Dissection of a Triangle into Similar Triangles

We prove that the number of non-similar triangles T which can be dissected into two, three or five similar non-right triangles is equal to zero, one and nine, respectively. We find all these triangles. Moreover, every triangle can be dissected into n similar triangles whenever n = 4 or n ≥ 6. In the last section we allow dissections into right-triangles but we add another restriction. We prove that in any perfect, prime and simplicial dissection into at least three tiles, the tiles must have one of only three possible shapes.     

Spherical f-Tilings by (Equilateral and Isosceles) Triangles and Isosceles Trapezoids

Dihedral f-tilings by spherical parallelograms and spherical triangles were obtained in [3–5]. In this paper we extend these results presenting the study of all dihedral f-tilings of the sphere S ², whose prototiles are a spherical equilateral or isosceles triangle and a spherical isosceles trapezoid. The combinatorial structure, including the symmetry group of each tiling, is given in Table 1.     

Isosceles Triangles Determined by a Planar Point Set

 It is proved that, for any ɛ>0 and n>n ₀(ɛ), every set of n points in the plane has at most triples that induce isosceles triangles. (Here e denotes the base of the natural logarithm, so the exponent is roughly 2.136.) This easily implies the best currently known lower bound, , for the smallest number of distinct distances determined by n points in the plane, due to Solymosi–Cs. Tóth and Tardos. Received: February, 2002 Final version received: September 15, 2002 RID="*" ID="*" Supported by NSF grant CCR-00-86013, PSC-CUNY Research Award 63382-00-32, and OTKA-T-032452 RID="†" ID="†" Supported by OTKA-T-030059 and AKP 2000-78-21     

矩形域及三角域上Bezier曲面的转换

本文讨论了矩形域及三角域上Ｂｅｚｉｅｒ曲面的相互关系，从几何变换的角度给出了显式的转换公式及几何解释，同时也给出了相应的算法。     

Cutting Polygons Composed of Equal Rectangles into Similar Rectangles

Mathematical Notes - Assume that a polygon is composed of equal rectangles. We find all quadratic irrationalities $$r$$ for which the polygon under study can be cut into similar rectangles with...     

Packing Rectangles into the Unit Square

Every sequence of rectangles of side lengths not greater than 1, whose total area is smaller than or equal to can be packed into the unit square.     

On the Isosceles Orthogonally Exponential Mappings

We prove that if X is a real linear normed space and dim X > 1, then, for every isosceles orthogonally exponential mapping f of X into a division ring, either f(X\{0}) = {0} or 0 ∉ f(X). As a consequence of this fact we obtain the following theorem: If X is not an inner product space and dim X > 2, then every isosceles orthogonally exponential mapping of X into a (commutative) field is exponential. We also generalize some results concerning the orthogonally additive mappings. This revised version was published online in June 2006 with corrections to the Cover Date.     

On Some Rational Triangles

We prove that every set of n ≥ 3 points in mathbbR²{mathbb{R}^2} can be slightly perturbed to a set of n points in mathbbQ²{mathbb{Q}^2} so that at least 3(n − 2) of mutual distances between those new points are rational numbers. Some special rational triangles that are arbitrarily close to a given triangle are also considered. Given a triangle ABC, we show that for each ε > 0 there is a triangle A′B′C′ with rational sides and at least one rational median such that |AA′|, |BB′|, |CC′| < ε and a Heronian triangle A′′B′′C′′ with three rational internal angle bisectors such that A^￠￠, B^￠￠, C^￠￠ ? mathbbQ²{A^{primeprime}, B^{primeprime}, C^{primeprime} in mathbb{Q}^2} and |AA′′|, |BB′′|, |CC′′| < ε.     

关于一类特殊梯形的等面积三角形划分

1970年 Monsky证明了正方形不能划分为奇数个面积相等的三角形 .Stein等人对梯形的等面积三角形划分作了深入的研究 ,得到了大量结果 .本文就未解决的问题作了进一步的讨论 ,即讨论一类特殊梯形的等面积三角形划分问题 .     

On Sequences of Nested Triangles

Summary For a given triangle, we consider several sequences of nested triangles obtained via iterative procedures. We are interested in the limiting behavior of these sequences. We briefly mention the relevant known results and prove that the triangle determined by the feet of the angle bisectors converges in shape towards an equilateral one. This solves a problem raised by Trimble~[5].     

关于任意三角形区域上的Durrmeyer算子

本文将区间[0,1]上的Durrmeyer算子推广到平面上的任意三角形区域中去,并在空间C(T),C^k(T)(k≥1),L_p(T)以及Sobolev空间W_pr(T)中研究了它的收敛性及逼近度估计,这里T是平面上的三角形区域。     

On the Uniqueness of Isosceles Orthogonality in Normed Linear Spaces

Via studying the relation between isosceles orthogonality and the lengths of segments contained in the unit sphere, existing results on the uniqueness of isosceles orthogonality are improved.     

四边形的等积三角剖分

1970年Monsky证明了著名的Richman猜想: 正方形不能剖分成奇数个面积相等的三角形。近年来Stein等人研究一类特殊类型的四边形的等积三角剖分问题，获得了许多重要结果。该文进一步研究四边形等积三角剖分的待解决问题。 