Geodesic
The dilation factor of the Peano--Hilbert curve
Matematičeskie zametki, Tome 80 (2006) no. 5, pp. 643-656

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It is proved that the maximum value of the ratio $|p(x)-p(y)|^2/|x-y|$ for the Peano–Hilbert curve $p\colon[0,1]=I\to I^2$ is equal to 6. 
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     author = {K. E. Bauman},
     title = {The dilation factor of the {Peano--Hilbert} curve},
     journal = {Matemati\v{c}eskie zametki},
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     publisher = {mathdoc},
     volume = {80},
     number = {5},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2006_80_5_a0/}
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K. E. Bauman. The dilation factor of the Peano--Hilbert curve. Matematičeskie zametki, Tome 80 (2006) no. 5, pp. 643-656. http://geodesic.mathdoc.fr/item/MZM_2006_80_5_a0/