Circle packing in an equilateral triangle

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Unsolved problem in mathematics:
What is the smallest possible equilateral triangle which an amount n of unit circles can be packed into?

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
Triangle
number
LengthAreaFigure
1Yes = 3.464...5.196...
CircleInEquilateralTrianglePacking(1).png
2 = 5.464...12.928...
Circle packing in equilateral triangle for 2 circles.png
3Yes = 5.464...12.928...
Circle packing in equilateral triangle for 3 circles.png
4 = 6.928...20.784...
Circle packing in equilateral triangle for 4 circles.png
5 = 7.464...24.124...
Circle packing in equilateral triangle for 5 circles.png
6Yes = 7.464...24.124...
Circle packing in equilateral triangle for 6.png
7 = 8.928...34.516...
Circle packing in equilateral triangle for 7 circles.png
8 = 9.293...37.401...
Circle packing in equilateral triangle for 8 circles.png
9 = 9.464...38.784...
Circle packing in equilateral triangle for 9 circles.png
10Yes = 9.464...38.784...
Circle packing in equilateral triangle for 10 circles.png
11 = 10.730...49.854...
Ircle packing in equilateral triangle for 11 circles.png
12 = 10.928...51.712...
Circle packing in equilateral triangle for 12 circles.png
13 = 11.406...56.338... Circle packing in equilateral triangle for 13 circles.png
14 = 11.464...56.908... Circle packing in equilateral triangle for 14 circles.png
15Yes = 11.464...56.908... Circle packing in equilateral triangle for 15 circles.png

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]

See also

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References

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