Peano-Gosper curves and the local isomorphism property

2017/05/02 by Francis Oger, Oger, Francis
Engineering · Mathematics · #05B45 (Primary) 52C20 #52C23 (Secondary) #Advanced Numerical Analysis Techniques #Combinatorics (math.CO) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1705.00787

openalex publication_date 2017/05/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider unbounded curves without endpoints. Isomorphism is equivalence up to translation. Self-avoiding plane-filling curves cannot be periodic, but they can satisfy the local isomorphism property: We obtain a set Ω of coverings of the plane by sets of disjoint self-avoiding nonoriented curves, generalizing the Peano-Gosper curves, such that: 1) each C ∈ Ω satisfies the local isomorphism property; any set of curves locally isomorphic to C belongs to Ω; 2) Ω is the union of 2^ω equivalence classes for the relation "C locally isomorphic to D"; each of them contains 2^ω (resp. 2^ω, 4, 0) isomorphism classes of coverings by 1 (resp. 2, 3, ≥ 4) curves. Each C ∈ Ω gives exactly 2 coverings by sets of oriented curves which satisfy the local isomorphism property. They have opposite orientations.

Citations

• The Fractal Geometry of Nature

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