Abstract
Using transversality and a dimension reduction argument, a result of Bezdek and Kuperberg is applied to polycylinders, showing that the optimal packing density of \(\mathbb {D}^2\times \mathbb {R}^n\) equals \(\pi /\sqrt{12}\) for all natural numbers n.
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1 Introduction
Open and closed Euclidean unit nballs will be denoted by \(\mathbb {B}^n\) and \(\mathbb {D}^n\) respectively. The closed unit interval is denoted by \(\mathbb {I}\). A general polycylinder C is a set congruent to \(\prod _{i=1}^{i=m}\lambda _i\mathbb {D}^{k_i}\) in \(\mathbb {R}^{ k_1+\dots + k_m}\), where \(\lambda _i\) is in \([0,\infty ]\). For this article, the term polycylinder refers to the special case of an infinite polycylinder over a twodimensional disk of unit radius. A polycylinder is a set congruent to \(\mathbb {D}^2 \times \mathbb {R}^n \) in \( \mathbb {R}^{n+2}\). A polycylinder packing of \(\mathbb {R}^{n+2}\) is a family \(\mathscr {C} = \{C_i\}_{i \in I}\) of polycylinders \(C_i \subset \mathbb {R}^{n+2}\) with mutually disjoint interiors. The upper density \(\delta ^+ (\mathscr {C})\) of a packing \(\mathscr {C}\) of \(\mathbb {R}^n\) is defined to be
The upper packing density \(\delta ^+ (C)\) of an object C is the supremum of \(\delta ^+ (\mathscr {C})\) over all packings \(\mathscr {C}\) of \(\mathbb {R}^n\) by C.
This article proves the following sharp bound for the packing density of infinite polycylinders:
Theorem 1
\(\delta ^+(\mathbb {D}^2 \times \mathbb {R}^n) = \pi /\sqrt{12} \) for all natural numbers n.
Theorem 1 generalizes a result of Bezdek and Kuperberg [1] and improves on results that may be computed using a method of Fejes Tóth and Kuperberg [3], cf. [2, 5]; it gives some of the first sharp upper bounds for packing density in high dimensions.
2 Transversality
This section introduces the required transversality arguments from affine geometry. A dflat is a ddimensional affine subspace of \(\mathbb {R}^n\). The parallel dimension \(\mathrm{dim}_\parallel \{F,\dots , G\}\) of a collection of flats \(\{F, \dots , G\}\) is the dimension of their maximal parallel subflats. The notion of parallel dimension can be interpreted in several ways, allowing a modest abuse of notation.

For a collection of flats \(\{F, \dots , G\}\), consider their tangent cones at infinity \(\{F_\infty , \dots , G_\infty \}\). The parallel dimension of \(\{F, \dots , G\}\) is the dimension of the intersection of these tangent cones. This may be viewed as the limit of a rescaling process \(\mathbb {R}^n \rightarrow r\mathbb {R}^n\) as r tends to 0, leaving only the scaleinvariant information.

For a collection of flats \(\{F, \dots , G\}\), consider each flat as a system of linear equations. The corresponding homogeneous equations determine a collection of linear subspaces \(\{F_\infty , \dots , G_\infty \}\). The parallel dimension is the dimension of their intersection \(F_\infty \,\cap \dots \cap \, G_\infty \).
Two disjoint dflats are parallel if their parallel dimension is d, that is, if every line in one is parallel to a line in the other.
Lemma 1
A pair of disjoint nflats in \(\mathbb {R}^{n+k}\) with \(n\ge k\), has parallel dimension strictly greater than \(nk.\)
Proof
Let F and G be such a pair. By homogeneity of \(\mathbb {R}^{n+k}\), let \(F=F_\infty .\) As \(F_\infty \) and G are disjoint, G contains a nontrivial vector \(\mathbf{v}\) such that \(G = G_\infty +\mathbf{v}\) and \(\mathbf{v}\) is not in \(F_\infty + G_\infty .\) It follows that
Count dimensions to find \(n+k > n +n  \mathrm{dim}_\parallel (F_\infty ,G_\infty ).\) \(\square \)
Corollary 1
A pair of disjoint nflats in \(\mathbb {R}^{n+2}\) has parallel dimension at least \(n1\).
3 Dimension Reduction
3.1 Pairwise Foliations
The core \(a_i\) of a polycylinder \(C_i\) congruent to \(\mathbb {D}^2 \times \mathbb {R}^n\) in \(\mathbb {R}^{n+2}\) is the distinguished nflat defining \(C_i\) as the set of points at most distance 1 from \(a_i\). In a packing \(\mathscr {C}\) of \(\mathbb {R}^{n+2}\) by polycylinders, Corollary 1 shows that, for every pair of polycylinders \(C_i\) and \(C_j\), one can choose parallel \((n1)\)dimensional subflats \(b_i \subset a_i\) and \(b_j \subset a_j\) and define a product foliation
with \(\mathbb {R}^3\) leaves that are orthogonal to \(b_i\) and to \(b_j\). Given a point x in \(a_i\), there is a distinguished \(\mathbb {R}^3\) leaf \(F_x^{b_i,b_j}\) that contains the point x. The foliation \(\mathscr {F}^{b_i,b_j}\) restricts to foliations of \(C_i\) and \(C_j\) with rightcircularcylinder leaves.
3.2 The Dirichlet Slice
In a packing \(\mathscr {C}\) of \(\mathbb {R}^{n+2}\) by polycylinders, the Dirichlet cell \(D_i\) associated with a polycylinder \(C_i\) is the set of points in \(\mathbb {R}^{n+2}\) which lie no further from \(C_i\) than from any other polycylinder in \(\mathscr {C}\). The Dirichlet cells of a packing partition \(\mathbb {R}^{n+2},\) as \(C_i \subset D_i\) for all polycylinders \(C_i\). To bound the density \(\delta ^+(\mathscr {C})\), it is enough to fix an i in I and consider the density of \(C_i\) in \(D_i\).
Consider the following slicing of the Dirichlet cell \(D_i\). Given a fixed polycylinder \(C_i\) in a packing \(\mathscr {C}\) of \(\mathbb {R}^{n+2}\) by polycylinders and a point x on the core \(a_i\), the plane \(p_x\) is the 2flat orthogonal to \(a_i\) and containing the point x. The Dirichlet slice \(d_x\) is the intersection of \(D_i\) and \(p_x.\)
Note that \(p_x\) is a subflat of \(F_x^{b_i,b_j}\) for all j in I.
3.3 Bezdek–Kuperberg Bound
For any point x on the core \(a_i\) of a polycylinder \(C_i\), the results of Bezdek and Kuperberg [1] apply to the Dirichlet slice \(d_x\).
Lemma 2
A Dirichlet slice is convex and, if bounded, a parabolasided polygon.
Proof
Construct the Dirichlet slice \(d_x\) as an intersection. Define \(d^j\) to be the set of points in \(p_x\) which lie no further from \(C_i\) than from \(C_j\). Then the Dirichlet slice \(d_x\) is realized as
Each arc of the boundary of \(d_x\) in \(p_x\) is given by an arc of the boundary of some \(d^j\) in \(p_x.\) The boundary of \(d^j\) in \(p_x\) is the set of points in \(p_x\) equidistant from \(C_i\) and \(C_j.\) Since the foliation \(\mathscr {F}^{b_i,b_j}\) is a product foliation, the arc of the boundary of \(d^j\) in \(p_x\) is also the set of points in \(p_x\) equidistant from the leaf \(C_i \cap F_x^{b_i,b_j}\) of \(\mathscr {F}^{b_i,b_j}_{C_i}\) and the leaf \(C_j \cap F_x^{b_i,b_j}\) of \(\mathscr {F}^{b_i,b_j}_{C_j}\). This reduces the analysis to the case of a pair of cylinders in \(\mathbb {R}^3.\) From [1], it follows that \(d^j\) is convex and the boundary of \(d_j\) in \(p_x\) is a parabola; the intersection of such sets \(d^j\) in \(p_x\) is convex, and a parabolasided polygon if bounded. \(\square \)
Let \(S_x(r)\) be the circle of radius r in \(p_x\) centered at x.
Lemma 3
The vertices of \(d_x\) are not closer to \(S_x(1)\) than the vertices of a regular hexagon circumscribed about \(S_x(1).\)
Proof
A vertex of \(d_x\) occurs where three or more polycylinders are equidistant, so the vertex is the center of a \((n+2)\)ball B tangent to three polycylinders. Thus B is tangent to three disjoint unit \((n+2)\)balls \(B_1\), \(B_2\), \(B_3\). By projecting into the affine hull of the centers of \(B_1\), \(B_2\), \(B_3\), it is immediate that the radius of B is no less than \(2/\sqrt{3} 1.\) \(\square \)
Lemma 4
Let y and z be points on the circle \(S_x(2/\sqrt{3})\). If each of y and z is equidistant from \(C_i\) and \(C_j\), then the angle yxz is smaller than or equal to \(2\arccos (\sqrt{3} 1) = 85.8828\dots ^\circ .\)
Proof
Following [1, 4], the existence of a supporting hyperplane of \(C_i\) that separates \({{\mathrm{int}}}(C_i)\) from \({{\mathrm{int}}}(C_j)\) suffices. \(\square \)
In [1], it is shown that planar objects satisfying Lemmas 2, 3 and 4 have area no less than \(\sqrt{12}.\) As the bound holds for all Dirichlet slices, it follows that \(\delta ^+(\mathbb {D}^2\times \mathbb {R}^n) \le \pi /\sqrt{12}\) in \(\mathbb {R}^{n+2}.\) The product of the dense disk packing in the plane with \(\mathbb {R}^n\) gives a polycylinder packing in \(\mathbb {R}^{n+2}\) that achieves this density. Combining this with the result of Thue [6] for \(n=0\) and the result of Bezdek and Kuperberg [1] for \(n=1\), Theorem 1 follows.
References
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Acknowledgments
Thanks to Thomas Hales, Włodzimierz Kuperberg and Robert Kusner. The author was supported by Austrian Science Fund (FWF) Project 5503 and National Science Foundation (NSF) Grant No. 1104102.
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Kusner, W. On the Densest Packing of Polycylinders in Any Dimension. Discrete Comput Geom 55, 638–641 (2016). https://doi.org/10.1007/s0045401697666
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DOI: https://doi.org/10.1007/s0045401697666