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 Tr. Mat. Inst. Steklova, 2008, Volume 263, Pages 251–271 (Mi tm795)

Minimal Peano Curve

E. V. Shchepina, K. E. Baumanb

a Steklov Mathematical Institute, Russian Academy of Sciences
b M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: A Peano curve $p(x)$ with maximum square-to-linear ratio $\frac{|p(x)-p(y)|^2}{|x-y|}$ equal to $5\frac23$ is constructed; this ratio is smaller than that of the classical Peano–Hilbert curve, whose maximum square-to-linear ratio is 6. The curve constructed is of fractal genus 9 (i.e., it is decomposed into nine fragments that are similar to the whole curve) and of diagonal type (i.e., it intersects a square starting from one corner and ending at the opposite corner). It is proved that this curve is a unique (up to isometry) regular diagonal Peano curve of fractal genus 9 whose maximum square-to-linear ratio is less than 6. A theory is developed that allows one to find the maximum square-to-linear ratio of a regular Peano curve on the basis of computer calculations.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2008, 263, 236–256

Bibliographic databases:

UDC: 519.6

Citation: E. V. Shchepin, K. E. Bauman, “Minimal Peano Curve”, Geometry, topology, and mathematical physics. I, Collected papers. Dedicated to Academician Sergei Petrovich Novikov on the occasion of his 70th birthday, Tr. Mat. Inst. Steklova, 263, MAIK Nauka/Interperiodica, Moscow, 2008, 251–271; Proc. Steklov Inst. Math., 263 (2008), 236–256

Citation in format AMSBIB
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This publication is cited in the following articles:
1. K. E. Bauman, “One-side Peano curves of fractal genus $9$”, Proc. Steklov Inst. Math., 275 (2011), 47–59
2. K. E. Bauman, “Lower estimate of the square-to-linear ratio for regular Peano curves”, Discrete Math. Appl., 24:3 (2014), 123–128
3. D. K. Shalyga, “O tochnom vychislenii kubo-lineinogo otnosheniya krivykh Peano”, Preprinty IPM im. M. V. Keldysha, 2014, 088, 13 pp.
4. E. V. Shchepin, “Attainment of Maximum Cube-to-Linear Ratio for Three-Dimensional Peano Curves”, Math. Notes, 98:6 (2015), 971–976
5. A. A. Korneev, E. V. Shchepin, “$L_\infty$-locality of three-dimensional Peano curves”, Proc. Steklov Inst. Math., 302 (2018), 217–249
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