### Abstract

Voronoi diagram of points in the Euclidean plane and its computation is foundational to computational geometry. Polynomial root-finding is the origin of fundamental discoveries in all of mathematics and sciences. There is an intrinsic connection between polynomial root-finding in the complex plane and the approximation of Voronoi cells of its roots via a fundamental family of iteration functions, the Basic Family. For instance, the immediate basin of attraction of a root of a complex polynomial under Newton's method is a rough approximation to its Voronoi cell. We formally introduce these connections through the Basic Family of iteration functions, its properties with respect to Voronoi diagrams, and a corresponding visualization called polynomiography. Polynomiography is a medium for art, math, education and science. By making use of the Basic Family we introduce a layering of the points within each Voronoi cell of a polynomial root and study its properties and potential applications. In particular, we prove some novel results about the Basic Family in connection with Voronoi diagrams.

Original language | English (US) |
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Title of host publication | 6th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2009 |

Pages | 31-40 |

Number of pages | 10 |

DOIs | |

State | Published - Dec 1 2009 |

Event | 6th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2009 - Copenhagen, Denmark Duration: Jun 23 2009 → Jun 26 2009 |

### Publication series

Name | 6th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2009 |
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### Other

Other | 6th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2009 |
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Country | Denmark |

City | Copenhagen |

Period | 6/23/09 → 6/26/09 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Information Systems
- Biomedical Engineering
- Applied Mathematics

### Keywords

- Complex polynomials
- Dynamical systems
- Fractal
- Iteration functions
- Newton's method
- Voronoi diagrams
- Zeros

### Cite this

*6th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2009*(pp. 31-40). [5362425] (6th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2009). https://doi.org/10.1109/ISVD.2009.17

}

*6th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2009.*, 5362425, 6th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2009, pp. 31-40, 6th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2009, Copenhagen, Denmark, 6/23/09. https://doi.org/10.1109/ISVD.2009.17

**Voronoi diagrams and polynomial root-finding.** / Kalantari, Bahman.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - Voronoi diagrams and polynomial root-finding

AU - Kalantari, Bahman

PY - 2009/12/1

Y1 - 2009/12/1

N2 - Voronoi diagram of points in the Euclidean plane and its computation is foundational to computational geometry. Polynomial root-finding is the origin of fundamental discoveries in all of mathematics and sciences. There is an intrinsic connection between polynomial root-finding in the complex plane and the approximation of Voronoi cells of its roots via a fundamental family of iteration functions, the Basic Family. For instance, the immediate basin of attraction of a root of a complex polynomial under Newton's method is a rough approximation to its Voronoi cell. We formally introduce these connections through the Basic Family of iteration functions, its properties with respect to Voronoi diagrams, and a corresponding visualization called polynomiography. Polynomiography is a medium for art, math, education and science. By making use of the Basic Family we introduce a layering of the points within each Voronoi cell of a polynomial root and study its properties and potential applications. In particular, we prove some novel results about the Basic Family in connection with Voronoi diagrams.

AB - Voronoi diagram of points in the Euclidean plane and its computation is foundational to computational geometry. Polynomial root-finding is the origin of fundamental discoveries in all of mathematics and sciences. There is an intrinsic connection between polynomial root-finding in the complex plane and the approximation of Voronoi cells of its roots via a fundamental family of iteration functions, the Basic Family. For instance, the immediate basin of attraction of a root of a complex polynomial under Newton's method is a rough approximation to its Voronoi cell. We formally introduce these connections through the Basic Family of iteration functions, its properties with respect to Voronoi diagrams, and a corresponding visualization called polynomiography. Polynomiography is a medium for art, math, education and science. By making use of the Basic Family we introduce a layering of the points within each Voronoi cell of a polynomial root and study its properties and potential applications. In particular, we prove some novel results about the Basic Family in connection with Voronoi diagrams.

KW - Complex polynomials

KW - Dynamical systems

KW - Fractal

KW - Iteration functions

KW - Newton's method

KW - Voronoi diagrams

KW - Zeros

UR - http://www.scopus.com/inward/record.url?scp=77951492777&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77951492777&partnerID=8YFLogxK

U2 - 10.1109/ISVD.2009.17

DO - 10.1109/ISVD.2009.17

M3 - Conference contribution

AN - SCOPUS:77951492777

SN - 9780769537818

T3 - 6th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2009

SP - 31

EP - 40

BT - 6th International Symposium on Voronoi Diagrams in Science and Engineering, ISVD 2009

ER -