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... File:6 cirkloj en 45 45 90 triangulo.png
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	$5 + 3 \sqrt{2} + \frac{1}{3} \sqrt{6}$ = 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.

Stub icon

This elementary geometry-related article is a stub. You can help Wikipedia by expanding it.

[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.

[1]

[2]

[3]

[4]

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

Circle packing in an isosceles right triangle

References

Navigation menu

Page actions

Page actions

Personal tools

Navigation

Search

Tools

In other projects

In other languages