Packing congruent hyperspheres into a hypersphere

Abstract

The paper considers a problem of packing the maximal number of congruent nD hyperspheres of given radius into a larger nD hypersphere of given radius where n = 2, 3, . . . , 24. Solving the problem is reduced to solving a sequence of packing subproblems provided that radii of hyperspheres are variable. Mathematical models of the subproblems are constructed. Characteristics of the mathematical models are investigated. On the ground of the characteristics we offer a solution approach. For n ≤ 3 starting points are generated either in accordance with the lattice packing of circles and spheres or in a random way. For n > 3 starting points are generated in a random way. A procedure of perturbation of lattice packings is applied to improve convergence. We use the Zoutendijk feasible direction method to search for local maxima of the subproblems. To compute an approximation to a global maximum of the problem we realize a non-exhaustive search of local maxima. Our results are compared with the benchmark results for n = 2. A number of numerical results for 2 ≤ n ≤ 24 are given.

Packing circles within ellipses

“…Every circle must be inside the ellipse and at a distance not smaller than r to the ellipse's border. The largest absolute violation of this limit is given by (27). In fact, the exact calculation of the distance to the ellipse's border in (27) depends on the fulfillment of h ξ i (u, v, s) = 0 in (19).…”

“…The largest absolute violation of this limit is given by (27). In fact, the exact calculation of the distance to the ellipse's border in (27) depends on the fulfillment of h ξ i (u, v, s) = 0 in (19). Therefore, the expression in (27) is an approximation of the maximum violation of the ellipse's border.…”

“…In fact, the exact calculation of the distance to the ellipse's border in (27) depends on the fulfillment of h ξ i (u, v, s) = 0 in (19). Therefore, the expression in (27) is an approximation of the maximum violation of the ellipse's border. Table 7 summarizes these results pointing to the illustration of each solution within the text.…”

“…This lattice is the densest 2D lattice and non-lattice packing in the (infinite) plane with density 1 6 π √ 3 ≈ 0.9068996821 [30]. In a similar context, lattices were also considered in [27] to generate initial guesses for an optimization procedure.…”

Packing ellipsoids by nonlinear optimization

“…In particular, many works have tackled the problem with optimization tools. See, for example, [7,9,17,18,19,23,37,50,51,52,53] and the references therein. On the other hand, the problem of packing ellipsoids has received more attention only in the past few years.…”