arXiv research

Study geodesics on spherical polyhedra, estimating their number.

Introduces Coxeter polyhedra in various geometries.

Classical H.Minkowski theorems on existence and uniqueness of convex polyhedra with prescribed directions and areas of faces as well as the well-known generalization of H.Minkowski uniqueness theorem due to A.D.Alexandrov are extended to a class of nonconvex polyhedra which are called polyhedral herissons and may be de…

Polyhedra's structure is uniquely defined by edge lengths and dihedral angles, even nonconvex.

Formula for transgressions on polyhedral manifolds, linking face volumes and outer angles.

A spherical topological manifold of dimension n-1 forms a prototile on its cover, the (n-1)-sphere. The tiling is generated by the fixpoint-free action of the group of deck transformations. By a general theorem, this group is isomorphic to the first homotopy group. Multiplicity and selection rules appear in the form of…

From the homotopy groups of two cubic spherical 3-manifolds we construct the isomorphic groups of deck transformations acting on the 3-sphere. These groups become the cyclic group of order eight and the quaternion group respectively. By reduction of representations from the orthogonal group to the identity representati…

The paper is centered around a new proof of the infinitesimal rigidity of convex polyhedra. The proof is based on studying derivatives of the discrete Hilbert-Einstein functional on the space of "warped polyhedra" with a fixed metric on the boundary. This approach is in a sense dual to using derivatives of the volume i…

By a result of W.~P. Thurston, the moduli space of flat metrics on the sphere with $n$ cone singularities of prescribed positive curvatures is a complex hyperbolic orbifold of dimension $n-3$. The Hermitian form comes from the area of the metric. Using geometry of Euclidean polyhedra, we observe that this space has a n…

Constructs new steady gradient Ricci solitons for higher dimensions.

Geodesic loops on tetrahedra are studied in spherical and hyperbolic spaces.

Discrete Laplacians defined for spherical and hyperbolic surfaces.

Euler explored spherical geometry using trigonometric formulae and solid geometry methods.

The classical Schläfli formula, and its ``higher'' analogs given in [SS03], are relations between the variations of the volumes and ``curvatures'' of faces of different dimensions of a polyhedra (which can be Euclidean, spherical or hyperbolic) under a first-order deformation. We describe here analogs of those formulas…

We study surfaces with decorations and prove uniformization in non-Euclidean geometries.

We explore some generalizations of fullerenes F_v (simple polyhedra with v vertices and only 5- and 6-gonal faces) seen as (d-1)-dimensional simple manifolds (preferably, spherical or polytopal) with only 5- and 6-gonal 2-faces. First, finite and planar (infinite) 3-fullerenes are described. Three infinite families of …

Löbell polyhedra have small systoles and are quasi-arithmetic.

New hyperbolic polyhedra with $π/3$ angles and volumes calculated.

The paper sets new limits on hyperbolic polyhedra volumes.

The study sets limits on dihedral angles of large hyperbolic polyhedra.

Improved volume estimates for right-angled polyhedra in hyperbolic space.

We study convex polyhedra in $\mathbb{R}\mathbb{P}^3$ with all their vertices on a sphere. We do not require, in particular, that the polyhedra lie in the interior of the sphere, hence the term "weakly inscribed". Such polyhedra can be interpreted as ideal polyhedra, if we regard $\mathbb{R}\mathbb{P}^3$ as a combinati…

We review several results related to the characterization of polyhedra in hyperbolic 3-space. In particular we present Rivin's theorem that gives a characterization of compact convex hyperbolic polyhedra, and Hodgson's proof of the Adreev's theorem. We also review the analogous characterization of ideal polyhedra, and …

Improved bounds on ideal vertices in right-angled hyperbolic polyhedra.

Study calculates mass of special polyhedra in hyperbolic space.

Mass in relativity linked to polyhedra geometry.

Polyhedra volume conjecture supports Stoker conjecture weakly.

In this paper we consider a class of right-angled polyhedra in three-dimensional Lobachevsky space, all vertices of which lie on the absolute. New upper bounds on volumes in terms the number of faces of the polyhedron are obtained. Volumes of polyhedra with at most 23 faces are computed. It is shown that the minimum vo…

This article defines a pair of combinatorial operations on the combinatorial structure of compact right-angled hyperbolic polyhedra in dimension three called decomposition and edge surgery. It is shown that these operations simplify the combinatorics of such a polyhedron, while keeping it within the class of right-angl…

Intermediate logic of all convex polyhedra is axiomatized.

Study approximates Riemannian manifolds using polyhedra.

The study examines Riemannian surfaces with simple singularities.

We present a notion of mutation of hyperbolic polyhedra, analogous to mutation in knot theory, and then present a general question about commensurability of mutant pairs of polyhedra. We motivate that question with several concrete examples of mutant pairs for which commensurability is unknown. The polyhedra we conside…

We prove that, both in the hyperbolic and spherical 3-spaces, there exist nonconvex compact boundary-free polyhedral surfaces without selfintersections which admit nontrivial continuous deformations preserving all dihedral angles and study properties of such polyhedral surfaces. In particular, we prove that the volume …

New families of hyperbolic polyhedra yield infinitely many unique reflection groups.

The paper solves three problems related to monostable polyhedra.

We describe the first-order variations of the angles of Euclidean, spherical or hyperbolic polygons under infinitesimal deformations such that the lengths of the edges do not change. Using this description, we introduce a vector-valued quadratic invariant $b$ on the space of those isometric deformations which, for conv…

In this article we establish the relation between the spines of 3-manifolds and the polyhedra with identified faces. We do this by showing that the spines of the closed, connected, orientable 3-manifolds can be presented through polyhedra with identified faces in a very natural way. We also prove the equivalence betwee…

A variant of the Circle Packing Theorem states that the combinatorial class of any convex polyhedron contains elements midscribed to the unit sphere centered at the origin, and that these representatives are unique up to Möbius transformations of the sphere. Motivated by this result, various papers investigate the prob…

Paper solves long-standing problem of infinite ideal polyhedra in hyperbolic space.

Mass in relativity linked to polyhedra geometry.

We consider a natural question: "Is it true that each homotopy domination of a polyhedron over itself is a homotopy equivalence?" and a strongly related problem of K. Borsuk (1967): "Is it true that two ANR's homotopy dominating each other have the same homotopy type?" The answer was earlier known to be positive for ma…

An algorithm for determining the list of smallest volume right-angled hyperbolic polyhedra in dimension 3 is described. This algorithm has been implemented on computer using the program Orb to compute volumes, and the first 825 polyhedra in the list have been determined.

New periodic polyhedra found in curved spaces.

Software finds ideal polyhedra with rational dihedral angles and volume maxima.

The paper proves rigidity and uniformization theorems for infinite circle patterns and convex polyhedra in hyperbolic 3-space.

We prove that there are thirteen Archimedean/semiregular polyhedra by using Euler's polyhedral formula.

Polyhedra rigidity theorem in hyperbolic space proved.