Seven mutually touching infinite cylinders

TL;DR

This paper proves the existence of two configurations of seven infinite cylinders of radius one that mutually touch, using algebraic, symbolic, and numerical computational techniques to solve a complex polynomial system.

Contribution

It introduces an algebraic approach with homotopy continuation to solve a high-dimensional polynomial system for cylinder arrangements, finding explicit solutions for the longstanding problem.

Findings

Two explicit solutions of mutually touching cylinders are identified.

The approach confirms the existence of such configurations through rigorous numerical verification.

The method demonstrates the feasibility of solving large polynomial systems in geometric problems.

Abstract

We solve a problem of Littlewood: there exist seven infinite circular cylinders of unit radius which mutually touch each other. In fact, we exhibit two such sets of cylinders. Our approach is algebraic and uses symbolic and numerical computational techniques. We consider a system of polynomial equations describing the position of the axes of the cylinders in the 3 dimensional space. To have the same number of equations (namely 20) as the number of variables, the angle of the first two cylinders is fixed to 90 degrees, and a small family of direction vectors is left out of consideration. Homotopy continuation method has been applied to solve the system. The number of paths is about 121 billion, it is hopeless to follow them all. However, after checking 80 million paths, two solutions are found. Their validity, i.e., the existence of exact real solutions close to the approximate solutions at hand, was verified with the alphaCertified method as well as by the interval Krawczyk method.