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	$2+{\sqrt {2}}$ = 3.414...
2	$2{\sqrt {2}}$ = 4.828...
3	$4+{\sqrt {2}}$ = 5.414...
4	$2+3{\sqrt {2}}$ = 6.242...
5	$4+{\sqrt {2}}+{\sqrt {3}}$ = 7.146...
6	$6+{\sqrt {2}}$ = 7.414...
7	$4+{\sqrt {2}}+{\sqrt {2+4{\sqrt {2}}}}$ = 8.181...
8	$2+3{\sqrt {2}}+{\sqrt {6}}$ = 8.692...
9	$2+5{\sqrt {2}}$ = 9.071...
10	$8+{\sqrt {2}}$ = 9.414...
11	$5+3{\sqrt {2}}+{\dfrac {1}{3}}{\sqrt {6}}$ = 10.059...
12	10.422...
13	10.798...
14	$2+3{\sqrt {2}}+2{\sqrt {6}}$ = 11.141...
15	$10+{\sqrt {2}}$ = 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. S2CID 189916723.
^ 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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[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.

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v t e Packing problems
Abstract packing	Bin Set
Circle packing	In a circle / equilateral triangle / isosceles right triangle / square Apollonian gasket Circle packing theorem Tammes problem (on sphere)
Sphere packing	Apollonian Finite In a sphere In a cube In a cylinder Close-packing Kissing number Sphere-packing (Hamming) bound
Other 2-D packing	Square packing
Other 3-D packing	Tetrahedron Ellipsoid
Puzzles	Conway Slothouber–Graatsma