lofting, respectively, have been discussed.

Perturbation of transportation polytopes

International audience We describe a perturbation method that can be used to compute the multivariate generating function (MGF) of a non-simple polyhedron, and then construct a perturbation that works for any transportation polytope. Applying this perturbation to the family of central transportation polytopes of order $kn \times n$, we obtain formulas for the MGF of the polytope. The formulas we obtain are enumerated by combinatorial objects. A special case of the formulas recovers the results on Birkhoff polytopes given by the author and De Loera and Yoshida. We also recover the formula for the number of maximum vertices of transportation polytopes of order $kn \times n$. Nous décrivons une méthode de perturbation qui peut être utilisée pour calculer la fonction génératrice multivariée (MGF) d'un polyèdre non-simple, et ensuite construire une perturbation qui fonctionne pour tout polytope de transport. Appliquant cette perturbation à la famille des centraux de transport polytopes de l'ordre $kn \times n$, nous obtenons des formules pour le MGF du polytope. Les formules que nous obtenons sont énumérées par les objets combinatoires. Un cas spécial des formules récupère les résultats sur des polytopes de Birkhoff donnés par l'auteur et De Loera et Yoshida. Nous récupérons également la formule pour le nombre de sommets maximum des de transport polytopes d'ordre $kn \times n$.

Synthesis and Characterization of Mn Quantum Dots by Bioreduction with Water Hyacinth

The bio-reduction method is reported as a part of a complimentary self-sustained technology, where bioremediation and metal particle production are related. The use of the characterization methods in this self sustainable technique open the expectative to be used for several other elements and with other plants, which will be discussed. However, the particular case of Mn nanoparticles involves an important option to generate nanoparticles in the range of 1–4 nanometers with a well controlled size and with a structure based on an fcc-like geometry for the smallest clusters and with more complex arrays for cluster greater than four shells, which involves magnetic moments significantly related to their atomistic configuration. At the same time, the use of the characterization methods establishes the dependence of the nanoparticle's size on the pH conditions used during the synthesis; small clusters in the range of 1–2 nm were generated using pH = 5, and it was shown that for the smallest aggregates, simple polyhedron shapes are stable.

ON RECONSTRUCTING POLYHEDRA FROM PARALLEL SLICES

The problem of reconstructing a three-dimensional object from parallel slices has application in computer vision and medicine. Here we explore a specific existence question: given two polygons in parallel planes, is it always possible to find a polyhedron that has those polygons as faces, and whose vertices are precisely the vertices of the two polygons? We answer this question in the negative by providing an example of two polygons that cannot be connected to form a simple polyhedron. One polygon is a triangle, the other a somewhat complicated shape with spiraling pockets.

The energy of twin-crystals

Vernadsky has pointed out that in certain cases a twin-crystal appears to be a more stable form than the simple polyhedron, as proved by Scacchi's observation that twins sometimes exceed in magnitude and regularity simple crystals growing in the same solution. He suggests that an explanation can be found in the fact that the surfaceenergy may be tess for a twin than for a simple crystal of the same volume. The suggestion is a valuable one, but Vernadsky applies no quantitative test to his theory.