Related papers: On inside-out Dissections of Polygons and Polyhedr…

We study the problem of partitioning a given simple polygon $P$ into a minimum number of connected polygonal pieces, each of bounded size. We describe a general technique for constructing such partitions that works for several notions of…

This article gives a short proof that all ideal polygons admit a short orthogeodesic decomposition. Specifically, all $n$-gons admit an orthogeodesic decomposition with orthogeodesics all of length at most $\sim 2 \log(n)$, and this is…

We study the geometry of billiard orbits on rectangular billiards. A truncated billiard orbit induces a partition of the rectangle into polygons. We prove that thirteen is a sharp upper bound for the number of different areas of these…

We prove that two polygons $A$ and $B$ have a reversible hinged dissection (a chain hinged dissection that reverses inside and outside boundaries when folding between $A$ and $B$) if and only if $A$ and $B$ are two noncrossing nets of a…

We study the problems of covering or partitioning a polygon $P$ (possibly with holes) using a minimum number of small pieces, where a small piece is a connected sub-polygon contained in an axis-aligned unit square. For covering, we seek to…

Counting interior-disjoint empty convex polygons in a point set is a typical Erd\H{o}s-Szekeres-type problem. We study this problem for 4-gons. Let $P$ be a set of $n$ points in the plane and in general position. A subset $Q$ of $P$, with…

It is well-known that every isosceles tetrahedron (disphenoid) admits infinitely many simple closed geodesics on its surface. They can be naturally enumerated by pairs of co-prime integers $n > m > 1$ with two additional cases $(1,0)$ and…

We prove that, given a polyhedron $\mathcal P$ in $\mathbb{R}^3$, every point in $\mathbb R^3$ that does not see any vertex of $\mathcal P$ must see eight or more edges of $\mathcal P$, and this bound is tight. More generally, this remains…

In this work, we show the geometric properties of a family of polyhedra obtained by folding a regular tetrahedron along regular triangular grids. Each polyhedron is identified by a pair of nonnegative integers. The polyhedron can be cut…

In this paper outer, or dual, billiards outside regular polygons are studied; in particular, periodic points for cases of strictly convex "tables" and for regular n-gons with n = 3,4,6,8,12 are discussed. The main results of the paper are:…

We give a short proof for a formula for the number of divisions of a convex (sn+2)-gon along non-crossing diagonals into (sj+2)-gons, where 1<=j<=n-1. In other words, we consider dissections of an (sn+2)-gon into pieces which can be further…

This paper proposed a method to judge whether the point is inside or outside of the simple convex polygon by the intersection of the vertical line. It determined the point to an area enclosed by two straight lines, then convert the problem…

A polygon is \textit{small} if it has unit diameter. The maximal area of a small polygon with a fixed number of sides $n$ is not known when $n$ is even and $n\geq14$. We determine an improved lower bound for the maximal area of a small…

We address the question: Given a positive integer $N$, can any 2D convex polygonal region be partitioned into $N$ convex pieces such that all pieces have the same area and same perimeter? The answer to this question is easily `yes' for…

We consider the combinatorial question of how many convex polygons can be made by using the edges taken from a fixed triangulation of n vertices. For general triangulations, there can be exponentially many: we show a construction that has…

A planar set $P$ is said to be cover-decomposable if there is a constant $k=k(P)$ such that every $k$-fold covering of the plane with translates of $P$ can be decomposed into two coverings. It is known that open convex polygons are…

We pose and answer several questions concerning the number of ways to fold a polygon to a polytope, and how many polytopes can be obtained from one polygon; and the analogous questions for unfolding polytopes to polygons. Our answers are,…

What is the maximum number of intersections of the boundaries of a simple $m$-gon and a simple $n$-gon, assuming general position? This is a basic question in combinatorial geometry, and the answer is easy if at least one of $m$ and $n$ is…

An orthogonal polygon is called an ortho-unit polygon if its vertices have integer coordinates, and all of its edges have length one. In this paper we prove that any ortho-unit polygon with $n \geq 12$ vertices can be guarded with at most…

We investigate the decomposition problem of balls into finitely many congruent pieces in dimension $d=2k$. In addition, we prove that the $d$ dimensional unit ball $B_d$ can be divided into finitely many congruent pieces if $d=4$ or $d\ge…