On the minimum diameter of plane integral point sets

Since ancient times mathematicians consider geometrical objects with integral side lengths. We consider plane integral point sets $\mathcal{P}$, which are sets of $n$ points in the plane with pairwise integral distances where not all the points are collinear. The largest occurring distance is called its diameter. Naturally the question about the minimum possible diameter $d(2,n)$ of a plane integral point set consisting of $n$ points arises. We give some new exact values and describe state-of-the-art algorithms to obtain them. It turns out that plane integral point sets with minimum diameter consist very likely of subsets with many collinear points. For this special kind of point sets we prove a lower bound for $d(2,n)$ achieving the known upper bound $n^{c_2\log\log n}$ up to a constant in the exponent. A famous question of Erd\H{o}s asks for plane integral point sets with no 3 points on a line and no 4 points on a circle. Here, we talk of point sets in general position and denote the corresponding minimum diameter by $\dot{d}(2,n)$. Recently $\dot{d}(2,7)=22 270$ could be determined via an exhaustive search.Comment: 12 pages, 5 figure

There are integral heptagons, no three points on a line, no four on a circle

We give two configurations of seven points in the plane, no three points in a line, no four points on a circle with pairwise integral distances. This answers a famous question of Paul Erd\H{o}s.Comment: 4 pages, 1 figur

On the characteristic of integral point sets in Em\mathbb{E}^m

We generalise the definition of the characteristic of an integral triangle to integral simplices and prove that each simplex in an integral point set has the same characteristic. This theorem is used for an efficient construction algorithm for integral point sets. Using this algorithm we are able to provide new exact values for the minimum diameter of integral point sets.Comment: 9 pages, 1 figur

Constructing 77-clusters

A set of $n$-lattice points in the plane, no three on a line and no four on a circle, such that all pairwise distances and all coordinates are integral is called an $n$-cluster (in $\mathbb{R}^2$). We determine the smallest existent $7$-cluster with respect to its diameter. Additionally we provide a toolbox of algorithms which allowed us to computationally locate over 1000 different $7$-clusters, some of them having huge integer edge lengths. On the way, we exhaustively determined all Heronian triangles with largest edge length up to $6\cdot 10^6$.Comment: 18 pages, 2 figures, 2 table

Enumeration of integral tetrahedra

We determine the numbers of integral tetrahedra with diameter $d$ up to isomorphism for all $d\le 1000$ via computer enumeration. Therefore we give an algorithm that enumerates the integral tetrahedra with diameter at most $d$ in $O(d^5)$ time and an algorithm that can check the canonicity of a given integral tetrahedron with at most 6 integer comparisons. For the number of isomorphism classes of integral $4\times 4$ matrices with diameter $d$ fulfilling the triangle inequalities we derive an exact formula.Comment: 10 pages, 1 figur

Squarepants in a Tree: Sum of Subtree Clustering and Hyperbolic Pants Decomposition

We provide efficient constant factor approximation algorithms for the problems of finding a hierarchical clustering of a point set in any metric space, minimizing the sum of minimimum spanning tree lengths within each cluster, and in the hyperbolic or Euclidean planes, minimizing the sum of cluster perimeters. Our algorithms for the hyperbolic and Euclidean planes can also be used to provide a pants decomposition, that is, a set of disjoint simple closed curves partitioning the plane minus the input points into subsets with exactly three boundary components, with approximately minimum total length. In the Euclidean case, these curves are squares; in the hyperbolic case, they combine our Euclidean square pants decomposition with our tree clustering method for general metric spaces.Comment: 22 pages, 14 figures. This version replaces the proof of what is now Lemma 5.2, as the previous proof was erroneou

On the generation of Heronian triangles

We describe several algorithms for the generation of integer Heronian triangles with diameter at most $n$. Two of them have running time $\mathcal{O}\left(n^{2+\varepsilon}\right)$. We enumerate all integer Heronian triangles for $n\le 600000$ and apply the complete list on some related problems.Comment: 10 pages, 2 figures, 2 tables. http://sci-gems.math.bas.bg/jspui/handle/10525/38