Related papers: Regular polyhedra in the 3-torus

We study three-dimensional path geometries with nontrivial torsion of maximal rank. We introduce the notion of constant torsion and show that such path geometries are in one-to-one correspondence with certain cone structures modeled on…

We give a complete coarse classification of Legendrian and transverse torus knots in any contact structure on $S^3$.

We investigate a novel setting for polytope rigidity, where a flex must preserve edge lengths and the planarity of faces, but is allowed to change the shapes of faces. For instance, the regular cube is flexible in this notion. We present…

We characterize all groups which can occur as the topological symmetry group or the orientation preserving topological symmetry group of some embedding of the Petersen graph in S^3.

We prove that any constant mean curvature embedded torus in the three dimensional sphere is axially symmetric, and use this to give a complete classification of such surfaces for any given value of the mean curvature.

We prove that if the dimension of the first cohomology group of a $RCD^*(0,N)$ space is $N$, then the space is a flat torus. This generalizes a classical result due to Bochner to the non-smooth setting and also provides a first example…

The non-existence of non-trivial conformally symmetric manifolds in the three-dimensional Riemannian setting is shown. In Lorentzian signature, a complete local classification is obtained. Furthermore, the isometry classes are examined.

We give a classification of homogeneous Riemannian structures on (non locally symmetric) $3$-dimensional Lie groups equipped with left invariant Riemannian metrics. This work together with classifications due to previous works yields a…

We present a conjecture, based on computational results, on the area minimizing way to enclose and separate two arbitrary volumes in the flat cubic 3-torus. For comparable small volumes, we prove that an area minimizing double bubble in the…

We study theta-curves embedded in a standard torus in the 3-sphere. We show that each nontrivial torus knot together with an essential arc determines a prime theta-curve, yielding explicit infinite families of prime theta-curves. We compute…

This paper is a continuation of my paper "Lattices of flats for symplectic matroids". We explore geometric constructions originating from the lattice of flats of ranked symplectic matroids. We observe that a ranked symplectic matroid always…

We obtain, by a direct computation, explicit descriptions of all principally polarized semi-abelic varieties of torus rank up to 3. We describe the geometry of their symmetric theta divisors and obtain explicit formulas for the involution…

We introduce and study the notion of orthosymmetric spaces over an Archimedean vector lattice as a generalization of finite-dimentional Euclidean inner spaces. A special attention has been paid to linear operators on these spaces.

It is known that the space of convex polygons in the Euclidean plane with fixed normals, up to homotheties and translations, endowed with the area form, is isometric to a hyperbolic polyhedron. In this note we show a class of convex…

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…

We give a classification of the lattices of rank r=4, r=8 and r=12 over \Q(\sqrt{-3}), which are even and unimodular \Z-lattices. Using this classification we construct the associated theta series, which are Hermitian modular forms, and…

In this article we obtain the classification of the congruences of lines with one-dimensional focal locus. It turns out that one can restrict to study the case of $\mathbb{P}^3$.

This paper continues the study initiated in "The aithmetic of Triangles." We begin by examining a set of similar tetrahedra with parallel sides, together with a set of points in three-dimensional space. It turns out that the set…

Using elementary graded automorphisms of polytopal algebras (essentially the coordinate rings of projective toric varieties) polyhedral versions of the group of elementary matrices and the Steinberg and Milnor groups are defined. They…

Regular polytopes, the generalization of the five Platonic solids in 3 space dimensions, exist in arbitrary dimension $n\geq-1$; now in {\rm dim}. 2, 3 and 4 there are \emph{extra} polytopes, while in general dimensions only the…