The Dense Packing of 13 Congruent Circles in a Circle

The densest packings of n congruent circles in a circle are known for n 12 and n = 19. In this paper we exhibit the densest packings of 13 congruent circles in a circle. We show that the optimal con gurations are identical to Kravitz's [11] conjecture. We use a technique developed from a method of Bateman and Erd}os [1] which proved fruitful in investigating the cases n = 12 and 19 by the author [6, 7] . 1. Preliminaries and Results We shall denote the points of the Euclidean plane E by capitals, sets of points by script capitals, and the distance of two points by d(P;Q). We use PQ for the line through P , Q, and PQ for the segment with endpoints P , Q. \POQ denotes the angle determined by the three points P ,O,Q in this order. C(r) means the closed disc of radius r with center O. By an annulus r < s we mean all points P such that r < d(P;O) s. We utilize the linear structure of E by identifying each point P with the vector ~ OP , where O is the origin. For a point P and a vector ~a by P + ~a we always mean the vector ~ OP + ~a. The problem of nding the densest packing of congruent circles in a circle arose in the 1960s. The question was to nd the smallest circle in which we can pack n congruent unit circles, or equivalently, the smallest circle in which we can place n points with mutual distances at least 1. Dense circle packings were rst given by Kravitz [11] for n = 2; : : : ; 16. Pirl [14] proved that the these arrangements are optimal for n 9 and he also found the optimal con guration for n = 10. Pirl also conjectured dense con gurations for 11 n 19. For n 6 proofs were given independently by Graham [3]. A proof for n = 6 and 7 was also given by Crilly and Suen [4]. Subsequent improvements were presented by Goldberg [8] for n = 14; 16 and 17. He also found a new packing with 20 circles. In 1975 Reis [15] used a mechanical argument to generate remarkably good packings up to 25 circles. Recently, Graham et al. [9, 10] using computers established packings with more than 100 circles and improved the packing of 25 circles. In 1994 Melissen [12] proved Pirl's conjecture for n = 11 and the author [6, 7] proved it for n = 19 and n = 12. The problem of nding the densest packing of equal circles in a circle is also mentioned as an unsolved problem in the book of Croft, Falconer and Guy [5]. Packings of congruent circles in hyperbolic plane were treated by K. Bezdek [2]. Analogous results of packing n equal circles in an equilateral triangle and square can be traced down in the doctoral dissertation of Melissen [13]. 1991 Mathematics Subject Classi cation. Primary52C15.