Related papers: Classification of Reflexive Polyhedra in Three Dim…

Toric surface codes are a class of error-correcting codes coming from a lattice polytope defining a two-dimensional toric variety. Previous authors have mostly completed classifications of these toric surface codes with dimension up to $k =…

The extension complexity of a polytope measures its amenability to succinct representations via lifts. There are several versions of extension complexity, including linear, real semidefinite, and complex semidefinite. We focus on the last…

Regular polygonal complexes in euclidean 3-space are discrete polyhedra-like structures with finite or infinite polygons as faces and with finite graphs as vertex-figures, such that their symmetry groups are transitive on the flags. The…

In view of solving problems of geometric realizability of polyhedra with given geometric constraints, we describe the space of geometric realizations of a simply-connected triangulated euclidean polyhedron in $\mathbb{R}^3$ up to similarity…

Edge-to-edge tilings of the sphere by congruent quadrilaterals are completely classified in a series of three papers. This second one applies the powerful tool of trigonometric Diophantine equations to classify the case of…

We study the problem of symmetry detection of 3D shapes from single-view RGB-D images, where severely missing data renders geometric detection approach infeasible. We propose an end-to-end deep neural network which is able to predict both…

A $3$-dimensional polytope $P$ is $k$-equiprojective when the projection of $P$ along any line that is not parallel to a facet of $P$ is a polygon with $k$ vertices. In 1968, Geoffrey Shephard asked for a description of all equiprojective…

Bosse et al. conjectured that for every natural number $d \ge 2$ and every $d$-dimensional polytope $P$ in $\real^d$ there exist $d$ polynomials $p_0(x),...,p_{d-1}(x)$ satisfying $P=\{x \in \mathbb{R}^d : p_0(x) \ge 0, >..., p_{d-1}(x) \ge…

Let $K/k$ be a finite Galois extension, $G=\text{Gal}(K/k)$, $\Sigma$ be a fan in a lattice $N$ and $X_{\Sigma}$ be an associated toric variety over $k$. It is well known that the set of $K/k$-forms of $X_{\Sigma}$ is in bijection with…

In this paper, we classify the perfect lattices in dimension 8. There are 10916 of them. Our classification heavily relies on exploiting symmetry in polyhedral computations. Here we describe algorithms making the classification possible.

This paper presents a full catalogue, up to conjugacy and subgroups of finite index, of all matrix groups $H < {\rm GL}(3,\mathbb{R})$ that give rise to a continuous wavelet transform with associated irreducible quasi-regular…

We show a one-to-one correspondence between arrangements of d lines in the projective plane, and lines in P^{d-2}. We apply this correspondence to classify (3,q)-nets over the complex numbers for all q<=6. When q=6, we have twelve possible…

Generic polyhedra are interesting mathematical objects to study in their own right. In this paper, we initialize a systematic study of two-dimensional generic polyhedra with an eye towards applications to low-dimensional topology,…

This paper proposes a novel neural-network-based adaptive hybrid-reflectance three-dimensional (3-D) surface reconstruction model. The neural network combines the diffuse and specular components into a hybrid model. The proposed model…

We construct compact polyhedra with $m$-gonal faces whose links are generalized 3-gons. It gives examples of cocompact hyperbolic bildings of type $P(m,3)$. For $m=3$ we get compact spaces covered by Euclidean buildings of type $A_2$.

We will describe some results regarding the algorithmic nature of homeomorphism problems for manifolds; in particular, the following theorem. Theorem 1: Every PL or smooth simply connected manifold M^n of dimension n at least 5 can be…

Let $X\subset \P^5$ be a smooth cubic fourfold. A well known conjecture asserts that $X$ is rational if and only if there an Hodge theoretically associated K3 surface $S$. The surface $S$ can be associated to $X$ in two other different…

Pogorelov proved in 1949 that every every convex polyhedron has at least three simple closed quasigeodesics. Whereas a geodesic has exactly pi surface angle to either side at each point, a quasigeodesic has at most pi surface angle to…

In this note we characterize all regular tetrahedra whose vertices in R^3 have integer coordinates. The main result is a consequence of the characterization of all equilateral triangles having integer coordinates contained in previous work.…

We study the cohomology of reflexive rank 2 sheaves on smooth projective threefolds. Applications are given to the moduli space of reflexive sheaves.