Circle packing in an equilateral triangle
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.[1][2][3]
A conjecture of Paul Erdős and Norman Oler states that, if n is a triangular number, then the optimal packings of n − 1 and of n circles have the same side length: that is, according to the conjecture, an optimal packing for n − 1 circles can be found by removing any single circle from the optimal hexagonal packing of n circles.[4] This conjecture is now known to be true for n ≤ 15.[5]
Minimum solutions for the side length of the triangle:[1]
| Number of circles | Length |
|---|---|
| 1 | 3.464... |
| 2 | 5.464... |
| 3 | 5.464... |
| 4 | 6.928... 120x120px |
| 5 | 7.464...  130x130px |
| 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.[6]
See also
- Circle packing in an isosceles right triangle
- Malfatti circles, a construction giving the optimal solution for three circles in an equilateral triangle
References
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