On the Densest Packing of Polycylinders in Any Dimension

Discrete & Computational Geometry, Apr 2016

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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On the Densest Packing of Polycylinders in Any Dimension

Discrete Comput Geom On the Densest Packing of Polycylinders in Any Dimension Wöden Kusner 0 Polycylinders 0 Packing 0 Density 0 Slicing 0 Editor in Charge: János Pach 0 Institute of Analysis and Number Theory, Graz University of Technology, Steyrergasse 30/II , 8010 Graz , Austria Using transversality and a dimension reduction argument, a result of Bezdek and Kuperberg is applied to polycylinders, showing that the optimal packing density of D2 × Rn equals π/√12 for all natural numbers n. Open and closed Euclidean unit n-balls will be denoted by Bn and Dn respectively. The closed unit interval is denoted by I. A general polycylinder C is a set congruent to ii==1m λi Dki in Rk1+···+km , where λi is in [0, ∞]. For this article, the term polycylinder refers to the special case of an infinite polycylinder over a two-dimensional disk of unit radius. A polycylinder is a set congruent to D2 × Rn in Rn+2. A polycylinder packing of Rn+2 is a family C = {Ci }i∈I of polycylinders Ci ⊂ Rn+2 with mutually disjoint interiors. The upper density δ+(C ) of a packing C of Rn is defined to be 1 Introduction δ+(C ) = lim sup VoVlo(Cl(r∩BrnB)n ) . r →∞ The upper packing density δ+(C ) of an object C is the supremum of δ+(C ) over all packings C of Rn by C . This article proves the following sharp bound for the packing density of infinite polycylinders: Theorem 1 δ+(D2 × Rn) = π/√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 d-flat is a d-dimensional affine subspace of Rn. The parallel dimension dim {F, . . . , G} of a collection of flats {F, . . . , G} is the dimension of their maximal parallel sub-flats. The notion of parallel dimension can be interpreted in several ways, allowing a modest abuse of notation. – For a collection of flats {F, . . . , G}, consider their tangent cones at infinity {F∞, . . . , G∞}. The parallel dimension of {F, . . . , G} is the dimension of the intersection of these tangent cones. This may be viewed as the limit of a rescaling process Rn → r Rn as r tends to 0, leaving only the scale-invariant information. – For a collection of flats {F, . . . , G}, consider each flat as a system of linear equations. The corresponding homogeneous equations determine a collection of linear subspaces {F∞, . . . , G∞}. The parallel dimension is the dimension of their intersection F∞ ∩ · · · ∩ G∞. Two disjoint d-flats 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 n-flats in Rn+k with n ≥ k, has parallel dimension strictly greater than n − k. Proof Let F and G be such a pair. By homogeneity of Rn+k , let F = F∞. As F∞ and G are disjoint, G contains a non-trivial vector v such that G = G∞ + v and v is not in F∞ + G∞. It follows that dim(Rn+k ) ≥ dim F∞ + G∞ + span(v) > dim(F∞ + G∞) = dim(F∞) + dim(G∞) − dim(F∞ ∩ G∞). Count dimensions to find n + k > n + n − dim (F∞, G∞). Corollary 1 A pair of disjoint n-flats in Rn+2 has parallel dimension at least n − 1. 3 Dimension Reduction 3.1 Pairwise Foliations The core ai of a polycylinder Ci congruent to D2 × Rn in Rn+2 is the distinguished n-flat defining Ci as the set of points at most distance 1 from ai . In a packing C of Rn+2 by polycylinders, Corollary 1 shows that, for every pair of polycylinders Ci and C j , one can choose parallel (n − 1)-dimensional subflats bi ⊂ ai and b j ⊂ a j and define a product foliation F bi ,b j : Rn+2 → Rn−1 × R3 with R3 leaves that are orthogonal to bi and to b j . Given a point x in ai , there is a distinguished R3 leaf Fxbi ,b j that contains the point x . The foliation F bi ,b j restricts to foliations of Ci and C j with right-circular-cylinder leaves. 3.2 The Dirichlet Slice In a packing C of Rn+2 by polycylinders, the Dirichlet cell Di associated with a polycylinder Ci is the set of points in Rn+2 which lie no further from Ci than from any other polycylinder in C . The Dirichlet cells of a packing partition Rn+2, as Ci ⊂ Di for all polycylinders Ci . To bound the density δ+(C ), it is enough to fix an i in I and consider the density of Ci in Di . Consider the following slicing of the Dirichlet cell Di . Given a fixed polycylinder Ci in a packing C of Rn+2 by polycylinders and a point x on the core ai , the plane px is the 2-flat orthogonal to ai and containing the point x . The Dirichlet slice dx is the intersection of Di and px . Note that px is a sub-flat of Fxbi ,b j for all j in I. 3.3 Bezdek–Kuperberg Bound For any point x on the core ai of a polycylinder Ci , the results of Bezdek and Kuperberg [ 1 ] apply to the Dirichlet slice dx . Lemma 2 A Dirichlet slice is convex and, if bounded, a parabola-sided polygon. Proof Construct the Dirichlet slice dx as an intersection. Define d j to be the set of points in px which lie no further from Ci than from C j . Then the Dirichlet slice dx is realized as dx = j∈I d j . Each arc of the boundary of dx in px is given by an arc of the boundary of some d j in px . The boundary of d j in px is the set of points in px equidistant from Ci and C j . Since the foliation F bi ,b j is a product foliation, the arc of the boundary of d j in px is also the set of points in px equidistant from the leaf Ci ∩ Fxbi ,b j of F bi ,b j |Ci and the leaf C j ∩ Fxbi ,b j of F bi ,b j |C j . This reduces the analysis to the case of a pair of cylinders in R3. From [ 1 ], it follows that d j is convex and the boundary of d j in px is a parabola; the intersection of such sets d j in px is convex, and a parabola-sided polygon if bounded. Let Sx (r ) be the circle of radius r in px centered at x . Lemma 3 The vertices of dx are not closer to Sx (1) than the vertices of a regular hexagon circumscribed about Sx (1). 3 − 1. Proof A vertex of dx 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 B1, B2, B3. By projecting into the affine hull of the centers of B1, B2, B3, it is immediate that the radius of B is no less than √ 2/ √ Lemma 4 Let y and z be points on the circle Sx (2/ 3). If each of y and z is equidistant √ from Ci and C j , then the angle y x z is smaller than or equal to 2 arccos( 3 − 1) = 85.8828 . . .◦ . Proof Following [ 1, 4 ], the existence of a supporting hyperplane of Ci that separates int(Ci ) from int(C j ) suffices. In [ 1 ], it is shown that planar objects satisfying Lemmas 2, 3 and 4 have area no less tπh/a√n1√21i2n. ARsn+th2e. Tbohuenpdrohdouldcst ofofrthalel dDeinriscehldeitskslipcaecsk,iintgfoilnlotwhes tphlaatnδe+w(Dith2 ×RnRgni)ve≤s a polycylinder packing in Rn+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. 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. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. 1. Bezdek , A. , Kuperberg , W. : Maximum density space packing with congruent circular cylinders of infinite length . Mathematika 37 ( 1 ), 74 - 80 ( 1990 ) 2. Blichfeldt , H.F. : The minimum value of quadratic forms, and the closest packing of spheres . Math. Ann. 101 ( 1 ), 605 - 608 ( 1929 ) 3. Tóth , G.F. , Kuperberg , W. : Blichfeldt's density bound revisited . Math. Ann. 295 ( 1 ), 721 - 727 ( 1993 ) 4. Kusner , W.: Upper bounds on packing density for circular cylinders with high aspect ratio . Discrete Comput. Geom . 51 ( 4 ), 964 - 978 ( 2014 ) 5. Rankin , R.A. : On the closest packing of spheres in n dimensions . Ann. Math. 48 , 228 - 229 ( 1947 ) 6. Thue , A. : On the densest packing of congruent circles in the plane . Skr. Vidensk-Selsk, Christiania 1 , 3 - 9 ( 1910 )


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On the Densest Packing of Polycylinders in Any Dimension, Discrete & Computational Geometry, 2016, 638-641, DOI: 10.1007/s00454-016-9766-6