Circle packing in an isosceles right triangle

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]

References

1. ^ Specht, Eckard (2011-03-11). "The best known packings of equal circles in an isosceles right triangle". http://hydra.nat.uni-magdeburg.de/packing/crt/crt.html. 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.  edit
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. doi:10.1016/j.ejor.2011.04.024.  edit