### Abstract

We present an analytical method to explicitly enumerate all self-similar space-filling curves similar to Hilbert curve, and find their number grows with length L as Z<inf>L</inf> ∼ 1.35699<sup>L</sup>. This presents a first step in the exact characterization of the crumpled globule ensemble relevant for dense topologically constrained polymer matter and DNA folding. Moreover, this result gives a stringent lower bound on the number of Hamiltonian walks on a simple cubic lattice. Additionally, we compute the exact number of crumpled curves with arbitrary endpoints, and the closed crumpled curves on a cube 4 × 4 × 4 cube.

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

Article number | 195001 |

Pages (from-to) | 1-12 |

Number of pages | 12 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 48 |

Issue number | 19 |

DOIs | |

State | Published - Apr 15 2015 |

### Fingerprint

### Keywords

- Crumpled globule
- Enumeration
- Fractal
- Hamiltonian walk
- Hilbert curve
- Polymer
- Space-filling

### ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Modeling and Simulation
- Statistics and Probability

### Cite this

*Journal of Physics A: Mathematical and Theoretical*,

*48*(19), 1-12. [195001]. https://doi.org/10.1088/1751-8113/48/19/195001

**On enumeration of Hilbert-like curves.** / Smrek, Jan; Grosberg, Alexander Y.

Research output: Contribution to journal › Article

*Journal of Physics A: Mathematical and Theoretical*, vol. 48, no. 19, 195001, pp. 1-12. https://doi.org/10.1088/1751-8113/48/19/195001

}

TY - JOUR

T1 - On enumeration of Hilbert-like curves

AU - Smrek, Jan

AU - Grosberg, Alexander Y.

PY - 2015/4/15

Y1 - 2015/4/15

N2 - We present an analytical method to explicitly enumerate all self-similar space-filling curves similar to Hilbert curve, and find their number grows with length L as ZL ∼ 1.35699L. This presents a first step in the exact characterization of the crumpled globule ensemble relevant for dense topologically constrained polymer matter and DNA folding. Moreover, this result gives a stringent lower bound on the number of Hamiltonian walks on a simple cubic lattice. Additionally, we compute the exact number of crumpled curves with arbitrary endpoints, and the closed crumpled curves on a cube 4 × 4 × 4 cube.

AB - We present an analytical method to explicitly enumerate all self-similar space-filling curves similar to Hilbert curve, and find their number grows with length L as ZL ∼ 1.35699L. This presents a first step in the exact characterization of the crumpled globule ensemble relevant for dense topologically constrained polymer matter and DNA folding. Moreover, this result gives a stringent lower bound on the number of Hamiltonian walks on a simple cubic lattice. Additionally, we compute the exact number of crumpled curves with arbitrary endpoints, and the closed crumpled curves on a cube 4 × 4 × 4 cube.

KW - Crumpled globule

KW - Enumeration

KW - Fractal

KW - Hamiltonian walk

KW - Hilbert curve

KW - Polymer

KW - Space-filling

UR - http://www.scopus.com/inward/record.url?scp=84928540661&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84928540661&partnerID=8YFLogxK

U2 - 10.1088/1751-8113/48/19/195001

DO - 10.1088/1751-8113/48/19/195001

M3 - Article

VL - 48

SP - 1

EP - 12

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 19

M1 - 195001

ER -