On enumeration of Hilbert-like curves

Jan Smrek, Alexander Y. Grosberg

    Research output: Contribution to journalArticlepeer-review


    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.

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


    • 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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