Upper bounds for packings of cylinders around the unit sphere
Abstract
In this note we study the problem of packing infinite circular cylinders tangent to the unit sphere. For a given number of cylinders N, N≥ 3, the maximal common cylinder radius RN is the maximum R for which there exist N circular cylinders of radius R with non-overlapping interiors, all tangent to the unit sphere. In 1990, Kuperberg conjectured that there does not exist a packing of more than 6 cylinders of radius one tangent to the unit sphere, that is, R7<1. This conjecture was recently confirmed by an exhaustive computer-assisted case analysis. Here we prove new bounds for RN for small N by considering an auxiliary configuration of spherical quadrilaterals. In particular, we provide a short geometric proof of Kuperberg's conjecture and show that R7<0.9978.
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BibTeX
@misc{glazyrin2026upper,
title = {{Upper bounds for packings of cylinders around the unit sphere}},
author = {Alexey Glazyrin},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00251},
eprint = {2611.00251},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00251},
}
RIS
TY - UNPB
TI - Upper bounds for packings of cylinders around the unit sphere
AU - Alexey Glazyrin
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00251
UR - https://arxiv.org/abs/2611.00251
AB - In this note we study the problem of packing infinite circular cylinders tangent to the unit sphere. For a given number of cylinders N, N≥ 3, the maximal common cylinder radius R_N is the maximum R for which there exist N circular cylinders of radius R with non-overlapping interiors, all tangent to the unit sphere. In 1990, Kuperberg conjectured that there does not exist a packing of more than 6 cylinders of radius one tangent to the unit sphere, that is, R_7<1. This conjecture was recently confirmed by an exhaustive computer-assisted case analysis. Here we prove new bounds for R_N for small N by considering an auxiliary configuration of spherical quadrilaterals. In particular, we provide a short geometric proof of Kuperberg's conjecture and show that R_7<0.9978.
ER -