93,058 research outputs found
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 , which are
sets of 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 of a plane integral point set
consisting of 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 achieving the known upper bound 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
. Recently 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
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 -clusters
A set of -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 -cluster (in ). We determine the smallest existent
-cluster with respect to its diameter. Additionally we provide a toolbox of
algorithms which allowed us to computationally locate over 1000 different
-clusters, some of them having huge integer edge lengths. On the way, we
exhaustively determined all Heronian triangles with largest edge length up to
.Comment: 18 pages, 2 figures, 2 table
Enumeration of integral tetrahedra
We determine the numbers of integral tetrahedra with diameter up to
isomorphism for all via computer enumeration. Therefore we give an
algorithm that enumerates the integral tetrahedra with diameter at most in
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 matrices with diameter
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 . Two of them have running time
. We enumerate all integer Heronian
triangles for 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
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