Related papers: Integrality relations for polygonal dissections
Ergodic systems, being indecomposable are important part of the study of dynamical systems but if a system is not ergodic, it is natural to ask the following question: Is it possible to split it into ergodic systems in such a way that the…
In this paper, we show that an equilateral triangle cannot be dissected into finitely many smaller equilateral triangles, no two of which share two vertices. We do this without the use of Electrical Networks.
We introduce a notion of embedding codimension of an arbitrary local ring, establish some general properties, and study in detail the case of arc spaces of schemes of finite type over a field. Viewing the embedding codimension as a measure…
In this paper we consider planar polygons with parallel opposite sides. This type of polygons can be regarded as discretizations of closed convex planar curves by taking tangent lines at samples with pairwise parallel tangents. For this…
We describe a polynomial time algorithm that takes as input a polygon with axis-parallel sides but irrational vertex coordinates, and outputs a set of as few rectangles as possible into which it can be dissected by axis-parallel cuts and…
A frieze on a polygon is a map from the diagonals of the polygon to an integral domain which respects the Ptolemy relation. Conway and Coxeter previously studied positive friezes over $\mathbb{Z}$ and showed that they are in bijection with…
The open problem of whether or not every pair of equal-area polygons has a hinged dissection is discussed.
We propose to take a look at a new approach to the study of integral polyhedra. The main idea is to give an integral representation, or matrix model representation, for the key combinatorial characteristics of integral polytopes. Based on…
In 1998 A. Connes proposed an algebraic proof of Morley's trisector theorem. He observed that the points of intersection of the trisectors are the fixed points of pairwise products of rotations around vertices of the triangle with angles…
A point in the interior of a planar triangle determines a subdivision into six subtriangles. A triangle with angles commensurable with $\pi$ is called $\pi$-commensurable. For such a triangle a subdivision where each of the subtriangles are…
We show that any pseudo-effective divisor on a normal surface decomposes uniquely into its "integral positive" part and "integral negative" part, which is an integral analog of Zariski decompositions. By using this decomposition, we give…
Part I: The two-dimensional Pascal Triangle will be generalized into a three-dimensional Pascal Pyramid and four-, five- or whatsoever-dimensional hyper-pyramids. Part II: The Bilateral Binomial Theorem will be generalised into a Bilateral…
In this paper, we show that if we decompose a polygon into two smaller polygons, then by comparing the number of extremal vertices in the original polygon versus the sum of the two smaller polygons, we can gain at most two globally extremal…
Suppose that $I$ is a unit square. Let $T$ (resp. $\Delta$) be an isosceles right triangle (resp. an equilateral triangle). We prove that any collection of triangles homothetic to $T$ (resp. $\Delta$), whose total area does not exceed…
This article provides the basic algebraic background on infinitesimal deformations and presents the proof of the well-known fact that the non-trivial infinitesimal deformations of a $K$-algebra $R$ are parameterized by the elements of…
This paper deals with the history of the following problem: "Can an arbitrary rectangle be dissected into 3 non-rectangular congruent regions?" We present a new elementary proof that the answer is indeed no.
If E is any ellipse inscribed in a convex quadrilateral, D, then we prove that Area(E)/Area(D) is less than or equal to pi/4, and equality holds if and only if D is a parallelogram and E is tangent to the sides of D at the midpoints. This…
In this paper, we consider the problem of partitioning a polygon into a set of connected disjoint sub-polygons, each of which covers an area of a specific size. The work is motivated by terrain covering applications in robotics, where the…
Partitionings (or segmentations) divide a given domain into disjoint connected regions whose union forms again the entire domain. Multi-dimensional partitionings occur, for example, when analyzing parameter spaces of simulation models,…
We give a simple and complete description of those convex lattice polygons in the plane that can be dissected into lattice triangles of integer area. A new version of Sperner's Lemma plays a central role.