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	${\displaystyle 2+{\sqrt {2}}}$ = 3.414...
2	${\displaystyle 2{\sqrt {2}}}$ = 4.828...
3	${\displaystyle 4+{\sqrt {2}}}$ = 5.414...
4	${\displaystyle 2+3{\sqrt {2}}}$ = 6.242...
5	${\displaystyle 4+{\sqrt {2}}+{\sqrt {3}}}$ = 7.146...
6	${\displaystyle 6+{\sqrt {2}}}$ = 7.414...
7	${\displaystyle 4+{\sqrt {2}}+{\sqrt {2+4{\sqrt {2}}}}}$ = 8.181...
8	${\displaystyle 2+3{\sqrt {2}}+{\sqrt {6}}}$ = 8.692...
9	${\displaystyle 2+5{\sqrt {2}}}$ = 9.071...
10	${\displaystyle 8+{\sqrt {2}}}$ = 9.414...
11	${\displaystyle 5+3{\sqrt {2}}+{\dfrac {1}{3}}{\sqrt {6}}}$ = 10.059...
12	10.422...
13	10.798...
14	${\displaystyle 2+3{\sqrt {2}}+2{\sqrt {6}}}$ = 11.141...
15	${\displaystyle 10+{\sqrt {2}}}$ = 11.414...

References

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. S2CID 189916723.
3. 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.
4. 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.