Related papers: Space vectors forming rational angles

Consider an arbitrary $n$-dimensional lattice $\Lambda$ such that $\mathbb{Z}^n \subset \Lambda \subset \mathbb{Q}^n$. Such lattices are called {\it rational} and can always be obtained by adding $m \le n$ rational vectors to…

We consider properties of polynomials with coefficients in division rings. A theorem on the decomposition of a polynomial with coefficients in an arbitrary division ring is obtained. It is shown that if a non-central element is not a root…

We prove that the regular octahedron has the minimal surface area among 3-polytopes of given volume and having at most six vertices.

When the number of non-triangular faces adjacent to a vertex $v$ is less than or equal to three, the vertex $v$ will be called (\emph{combinatorially}) \emph{rigid}. We study the number of rigid vertices and suggest a conjecture on a…

Let $\mathbb{F}$ be a field of characteristic $\neq 2$ and $3$, let $V$ be a $\mathbb{F}$-vector space of dimension $6$, and let $\Omega \in \wedge ^2V^\ast $ be a non-degenerate form. A system of generators for polynomial invariant…

For certain polynomials we relate the number of roots inside the unit circle with the index of a non-degenerate isolated umbilic point on a real analytic surface in Euclidean 3-space. In particular, for $N>0$ we prove that for a certain…

It is well known that sin(a\pi/b), cos(a\pi/b), etc., are only rational numbers for a few select integers a and b. We show that this is equivalent to the fact that only for d = 1,2,3,4, and 6 is the primitive dth root of unity of degree 2…

We consider arrangements of tropical hyperplanes where the apices of the hyperplanes are taken to infinity in certain directions. Such an arrangement defines a decomposition of Euclidean space where a cell is determined by its `type' data,…

Let $n$ be a positive integer. We show that a unit rational space vector whose multiple by $n$ is an integer vector can be extended to a rational orthonormal basis whose all members have the same property.

We construct, for any positive integer n, a family of n congruent convex polyhedra in R^3, such that every pair intersects in a common facet. Previously, the largest such family contained only eight polytopes. Our polyhedra are Voronoi…

We introduce the theory of div point sets, which aims to provide a framework to study the combinatoric nature of any set of points in general position on an Euclidean plane. We then show that proving the unsatisfiability of some first-order…

We formulate and give partial answers to several combinatorial problems on volumes of simplices determined by $n$ points in 3-space, and in general in $d$ dimensions. (i) The number of tetrahedra of minimum (nonzero) volume spanned by $n$…

We give an elementary introduction to some recent polyhedral techniques for understanding and solving systems of multivariate polynomial equations. We provide numerous concrete examples and illustrations, and assume no background in…

A set of points in $\mathbb{R}^d$ is acute, if any three points from this set form an acute angle. In this note we construct an acute set in $\mathbb{R}^d$ of size at least $2^{d/2}$.

Elementary geometric arguments are used to compute the group of homotopy classes of maps from a 4-manifold X to the 3-sphere, and to enumerate the homotopy classes of maps from X to the 2-sphere. The former completes a project initiated by…

We prove that any set of points in $\mathbb{R}^d$, any three of which form an angle less than $\frac{\pi}{3} + c$, has size $(1+\Theta(c))^d$ for sufficiently small $c>0$. The proof is based on a refinement of an approach by Erd\H{o}s and…

We partition in classes the set of matroids of fixed dimension on a fixed vertex set. In each class we identify two special matroids, respectively with minimal and maximal h-vector in that class. Such extremal matroids also satisfy a…

We show an explicit construction in 3 dimensions for a convex, mono-monostatic polyhedron (i.e., having exactly one stable and one unstable equilibrium) with 21 vertices and 21 faces. This polyhedron is a 0-skeleton, with equal masses…

We analyze polyhedra composed of hexagons and triangles with three faces around each vertex, and their 3-regular planar graphs of edges and vertices, which we call "trihexes". Trihexes are analogous to fullerenes, which are 3-regular planar…

The space of shapes of a polyhedron with given total angles less than 2\pi at each of its n vertices has a Kaehler metric, locally isometric to complex hyperbolic space CH^{n-3}. The metric is not complete: collisions between vertices take…