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This work reveals the upper limit of touching cylinders in 3D as 10, indicating deep geometric connections.
Littlewood asked for the maximum number N of congruent infinite cylinders that can be arranged in R³ so that every pair touches. We improve upon the proof of the second author that N ≤ 18 to show that N ≤ 10. Together with the lower bound established by Bozóki, Lee, and Rónyai, this shows that N ∈ \7,8,9,10\. Our method is based on linear algebra and Ramsey theory, and makes partial use of computer verification. We also provide a completely computer-free proof that N ≤ 12.
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Dillon et al. (2025) studied this question.
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