Circle packing in an isosceles right triangle
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Short description: Two-dimensional packing problem
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 | [math]\displaystyle{ 2 + \sqrt {2} }[/math] = 3.414... |
| 2 | [math]\displaystyle{ 2 \sqrt {2} }[/math] = 4.828... |
| 3 | [math]\displaystyle{ 4 + \sqrt {2} }[/math] = 5.414... |
| 4 | [math]\displaystyle{ 2 + 3\sqrt {2} }[/math] = 6.242... |
| 5 | [math]\displaystyle{ 4 + \sqrt {2} + \sqrt{3} }[/math] = 7.146... |
| 6 | [math]\displaystyle{ 6 + \sqrt {2} }[/math] = 7.414... 
|
| 7 | [math]\displaystyle{ 4 + \sqrt {2} + \sqrt {2 + 4 \sqrt{2}} }[/math] = 8.181... |
| 8 | [math]\displaystyle{ 2 + 3 \sqrt {2} + \sqrt{6} }[/math] = 8.692... |
| 9 | [math]\displaystyle{ 2 + 5 \sqrt {2} }[/math] = 9.071... |
| 10 | [math]\displaystyle{ 8 + \sqrt {2} }[/math] = 9.414... |
| 11 | [math]\displaystyle{ 5 + 3 \sqrt {2} + \dfrac {1} {3} \sqrt {6} }[/math] = 10.059... |
| 12 | 10.422... |
| 13 | 10.798... |
| 14 | [math]\displaystyle{ 2 + 3 \sqrt {2} + 2 \sqrt{6} }[/math] = 11.141... |
| 15 | [math]\displaystyle{ 10 + \sqrt {2} }[/math] = 11.414... |
References
- ↑ Specht, Eckard (2011-03-11). "The best known packings of equal circles in an isosceles right triangle". http://hydra.nat.uni-magdeburg.de/packing/crt/crt.html.
- ↑ 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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