Contact numbers for congruent sphere packings in Euclidean 3-space. (English) Zbl 1259.52013
Let \(C(n)\) denote the largest possible number of touching pairs in a packing of \(n\) unit balls in \(3\)-dimensional Euclidean space.
The author proves that \[ 6n - 486^{2/3}\, n^{2/3} < C(n) < 6n - 0.695 n^{2/3} \] for all \(n\) of the form \(n=k(2 k^2+1)/3\) with \(k\geq 2\). He also provides similar estimates in the case when there are additional restrictions on the centers of balls in a packing.
The author proves that \[ 6n - 486^{2/3}\, n^{2/3} < C(n) < 6n - 0.695 n^{2/3} \] for all \(n\) of the form \(n=k(2 k^2+1)/3\) with \(k\geq 2\). He also provides similar estimates in the case when there are additional restrictions on the centers of balls in a packing.
Reviewer: A. E. Litvak (Edmonton)
MSC:
| 52C17 | Packing and covering in \(n\) dimensions (aspects of discrete geometry) |
Keywords:
congruent sphere packing; contact number; density; truncated Voronoi cell; union of balls; isoperimetric inequality; spherical cap packingSoftware:
kepler98Online Encyclopedia of Integer Sequences:
Octahedral numbers: a(n) = n*(2*n^2 + 1)/3.Decimal expansion of 486^(1/3).
References:
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