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Publication# What are the Longest Ropes on the Unit Sphere?

Abstract

We consider the variational problem of finding the longest closed curves of given minimal thickness on the unit sphere. After establishing the existence of solutions for any given thickness between 0 and 1, we explicitly construct for each given thickness Theta(n) := sin pi/(2n), n is an element of N, exactly phi(n) solutions, where. is Euler's totient function from number theory. Then we prove that these solutions are unique, and also provide a complete characterisation of sphere filling curves on the unit sphere; that is of those curves whose spherical tubular neighbourhood completely covers the surface area of the unit sphere exactly once. All of these results carry over to open curves as well, as indicated in the last section.

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Related concepts (33)

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Sphere

A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. Formally, a sphere is the set of points that are all at the same distance r from a given point in three-dimensional space. That given point is the centre of the sphere, and r is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians. The sphere is a fundamental object in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry.

Curve

In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that appeared more than 2000 years ago in Euclid's Elements: "The [curved] line is [...] the first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which [...

3-sphere

In mathematics, a 3-sphere, glome or hypersphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensions is an ordinary sphere (or 2-sphere, a two-dimensional surface), the boundary of a ball in four dimensions is a 3-sphere (an object with three dimensions). A 3-sphere is an example of a 3-manifold and an n-sphere.

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