Circle packing in an isosceles right triangle

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Circle packing in a right isosceles triangle is a packing problem where the objective is to pack n unit circles into the smallest possible isosceles right triangle.

Minimum solutions (lengths shown are length of leg) are shown in the table below.[1] Solutions to the equivalent problem of maximizing the minimum distance between n points in an isosceles right triangle, are known to be optimal for n< 8.[2] In 2011 a heuristic algorithm found 18 improvements on previously known optima, the smallest of which was for n=13.[3]

Number of circles Length
1 3.414...
2 4.828...
3 5.414...
4 6.242...
5 7.146...
6 7.414... 6 cirkloj en 45 45 90 triangulo.png
7 8.181...
8 8.692...
9 9.071...
10 9.414...
11 10.059...
12 10.422...
13 10.798...
14 11.141...
15 11.414...

References[edit]

  1. ^ Specht, Eckard (2011-03-11). "The best known packings of equal circles in an isosceles right triangle". Retrieved 2011-05-01. 
  2. ^ Xu, Y. (1996). "On the minimum distance determined by n (≤ 7) points in an isoscele right triangle". Acta Mathematicae Applicatae Sinica 12 (2): 169–175. doi:10.1007/BF02007736. 
  3. ^ López, C. O.; Beasley, J. E. (2011). "A heuristic for the circle packing problem with a variety of containers". European Journal of Operational Research 214 (3): 512. doi:10.1016/j.ejor.2011.04.024.