将正方形剖分为全等多边形

'Congruent Partitions' of Polygons - a Short Introduction

We introduce the problem of partitioning 2D regions (usually convex regions) into mutually congruent pieces ('tiles').

The smallest convex $K$-gon containing $N$ congruent disks

Consider the problem of fnding the smallest area convex $k$-gon containing $n\in\mathbb{N}$ congruent disks without an overlap. By using Wegner inequality in sphere packing theory we give a lower bound for the area of such polygons. For several cases where this bound is tight we construct corresponding optimal polygons. We also discuss its solution for some cases where this bound is not tight, e.g. $n = 2$ and $k$ is odd, and $n = 3$; $k = 4$. On the way to prove our results we prove a result on geometric invariants between two polygons whose sides are pairwise parallel, and give a new charact…

论将正多边形剖分为矩形

On Dissecting Polygons into Rectangles

本文研究正 n 边形重排为矩形的最少剖分块数 r(n)，给出 n=3 至 12 的构造上界，并在多数情形改进正方形剖分界。

Colored partitions of a convex polygon by noncrossing diagonals

For any positive integers $a$ and $b$, we enumerate all colored partitions made by noncrossing diagonals of a convex polygon into polygons whose number of sides is congruent to $b$ modulo $a$. For the number of such partitions made by a fixed number of diagonals, we give both a recurrence relation and an explicit representation in terms of partial Bell polynomials. We use basic properties of these polynomials to efficiently incorporate restrictions on the type of polygons allowed in the partitions.

球面由角度全等五边形进行边到边镶嵌

Edge-to-edge Tilings of the Sphere by Angle Congruent Pentagons

研究球面由全等五边形镶嵌时，仅通过角度信息确定镶嵌方式及简化过程对镶嵌特征的影响。

球面的侧向等边曲边三角形镶嵌

Side-to-side Tiling of the Sphere by Congruent Curvilinear Triangles

该研究分类了球面侧向等边曲边三角形镶嵌，发现其本质是直线三角形镶嵌的变形。