Related papers: Equable Triangles on the Eisenstein Lattice

We prove that almost every triangle can be dissected only into $n^2$ triangles which have to be equal one another. Moreover, such a dissection is unique for every $n$. It turns out that to solve this "simple" problem it is convenient to use…

We show that every tiling of a convex set in the Euclidean plane $\mathbb{R}^2$ by equilateral triangles of mutually different sizes contains arbitrarily small tiles. The proof is purely elementary up to the discussion of one family of…

We give a short proof of the contractibility of the space of geodesic triangulations with fixed combinatorial type of a convex polygon in the Euclidean plane. Moreover, for any $n>0$, we show that there exists a space of geodesic…

A rational triangle $T$ (one whose angles are rational multiples of $\pi$) unfolds to a translation surface $(X_T,\omega_T)$. The lattice triangle problem asks to classify those $T$ for which $(X_T,\omega_T)$ is a Veech (lattice) surface,…

We prove the existence of three non-round, non-isometric Einstein metrics with positive scalar curvature on the sphere $S^{10}.$ Previously, the only even-dimensional spheres known to admit non-round Einstein metrics were $S^6$ and $S^8.$

A polygon is equable if its area is equal to its perimeter. A pair of polygons is an amicable pair if the area of the first is equal to the perimeter of the second, and vice versa. A polygon is a lattice polygon if its vertices lie on the…

Every body knows that identical regular triangles or squares can tile the whole plane. Many people know that identical regular hexagons can tile the plane properly as well. In fact, even the bees know and use this fact! Is there any other…

We drastically simplify the problem of linearizing a general higher-order theory of gravity. We reduce it to the evaluation of its Lagrangian on a particular Riemann tensor depending on two parameters, and the computation of two derivatives…

Geodesic loops on polyhedra were studied only for Euclidean space and it was known that there are no simple geodesic loops on regular tetrahedra. Here we prove that: 1) On the spherical space, there are no simple geodesic loops on…

We consider module categories of path algebras of connected acyclic quivers. It is shown in this paper that the set of functorially finite torsion classes form a lattice if and only if the quiver is either Dynkin quiver of type A, D, E, or…

We show that if two lattice $3$-polytopes $P$ and $P'$ have the same Ehrhart function then they are $\operatorname{GL}_3({\mathbb Z})$-equidecomposable; that is, they can be partitioned into relatively open simplices $U_1,\dots, U_k$ and…

We classify triangles that can be tiled only into a square number of congruent triangles, settling Erd\H{o}s Problem 633.

We study the set of visible lattice points in multidimensional hypercubes. The problems we investigate mix together geometric, probabilistic and number theoretic tones. For example, we prove that almost all self-visible triangles with…

We classify, up to some lattice-theoretic equivalence, all possible configurations of rational double points that can appear on a surface whose minimal resolution is a complex Enriques surface.

Let (P) denote the problem of existence of a point in the plane of a given triangle T, that is at rational distance from all the vertices of T. In this article, we provide a complete solution to (P) for all equilateral triangles.

In this paper, we present a synthetic solution to a geometric open problem involving the radical axis of two strangely defined circumcircles. The solution encapsulates two generalizations, one of which uses a powerful projective result…

We construct an infinite series of simply transitive irreducible lattices in PGL_2(F_q((t))) \times PGL_2(F_q((t))) by means of a quaternion algebra over F_q(t). The lattices depend on an odd prime power q = p^r and a parameter \tau\ in…

Let $\cal T$ be a tiling of the plane with equilateral triangles no two of which share a side. We prove that if the side lengths of the triangles are bounded from below by a positive constant, then $\cal T$ is periodic and it consists of…

Let $ABC$ be an equilateral triangle. For certain triangles $T$ (the "tile") and certain $N$, it is possible to cut $ABC$ into $N$ copies of $T$. It is known that only certain shapes of $T$ are possible, but until now very little was known…

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…