On enumeration of Hilbert-like curves

Jan Smrek; Alexander Y. Grosberg

doi:10.1088/1751-8113/48/19/195001

On enumeration of Hilbert-like curves

Jan Smrek, Alexander Y. Grosberg

Research output: Contribution to journal › Article › peer-review

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_L ∼ 1.35699^L. 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	https://doi.org/10.1088/1751-8113/48/19/195001
State	Published - Apr 15 2015

Keywords

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

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Statistics and Probability
Modeling and Simulation
Mathematical Physics
General Physics and Astronomy

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10.1088/1751-8113/48/19/195001

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On enumeration of Hilbert-like curves

Abstract

Keywords

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