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, were known to be optimal for n < 8[2] and were extended up to n = 10.[3]
In 2011 a heuristic algorithm found 18 improvements on previously known optima, the smallest of which was for n = 13.[4]
| Number of circles | Length |
|---|---|
| 1 | = 3.414... |
| 2 | = 4.828... |
| 3 | = 5.414... |
| 4 | = 6.242... |
| 5 | = 7.146... |
| 6 | = 7.414...  |
| 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
- Specht, Eckard (2011-03-11). "The best known packings of equal circles in an isosceles right triangle". Retrieved 2011-05-01.
- 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.
- Harayama, Tomohiro (2000). Optimal Packings of 8, 9, and 10 Equal Circles in an Isosceles Right Triangle (Thesis). Japan Advanced Institute of Science and Technology. hdl:10119/1422.
- 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.
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