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 have been proved for n ≤ 15, and for any triangular number of circles, and conjectures are available for n ≤ 34.
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. This conjecture is now known to be true for n ≤ 15. In a paper by Graham and Lubachevsky concerning solutions for 22 ≤ n ≤ 34 they also conjectured seven infinite families of optimal solutions in addition to the one by Erdős and Oler. These families give conjectured solutions for many more numbers, including n = 37, 40, 42, 43, 46, 49.
Minimum solutions for the side length of the triangle:
| Number of circles | Triangle number | Length | Area | Figure |
|---|---|---|---|---|
| 1 | Yes | = 3.464... | 5.196... | |
| 2 | = 5.464... | 12.928... | ||
| 3 | Yes | = 5.464... | 12.928... | |
| 4 | = 6.928... | 20.784... | ||
| 5 | = 7.464... | 24.124... | ||
| 6 | Yes | = 7.464... | 24.124... | |
| 7 | = 8.928... | 34.516... | ||
| 8 | = 9.293... | 37.401... | ||
| 9 | = 9.464... | 38.784... | ||
| 10 | Yes | = 9.464... | 38.784... | |
| 11 | = 10.730... | 49.854... | ||
| 12 | = 10.928... | 51.712... | ||
| 13 | = 11.406... | 56.338... | ||
| 14 | = 11.464... | 56.908... | ||
| 15 | Yes | = 11.464... | 56.908... |
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, three circles of possibly unequal sizes packed into a triangle
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