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Circle packing in an equilateral triangle
Two-dimensional packing problem From Wikipedia, the free encyclopedia
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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.[1][2][3][4]
Unsolved problem in mathematics
What is the smallest possible equilateral triangle which an amount n of unit circles can be packed into?
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.[5] This conjecture is now known to be true for n ≤ 15.[6] 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[3]
Minimum solutions for the side length of the triangle:[1]
A closely related problem is to cover the equilateral triangle with a fixed number of equal circles, having as small a radius as possible.[7]
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See also
- Circle packing in an isosceles right triangle
- Malfatti circles, three circles of possibly unequal sizes packed into a triangle
References
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