313 research outputs found

    Highly saturated packings and reduced coverings

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    We introduce and study certain notions which might serve as substitutes for maximum density packings and minimum density coverings. A body is a compact connected set which is the closure of its interior. A packing P\cal P with congruent replicas of a body KK is nn-saturated if no n−1n-1 members of it can be replaced with nn replicas of KK, and it is completely saturated if it is nn-saturated for each n≥1n\ge 1. Similarly, a covering C\cal C with congruent replicas of a body KK is nn-reduced if no nn members of it can be replaced by n−1n-1 replicas of KK without uncovering a portion of the space, and it is completely reduced if it is nn-reduced for each n≥1n\ge 1. We prove that every body KK in dd-dimensional Euclidean or hyperbolic space admits both an nn-saturated packing and an nn-reduced covering with replicas of KK. Under some assumptions on K⊂EdK\subset \mathbb{E}^d (somewhat weaker than convexity), we prove the existence of completely saturated packings and completely reduced coverings, but in general, the problem of existence of completely saturated packings and completely reduced coverings remains unsolved. Also, we investigate some problems related to the the densities of nn-saturated packings and nn-reduced coverings. Among other things, we prove that there exists an upper bound for the density of a d+2d+2-reduced covering of Ed\mathbb{E}^d with congruent balls, and we produce some density bounds for the nn-saturated packings and nn-reduced coverings of the plane with congruent circles

    On packing spheres into containers (about Kepler's finite sphere packing problem)

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    In an Euclidean dd-space, the container problem asks to pack nn equally sized spheres into a minimal dilate of a fixed container. If the container is a smooth convex body and d≥2d\geq 2 we show that solutions to the container problem can not have a ``simple structure'' for large nn. By this we in particular find that there exist arbitrary small r>0r>0, such that packings in a smooth, 3-dimensional convex body, with a maximum number of spheres of radius rr, are necessarily not hexagonal close packings. This contradicts Kepler's famous statement that the cubic or hexagonal close packing ``will be the tightest possible, so that in no other arrangement more spheres could be packed into the same container''.Comment: 13 pages, 2 figures; v2: major revision, extended result, simplified and clarified proo
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