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• Circle Packings and the Sacred Lotus
Abstract

How must n non-overlapping equal circles be packed in a given circle so that the diameter of the circles will be as large as possible? This paper presents an account of this problem and its putative solutions and related configurations in lotus receptacles, classical Japanese mathematics (wasan) and traditional Japanese design. Particular emphasis is placed on the connection between the conjectural solutions of this discrete geometrical problem and the fruit arrangements in the receptacles of lotuses, because in most cases the actual fruit arrangements are identical to the mathematical solutions. As the lotus is an important symbol in Buddhism and lotus decorations are quite common in Japanese Buddhist art, packings of circles in a circle have been represented in Japanese art for centuries.

There are many practical situations in which close packing of a given number of equal discs can be achieved inside a larger circle. The best packings are intensively studied via rapidly developing computer-aided methods. In a recent paper [1] we showed how diverse the use of circle packings in a circle can be in science and art. We presented examples mainly from Western culture. We have since found, however, that circle packings in a circle also play an important role in Eastern culture and in particular in Japan.

Interestingly enough, the closest packings for small numbers of equal circles in a circle can be seen in books of classical Japanese mathematics, which is called wasan[2], and collections of Japanese family crests, called kamon[3], made long ago. Recently it was discovered that the carpels in the receptacles of lotus flowers are arranged in accordance with close packing [4]. This property is revealed even more clearly in the arrangement of lotus fruits in the receptacle. Because the lotus is an important flower in Japanese culture and an important symbol in Buddhism, numerous representations of lotus flowers are found in Japanese Buddhist art and design. So, packings of circles in a circle as decorations have been present in Buddhist art for a very long time.

The aims of this paper are (1) to make a brief comparison between lotus fruit arrangements and the closest packings of equal circles in a circle; (2) to present the related configurations in wasan and kamon design; and (3) to show examples of lotus flower representations in Japanese Buddhist art. In conclusion we put forward a proposition about the contribution of the geometric configurations of lotus receptacles to the conception of certain shapes seen in Japanese culture.

# Densest Packings

The design of ropes and cables has raised the following mathematical packing problem, which has become one of the classical problems of discrete geometry [5]: How must n non-overlapping equal circles be packed in a given circle so that the diameter of the circles will be as large as possible? The density of the packing is defined as the ratio of the total area of the circles to the area of the given circle. For the solution of this packing problem, both the diameter of the circles and the density are maximal, and their corresponding arrangement is called the optimum packing or the densest packing.

Proven solutions of the problem of densest packing of n non-overlapping equal circles in the unit circle have been established for up to n = 10 by Pirl [6], for n = 11 by Melissen [7] and for n = 12, 13 and

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Fig. 1.

Proven (n = 2 to 13 and n = 19) and conjectured (n = 14 to 18 and n = 20) best packings of up to 20 equal circles in a circle. Properties of packings are summarized in Table 1. (Reproduced from R.L. Graham, B.D. Lubachevsky, K.J. Nurmela and P.R.J. Östergard, "Dense Packings of Congruent Circles in a Circle," Discrete Mathematics 181 [1998] pp. 139-154, with permission from Elsevier Science B.V.)

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Table 1.

The best packings of up to 20 circles in a circle, characterized by the number n of circles, radius ratio r...