### Abstract

Given a simple polygon in the plane, a flip is defined as follows: consider the convex hull of the polygon. If there are no pockets do not perform a flip. If there are pockets then reflect one pocket across its line of support of the polygon to obtain a new simple polygon. In 1934 Paul Erds introduced the problem of repeatedly flipping all the pockets of a simple polygon simultaneously and he conjectured that the polygon would become convex after a finite number of flips. In 1939 Béla Nagy proved that if at each step only one pocket is flipped the polygon will become convex after a finite number of flips. The history of this problem is reviewed, and a simple elementary proof is given of a stronger version of the theorem. Variants, generalizations, and applications of the theorem of interest in computational knot theory, polymer physics and molecular biology are discussed. Several results in the literature are improved with the application of the theorem. For example, Grünbaum and Zaks recently showed that even non-simple (self-crossing) polygons may be convexified in a finite number of suitable flips. Their flips each take Θ( ^{n2}) time to determine. A simpler proof of this result is given that yields an algorithm that takes O(n) time to determine each flip. In the context of knot theory Millet proposed an algorithm for convexifying equilateral polygons in 3-dimensions with a generalization of a flip called a pivot. Here Millet's algorithm is generalized so that it works also in dimensions higher than three and for polygons containing edges with arbitrary lengths. A list of open problems is included.

Original language | English (US) |
---|---|

Pages (from-to) | 219-236 |

Number of pages | 18 |

Journal | Computational Geometry: Theory and Applications |

Volume | 31 |

Issue number | 3 |

DOIs | |

State | Published - Jun 1 2005 |

### Fingerprint

### Keywords

- Computational geometry
- Convexification
- Curve inflation
- Flips
- Flipturns
- Knot theory
- Molecular reconfiguration
- Pivots
- Polygonal linkages
- Polygons
- Polymer physics
- Robotics
- Self-avoiding walks

### ASJC Scopus subject areas

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

### Cite this

*Computational Geometry: Theory and Applications*,

*31*(3), 219-236. https://doi.org/10.1016/j.comgeo.2004.12.005

**The Erdös-Nagy theorem and its ramifications.** / Toussaint, Godfried.

Research output: Contribution to journal › Article

*Computational Geometry: Theory and Applications*, vol. 31, no. 3, pp. 219-236. https://doi.org/10.1016/j.comgeo.2004.12.005

}

TY - JOUR

T1 - The Erdös-Nagy theorem and its ramifications

AU - Toussaint, Godfried

PY - 2005/6/1

Y1 - 2005/6/1

N2 - Given a simple polygon in the plane, a flip is defined as follows: consider the convex hull of the polygon. If there are no pockets do not perform a flip. If there are pockets then reflect one pocket across its line of support of the polygon to obtain a new simple polygon. In 1934 Paul Erds introduced the problem of repeatedly flipping all the pockets of a simple polygon simultaneously and he conjectured that the polygon would become convex after a finite number of flips. In 1939 Béla Nagy proved that if at each step only one pocket is flipped the polygon will become convex after a finite number of flips. The history of this problem is reviewed, and a simple elementary proof is given of a stronger version of the theorem. Variants, generalizations, and applications of the theorem of interest in computational knot theory, polymer physics and molecular biology are discussed. Several results in the literature are improved with the application of the theorem. For example, Grünbaum and Zaks recently showed that even non-simple (self-crossing) polygons may be convexified in a finite number of suitable flips. Their flips each take Θ( n2) time to determine. A simpler proof of this result is given that yields an algorithm that takes O(n) time to determine each flip. In the context of knot theory Millet proposed an algorithm for convexifying equilateral polygons in 3-dimensions with a generalization of a flip called a pivot. Here Millet's algorithm is generalized so that it works also in dimensions higher than three and for polygons containing edges with arbitrary lengths. A list of open problems is included.

AB - Given a simple polygon in the plane, a flip is defined as follows: consider the convex hull of the polygon. If there are no pockets do not perform a flip. If there are pockets then reflect one pocket across its line of support of the polygon to obtain a new simple polygon. In 1934 Paul Erds introduced the problem of repeatedly flipping all the pockets of a simple polygon simultaneously and he conjectured that the polygon would become convex after a finite number of flips. In 1939 Béla Nagy proved that if at each step only one pocket is flipped the polygon will become convex after a finite number of flips. The history of this problem is reviewed, and a simple elementary proof is given of a stronger version of the theorem. Variants, generalizations, and applications of the theorem of interest in computational knot theory, polymer physics and molecular biology are discussed. Several results in the literature are improved with the application of the theorem. For example, Grünbaum and Zaks recently showed that even non-simple (self-crossing) polygons may be convexified in a finite number of suitable flips. Their flips each take Θ( n2) time to determine. A simpler proof of this result is given that yields an algorithm that takes O(n) time to determine each flip. In the context of knot theory Millet proposed an algorithm for convexifying equilateral polygons in 3-dimensions with a generalization of a flip called a pivot. Here Millet's algorithm is generalized so that it works also in dimensions higher than three and for polygons containing edges with arbitrary lengths. A list of open problems is included.

KW - Computational geometry

KW - Convexification

KW - Curve inflation

KW - Flips

KW - Flipturns

KW - Knot theory

KW - Molecular reconfiguration

KW - Pivots

KW - Polygonal linkages

KW - Polygons

KW - Polymer physics

KW - Robotics

KW - Self-avoiding walks

UR - http://www.scopus.com/inward/record.url?scp=84867967318&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84867967318&partnerID=8YFLogxK

U2 - 10.1016/j.comgeo.2004.12.005

DO - 10.1016/j.comgeo.2004.12.005

M3 - Article

VL - 31

SP - 219

EP - 236

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

SN - 0925-7721

IS - 3

ER -