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Wöden Kusner

Publications and source records attributed to Wöden Kusner.

8 recordsLinked to original sources
arXiv2020

Chirality for crooked curves

Chiral objects rotate when placed in a collimated flow or wind. We exploit this hydrodynamic intuition to construct a tensorial chirality measure for rigid filaments and curves. This tensor is trace-free, so if a curve has a right-handed twist about some axis, there is a perpendicular axis about which the twist is left-handed. Our measure places minimal requirements on the smoothness of the curve, hence it can be readily used to quantify chirality for biomolecules and polymers, polygonal and rectifiable curves, and other discrete geometrical structures.

cond-mat.soft↗
arXiv2019

A Gordian Pair of Links

We construct a pair of isotopic link configurations that are not thick isotopic while preserving total length.

math.GT↗
arXiv2018

Configuration Spaces of Equal Spheres Touching a Given Sphere: The Twelve Spheres Problem

The problem of twelve spheres is to understand, as a function of $r \in (0,r_{max}(12)]$, the configuration space of $12$ non-overlapping equal spheres of radius $r$ touching a central unit sphere. It considers to what extent, and in what fashion, touching spheres can be varied, subject to the constraint of always touching the central sphere. Such constrained motion problems are of interest in physics and materials science, and the problem involves topology and geometry. This paper reviews the history of work on this problem, presents some new results, and formulates some conjectures. It also presents general results on configuration spaces of $N$ spheres of radius $r$ touching a central unit sphere, with emphasis on $3 \le N \le 14$. The problem of determining the maximal radius $r_{max}(N)$ is a version of the Tammes problem, to which László Fejes Tóth made significant contributions.

math.MG↗
arXiv2017

On the densest packing of polycylinders in any dimension

Using transversality and a dimension reduction argument, a result of A. Bezdek and W. Kuperberg is applied to polycylinders $\mathbb{D}^2\times \mathbb{R}^n$, showing that the optimal packing density is $π/\sqrt{12}$ in any dimension.

math.MG↗
arXiv2016

Packings of Regular Pentagons in the Plane

We show that every packing of congruent regular pentagons in the Euclidean plane has density at most $(5-\sqrt5)/3$, which is about 0.92. More specifically, this article proves the pentagonal ice-ray conjecture of Henley (1986), and Kuperberg and Kuperberg (1990), which asserts that an optimal packing of congruent regular pentagons in the plane is a double lattice, formed by aligned vertical columns of upward pointing pentagons alternating with aligned vertical columns of downward pointing pentagons. The strategy is based on estimates of the areas of Delaunay triangles. Our strategy reduces the pentagonal ice-ray conjecture to area minimization problems that involve at most four Delaunay triangles. These minimization problems are solved by computer. The computer-assisted portions of the proof use techniques such as interval arithmetic, automatic differentiation, and a meet-in-the-middle algorithm.

math.MG↗
arXiv2015

The local optimality of the double lattice packing

This paper introduces a technique for proving the local optimality of packing configurations. Applying this technique to a general convex polygon, we prove that the construction of the optimal double lattice packing by Kuperberg and Kuperberg is also locally optimal in the full space of packings.

math.MG↗
arXiv2013

Upper bounds on packing density for circular cylinders with high aspect ratio

In the early 1990s, A. Bezdek and W. Kuperberg used a relatively simple argument to show a surprising result: The maximum packing density of circular cylinders of infinite length in $\mathbb{R}^3$ is exactly $π/\sqrt{12}$, the planar packing density of the circle. This paper modifies their method to prove a bound on the packing density of finite length circular cylinders. In fact, the maximum packing density for unit radius cylinders of length $t$ in $\mathbb{R}^3$ is bounded above by $π/\sqrt{12} + 10/t$.

math.MG↗