Circle packing in an isosceles right triangle

In this article, we will explore the impact of Circle packing in an isosceles right triangle on today's society. Since its emergence, Circle packing in an isosceles right triangle has captured the attention of people around the world, generating passionate and emotional discussions. Over the years, Circle packing in an isosceles right triangle has evolved and adapted to changes in society, influencing different aspects of daily life. Through this analysis, we will examine the role Circle packing in an isosceles right triangle has played in culture, politics, economics, and other areas, as well as its influence on the way people perceive the world around them. With a critical and reflective look, we will seek to better understand the impact that Circle packing in an isosceles right triangle has had and continues to have on our society.

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	${\displaystyle 2+{\sqrt {2}}}$ = 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.

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