Geodesic
Minimal Peano Curve
Informatics and Automation, Geometry, topology, and mathematical physics. I, Tome 263 (2008), pp. 251-271.

Voir la notice de l'article provenant de la source Math-Net.Ru

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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E. V. Shchepin; K. E. Bauman. Minimal Peano Curve. Informatics and Automation, Geometry, topology, and mathematical physics. I, Tome 263 (2008), pp. 251-271. http://geodesic.mathdoc.fr/item/TRSPY_2008_263_a16/

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