Circle packing in an equilateral 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 ...
in
discrete mathematics 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
equilateral triangle
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each othe ...
. Optimal solutions are known for and for any
triangular number
A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots in ...
of circles, and conjectures are available for .
[.]
A conjecture of
Paul Erdős
Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in ...
and Norman Oler states that, if is a triangular number, then the optimal packings of and of circles have the same side length: that is, according to the conjecture, an optimal packing for circles can be found by removing any single circle from the optimal hexagonal packing of circles. This conjecture is now known to be true for .
Minimum solutions for the side length of the triangle:
A closely related problem is to cover the equilateral triangle with a fixed number of equal circles, having as small a radius as possible.
[.]
See also
*
Circle packing in an isosceles right triangle
*
Malfatti circles
In geometry, the Malfatti circles are three circles inside a given triangle such that each circle is tangent to the other two and to two sides of the triangle. They are named after Gian Francesco Malfatti, who made early studies of the problem o ...
, a construction giving the optimal solution for three circles in an equilateral triangle
References
Circle packing
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