Circle packing in an equilateral triangle is a packing problem in discrete mathematics where the objective is to pack unit circles into the smallest possible equilateral triangle. Optimal solutions are known for and for any triangular number of circles, and conjectures are available for .[1] [2] [3]
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.[4] This conjecture is now known to be true for .[5]
Minimum solutions for the side length of the triangle:[1]
| Number of circles | Triangle number | Length | Area | Figure | |
|---|---|---|---|---|---|
| 1 | Yes | 2\sqrt{3} | 5.196... | ||
| 2 | 2+2\sqrt{3} | 12.928... | |||
| 3 | Yes | 2+2\sqrt{3} | 12.928... | ||
| 4 | 4\sqrt{3} | 20.784... | |||
| 5 | 4+2\sqrt{3} | 24.124... | |||
| 6 | Yes | 4+2\sqrt{3} | 24.124... | ||
| 7 | 2+4\sqrt{3} | 34.516... | |||
| 8 | 2+2\sqrt{3}+\tfrac{2}{3}\sqrt{33} | 37.401... | |||
| 9 | 6+2\sqrt{3} | 38.784... | |||
| 10 | Yes | 6+2\sqrt{3} | 38.784... | ||
| 11 | 4+2\sqrt{3}+\tfrac{4}{3}\sqrt{6} | 49.854... | |||
| 12 | 4+4\sqrt{3} | 51.712... | |||
| 13 | 4+\tfrac{10}{3}\sqrt{3}+\tfrac{2}{3}\sqrt{6} | 56.338... | |||
| 14 | 8+2\sqrt{3} | 56.908... | |||
| 15 | Yes | 8+2\sqrt{3} | 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.[6]