Dense Packings of Equal Disks in an Equilateral Triangle: From 22 to 34 and Beyond

Tilings of convex sets by mutually incongruent equilateral triangles contain arbitrarily small tiles

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 tilings of the full plane $\mathbb{R}^2$, which is based on a surprising connection to a random walk on a directed graph.

等边三角形的平铺

Tiling an Equilateral Triangle

本文研究等边三角形被特定形状的三角形平铺时，平铺数量$N$的可能取值，证明$N>3$时$N$不能为素数，并深入探讨瓷砖含$π/3$角的情形。

On the Optimality of Napoleon Triangles

An elementary geometric construction known as Napoleon's theorem produces an equilateral triangle built on the sides of any initial triangle: the centroids of each equilateral triangle meeting the original sides, all outward or all inward, comprise the vertices of the new equilateral triangle. In this note we observe that two Napoleon iterations yield triangles with useful optimality properties. Two inner transformations result in a (degenerate) triangle whose vertices coincide at the original centroid. Two outer transformations yield an equilateral triangle whose vertices are closest to the o…

关于埃尔多斯平方包装猜想

On a square packing conjecture of Erdős

本文证明了埃尔多斯关于正方形包装的猜想等价于某个级数的收敛性，并探讨了正方形、三角形和平行四边形包装的相似性。

Geometrically L^p-optimal lines of vertices of an equilateral triangle

We consider the distances between a line and a set of points in the plane defined by the L^p-norms of the vector consisting of the euclidian distance between the single points and the line. We determine lines with minimal geometric L^p-distance to the vertices of an equilateral triangle for all 1<= p<=\infty. The investigation of the L^p-distances for p\ne 1,2,\infty establishes the passage between the well-known sets of optimal lines for p=1,2,\infty. The set of optimal lines consists of three lines each parallel to one of the triangle sides for 1<= p < 4/3 and 2<p<=\infty and of the three pe…

Monochromatic triangles in two-colored plane

We prove that for any partition of the plane into a closed set $C$ and an open set $O$ and for any configuration $T$ of three points, there is a translated and rotated copy of $T$ contained in $C$ or in $O$. Apart from that, we consider partitions of the plane into two sets whose common boundary is a union of piecewise linear curves. We show that for any such partition and any configuration $T$ which is a vertex set of a non-equilateral triangle there is a copy of $T$ contained in the interior of one of the two partition classes. Furthermore, we give the characterization of these "polygonal" p…

双曲平面中的拿破仑构造

Napoleonic Constructions in the Hyperbolic Plane

本文研究双曲平面中的拿破仑构造问题，证明双曲平面中仅等边三角形具备拿破仑性质，且任意三角形的迭代拿破仑化会逐渐趋近等边并收敛至单点，揭示双曲情形与欧氏、球面情形的显著差异。

On the Multiple Packing Densities of Triangles

Given a convex disk $K$ and a positive integer $k$, let $δ_T^k(K)$ and $δ_L^k(K)$ denote the $k$-fold translative packing density and the $k$-fold lattice packing density of $K$, respectively. Let $T$ be a triangle. In a very recent paper, K. Sriamorn proved that $δ_L^k(T)=\frac{2k^2}{2k+1}$. In this paper, I will show that $δ_T^k(T)=δ_L^k(T)$.

基于相似弦三角形的球上拿破仑三角形

Napoleonic triangles on the sphere from similar chord triangles

该文研究单位球上拿破仑构造的变体，证明非等边球面三角形的外心构成等边三角形的充要条件，得到基于二次曲面与三次不等式的分类，还原特定定理并发现不同β下的解族。

The geometry of Minkowski spaces -- a survey. Part I

We survey elementary results in Minkowski spaces (i.e. finite dimensional Banach spaces) that deserve to be collected together, and give simple proofs for some of them. We place special emphasis on planar results. Many of these results have often been rediscovered as lemmas to other results. In Part I we cover the following topics: The triangle inequality and consequences such as the monotonicity lemma, geometric characterizations of strict convexity, normality (Birkhoff orthogonality), conjugate diameters and Radon curves, equilateral triangles and the affine regular hexagon construction, equ…

Incongruent equipartitions of the plane into quadrangles of equal perimeter

Motivated by a question of R.\ Nandakumar, we show that the Euclidean plane can be dissected into mutually incongruent convex quadrangles of the same area and the same perimeter. As a byproduct we obtain vertex-to-vertex dissections of the plane by mutually incongruent triangles of unit area that are arbitrarily close to the periodic vertex-to-vertex tiling by equilateral triangles.

平面上三角形的多重格填装与覆盖

Multiple Lattice Packings and Coverings of the Plane with Triangles

本文针对平面上三角形的多重格填装与覆盖问题，推导得出其j重格填装密度、覆盖密度公式及对应格的数量表达式，完善了凸圆盘多重格填装与覆盖的相关理论。

Two dynamical systems in the space of triangles

Let $M$ be the space of triangles, defined up to shifts, rotations and dilations. We define two maps $f:M\to M$ and $g:M\to M$. The map $f$ corresponds to a triangle of perimeter $π$ the triangle with angles numerically equal to edges of the initial triangle. The map $g$ corresponds to a triangle of perimeter $2π$ the triangle with \emph{exterior} angles numerically equal to edges of the initial triangle. For $p\in M$ the sequence $\{p,f(p),f(f(p)),\ldots\}$ converges to the equilateral triangle and the sequence $\{p,g(p),g(g(p)),\ldots\}$ converges to the "degenerate triangle" with angles $(0…

Pál不等式的三个定量版本

Three quantitative versions of the Pál inequality

本文研究Pál不等式的稳定性，证明三个定量版本，说明给定宽度平面凸集的面积接近最小值时其形状必接近等边三角形，并得到最小宽度约束下内半径的定量结果。

谢尔宾斯基三角形有多厚

How Thick Is the Sierpiński Triangle?

研究谢尔宾斯基三角形的厚度，证明其冯·吴厚度为√3/6，展示在任意点和尺度下，其邻近点能生成二维区域，且该常数最优。

Triangle Tiling I: The tile is similar to ABC or has a right angle

An N -tiling of triangle ABC by triangle T is a way of writing ABC as a union of N triangles congruent to T, overlapping only at their boundaries. The triangle T is the "tile'". The tile may or may not be similar to ABC . This paper is the first of several papers, which together seek a complete characterization of the triples (ABC,N,T) such that ABC can be N -tiled by T . In this paper, we consider the case in which the tile is similar to ABC, the case in which the tile is a right triangle, and the case when ABC is equilateral. We use (only) techniques from linear algebra and elementary field …

在宽度约束下形状优化：Cheeger常数与扭转刚度

Shape optimization under width constraint: the Cheeger constant and the torsional rigidity

本文证明等边三角形在固定最小宽度下最大化Cheeger常数并最小化扭转刚度，通过与简单形状的比较论证了非等边配置的不优性。

Isogonic and isodynamic points of a simplex in a real affine space

A non-equilateral triangle in a Euclidean plane has exactly two isogonic and two isodynamic points. There are a number of different but equivalent characterizations of these triangle centers. The aim of this paper is to work out characteristic properties of isogonic and isodynamic centers of simplices that can be transferred to higher dimensions. In addition, a geometric description of the Weiszfeld algorithm for calculating the Fermat point of a simplex is given.