### 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.

Original language | English |
---|---|

Pages (from-to) | 87-93 |

Number of pages | 7 |

Journal | Computational Geometry: Theory and Applications |

Volume | 48 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2015 |

### Fingerprint

### Keywords

- Alpha theory
- Certified solutions
- Homotopy method
- Polynomial system
- Touching cylinders

### ASJC Scopus subject areas

- Computational Theory and Mathematics
- Computer Science Applications
- Computational Mathematics
- Control and Optimization
- Geometry and Topology

### Cite this

*Computational Geometry: Theory and Applications*,

*48*(2), 87-93. https://doi.org/10.1016/j.comgeo.2014.08.007

**Seven mutually touching infinite cylinders.** / Bozóki, Sándor; Lee, Tsung Lin; Rónyai, L.

Research output: Contribution to journal › Article

*Computational Geometry: Theory and Applications*, vol. 48, no. 2, pp. 87-93. https://doi.org/10.1016/j.comgeo.2014.08.007

}

TY - JOUR

T1 - Seven mutually touching infinite cylinders

AU - Bozóki, Sándor

AU - Lee, Tsung Lin

AU - Rónyai, L.

PY - 2015

Y1 - 2015

N2 - 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.

AB - 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.

KW - Alpha theory

KW - Certified solutions

KW - Homotopy method

KW - Polynomial system

KW - Touching cylinders

UR - http://www.scopus.com/inward/record.url?scp=84907221214&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84907221214&partnerID=8YFLogxK

U2 - 10.1016/j.comgeo.2014.08.007

DO - 10.1016/j.comgeo.2014.08.007

M3 - Article

VL - 48

SP - 87

EP - 93

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

SN - 0925-7721

IS - 2

ER -