Circle packing in a right isosceles triangle is a
packing problem
Packing problems are a class of optimization problems in mathematics that involve attempting to pack objects together into containers. The goal is to either pack a single container as densely as possible or pack all objects using as few conta ...
where the objective is to pack
unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...
s into the smallest possible
isosceles right triangle
A special right triangle is a right triangle with some regular feature that makes calculations on the triangle easier, or for which simple formulas exist. For example, a right triangle may have angles that form simple relationships, such as 45° ...
.
Minimum solutions (lengths shown are length of leg) are shown in the table below. Solutions to the equivalent problem of maximizing the minimum distance between
points
Point or points may refer to:
Places
* Point, Lewis, a peninsula in the Outer Hebrides, Scotland
* Point, Texas, a city in Rains County, Texas, United States
* Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland
* Point ...
in an isosceles right triangle, were known to be
optimal
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
for and were extended up to .
In 2011 a
heuristic algorithm
In mathematical optimization and computer science, heuristic (from Greek εὑρίσκω "I find, discover") is a technique designed for solving a problem more quickly when classic methods are too slow for finding an approximate solution, or whe ...
found 18 improvements on previously known optima, the smallest of which was for .
References
Circle packing
{{elementary-geometry-stub