On enumeration of Hilbert-like curves

Jan Smrek, Alexander Y. Grosberg

    Research output: Contribution to journalArticle

    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 languageEnglish (US)
    Article number195001
    Pages (from-to)1-12
    Number of pages12
    JournalJournal of Physics A: Mathematical and Theoretical
    Volume48
    Issue number19
    DOIs
    StatePublished - Apr 15 2015

    Fingerprint

    Hamiltonians
    enumeration
    Enumeration
    Hilbert
    DNA
    Regular hexahedron
    Curve
    Polymers
    curves
    Space-filling Curves
    Closed curve
    Folding
    Analytical Methods
    Walk
    globules
    Ensemble
    cubic lattices
    folding
    Lower bound
    deoxyribonucleic acid

    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

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

    In: Journal of Physics A: Mathematical and Theoretical, Vol. 48, No. 19, 195001, 15.04.2015, p. 1-12.

    Research output: Contribution to journalArticle

    Smrek, Jan ; Grosberg, Alexander Y. / On enumeration of Hilbert-like curves. In: Journal of Physics A: Mathematical and Theoretical. 2015 ; Vol. 48, No. 19. pp. 1-12.
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