A new upper bound for mutually touching infinite cylinders

TL;DR

This paper establishes a new upper bound of 18 for the maximum number of congruent infinite cylinders in three-dimensional space that can all touch each other, advancing understanding of geometric arrangements.

Contribution

The paper improves the known upper bound for mutually touching infinite cylinders from 24 to 18, providing a tighter constraint on their possible configurations.

Findings

Proves that at most 18 congruent infinite cylinders can mutually touch in 3D space.

Improves previous upper bound of 24 to 18.

Contributes to the longstanding geometric problem posed by Littlewood.

Abstract

Let $N$ denote the maximum number of congruent infinite cylinders that can be arranged in $\mathbb{R}^3$ so that every pair of cylinders touches each other. Littlewood posed the question of whether $N=7$, which remains unsolved. In this paper, we prove that $N\leq 18$, improving the previously known upper bound of $24$ established by A. Bezdek.