PreprintCC BY 4.0arXiv
Insertion and expansion of cylinders around the unit ball
Abstract
A unit ball is surrounded with a ring of n equal infinite circular cylinders. Each cylinder touches the ball and its two neighbors. Can one more cylinder of the same radius be added if the configuration may be rearranged? Can the n cylinders instead be enlarged, and by how much? We prove that insertion is impossible for 3≤ n≤7 and possible for n≥9. The case n=8 remains open. The case n=6 is Kuperberg's problem from 1990, which was resolved recently. We prove that the ring of four cylinders cannot be expanded. This settles a conjecture of Ogievetsky and Shlosman from 2018. For n≥5 the largest possible expansion radius remains unknown.
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BibTeX
@misc{matic2026insertion,
title = {{Insertion and expansion of cylinders around the unit ball}},
author = {Ivan Matić and Radoš Radoičić and Andreas Spomer},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13972},
eprint = {2610.13972},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13972},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Insertion and expansion of cylinders around the unit ball
AU - Ivan Matić
AU - Radoš Radoičić
AU - Andreas Spomer
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.13972
UR - https://arxiv.org/abs/2610.13972
AB - A unit ball is surrounded with a ring of n equal infinite circular cylinders. Each cylinder touches the ball and its two neighbors. Can one more cylinder of the same radius be added if the configuration may be rearranged? Can the n cylinders instead be enlarged, and by how much? We prove that insertion is impossible for 3≤ n≤7 and possible for n≥9. The case n=8 remains open. The case n=6 is Kuperberg's problem from 1990, which was resolved recently. We prove that the ring of four cylinders cannot be expanded. This settles a conjecture of Ogievetsky and Shlosman from 2018. For n≥5 the largest possible expansion radius remains unknown.
ER -