Circle packing in an isosceles right triangle

{{Short description|Two-dimensional packing problem}}

Circle packing in a right isosceles triangle is a packing problem where the objective is to pack {{mvar|n}} unit circles into the smallest possible isosceles right triangle.

Minimum solutions (lengths shown are length of leg) are shown in the table below.{{cite web|url=http://hydra.nat.uni-magdeburg.de/packing/crt/crt.html|title=The best known packings of equal circles in an isosceles right triangle|first=Eckard|last=Specht|date=2011-03-11|access-date=2011-05-01}} Solutions to the equivalent problem of maximizing the minimum distance between {{mvar|n}} points in an isosceles right triangle, were known to be optimal for {{math|n < 8}}{{Cite journal | last1 = Xu | first1 = Y. | title = On the minimum distance determined by n (≤ 7) points in an isoscele right triangle | doi = 10.1007/BF02007736 | journal = Acta Mathematicae Applicatae Sinica | volume = 12 | issue = 2 | pages = 169–175 | year = 1996 | s2cid = 189916723 }} and were extended up to {{math|1=n = 10}}.{{cite thesis |last=Harayama |first=Tomohiro |date=2000 |title=Optimal Packings of 8, 9, and 10 Equal Circles in an Isosceles Right Triangle |publisher=Japan Advanced Institute of Science and Technology|hdl=10119/1422 }}

In 2011 a heuristic algorithm found 18 improvements on previously known optima, the smallest of which was for {{math|1=n = 13}}.{{Cite journal | last1 = López | first1 = C. O. | last2 = Beasley | first2 = J. E. | title = A heuristic for the circle packing problem with a variety of containers | doi = 10.1016/j.ejor.2011.04.024 | journal = European Journal of Operational Research | volume = 214 | issue = 3 | pages = 512 | year = 2011 }}

class="wikitable"
Number of circles ! Length
1 | $2\; +\; \backslash sqrt\; \{2\}$ = 3.414...
2 | $2\; \backslash sqrt\; \{2\}$ = 4.828...
3 | $4\; +\; \backslash sqrt\; \{2\}$ = 5.414...
4 | $2\; +\; 3\backslash sqrt\; \{2\}$ = 6.242...
5 | $4\; +\; \backslash sqrt\; \{2\}\; +\; \backslash sqrt\{3\}$ = 7.146...
6 | $6\; +\; \backslash sqrt\; \{2\}$ = 7.414... 120x120px
7 | $4\; +\; \backslash sqrt\; \{2\}\; +\; \backslash sqrt\; \{2\; +\; 4\; \backslash sqrt\{2\}\}$ = 8.181...
8 | $2\; +\; 3\; \backslash sqrt\; \{2\}\; +\; \backslash sqrt\{6\}$ = 8.692...
9 | $2\; +\; 5\; \backslash sqrt\; \{2\}$ = 9.071...
10 | $8\; +\; \backslash sqrt\; \{2\}$ = 9.414...
11 | $5\; +\; 3\; \backslash sqrt\; \{2\}\; +\; \backslash dfrac\; \{1\}\; \{3\}\; \backslash sqrt\; \{6\}$ = 10.059...
12 | 10.422...
13 | 10.798...
14 | $2\; +\; 3\; \backslash sqrt\; \{2\}\; +\; 2\; \backslash sqrt\{6\}$ = 11.141...
15 | $10\; +\; \backslash sqrt\; \{2\}$ = 11.414...