Circle packing in an equilateral triangle
Encyclopedia
Circle packing in an equilateral triangle is a packing problem
in discrete mathematics
where the objective is to pack n unit circles into the smallest possible equilateral triangle. Optimal solutions are known for n < 13 and for any triangular number
of circles, and conjectures are available for n < 28.
A conjecture of Paul Erdős
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 given number of circles, having as small a radius as possible.
Packing problem
Packing problems are a class of optimization problems in mathematics which involve attempting to pack objects together , as densely as possible. Many of these problems can be related to real life packaging, storage and transportation issues...
in discrete mathematics
Discrete mathematics
Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not...
where the objective is to pack n unit circles into the smallest possible equilateral triangle. Optimal solutions are known for n < 13 and for any triangular number
Triangular number
A triangular number or triangle number numbers the objects that can form an equilateral triangle, as in the diagram on the right. The nth triangle number is the number of dots in a triangle with n dots on a side; it is the sum of the n natural numbers from 1 to n...
of circles, and conjectures are available for n < 28.
A conjecture of Paul Erdős
Paul Erdos
Paul Erdős was a Hungarian mathematician. Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. He worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory...
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:
| Number of circles | Length |
|---|---|
| 1 | 3.464... |
| 2 | 5.464... |
| 3 | 5.464... |
| 4 | 6.928... |
| 5 | 7.464... |
| 6 | 7.464... |
| 7 | 8.928... |
| 8 | 9.293... |
| 9 | 9.464... |
| 10 | 9.464... |
| 11 | 10.730... |
| 12 | 10.928... |
| 13 | 11.406... |
| 14 | 11.464... |
| 15 | 11.464... |
A closely related problem is to cover the equilateral triangle with a given number of circles, having as small a radius as possible.
See also
- Circle packing in an isosceles right triangleCircle packing in an isosceles right triangleCircle packing in a right isosceles triangle is a packing problem where the objective is to pack n unit circles into the smallest possible isosceles right triangle.Minimum solutions are shown in the table below...
- Malfatti circlesMalfatti circlesIn 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...
, a construction giving the optimal solution for three circles in an equilateral triangle