Space-Filling curves like Peano, Osgood, Hilbert, Lance-Thomas ones.
Reference paper:
Hans Sagan, A geometrization of Lebesgue space-filling curve, Mathematical Intelligencer, Vol. 15, n. 4, 1993
Thanks.
Gianluca
This is a great question. In addition to helpful answers already given, there is a bit more to add.
The following RG research project on Hilbert space-filling curves may interest the followers of this thread:
https://www.researchgate.net/project/Hilbert-type-space-filling-curves
More importantly, see also
Article Context-based Space Filling Curves
New Gosper space-filling curves are introduced in
Conference Paper New Gosper Space Filling Curves
More to the point, for a 2018 paper on space-filling curves, see
https://arxiv.org/pdf/1801.07399.pdf
and the 2017 paper at
https://arxiv.org/pdf/1711.04473.pdf
Dear Gianluca,
Filling - Space curve is a fascinating phenomenon where such curve maps a one-dimensional interval into a two-dimensional area. Money people have thought it is impossible but using mathematical "analysis" and the "convergence" concepts, many mathematicians ( you mentioned some of them) succeed to construct such curves that cover 2- dimensional domains (cover areas). It is familiar to ask the student what is the length of the curve and now one can ask" what is the area covered by the given filling-space curve?
" Interesting!!
This equivalent to saying "we can use the pen to paint some picture (continuously) without using the paintbrush".
Best wishes
I don't know about "state-of the art" SFC but Hans Sagan has this wonderful book about them published in 1994:
https://www.amazon.com/Space-Filling-Curves-Universitext-Hans-Sagan/dp/0387942653/ref=sr_1_1?s=books&ie=UTF8&qid=1518011498&sr=1-1&keywords=Space+filling+curves
You can find more engineering oriented and more recent books:
https://www.amazon.com/Introduction-Optimization-Exploiting-Space-Filling-SpringerBriefs/dp/1461480418/ref=sr_1_3?s=books&ie=UTF8&qid=1518011498&sr=1-3&keywords=Space+filling+curves&dpID=51Djm4Aj0xL&preST=_SY291_BO1,204,203,200_QL40_&dpSrc=srch
https://www.amazon.com/Space-Filling-Curves-Introduction-Applications-Computational/dp/3642310451/ref=sr_1_2?s=books&ie=UTF8&qid=1518011498&sr=1-2&keywords=Space+filling+curves&dpID=41nKXDQQvXL&preST=_SY291_BO1,204,203,200_QL40_&dpSrc=srch
This is a great question. In addition to helpful answers already given, there is a bit more to add.
The following RG research project on Hilbert space-filling curves may interest the followers of this thread:
https://www.researchgate.net/project/Hilbert-type-space-filling-curves
More importantly, see also
Article Context-based Space Filling Curves
New Gosper space-filling curves are introduced in
Conference Paper New Gosper Space Filling Curves
More to the point, for a 2018 paper on space-filling curves, see
https://arxiv.org/pdf/1801.07399.pdf
and the 2017 paper at
https://arxiv.org/pdf/1711.04473.pdf
Here is a followup on @ Bruno Martin's post about Sagan's excellent 1994 book on space-filling curves.
A more recent book may be of interest to followers of this thread:
Michael Bader,
Space-Filling Curves An Introduction with Applications in Scientific Computing, Springer, 2013:
https://www.springer.com/gp/book/9783642310454
This book includes a hefty treatment of triangulation. Chapter 2 is on how to construct space-filling curves. For more about space-filling curves from a Mathematica perspective, see, for example, the
TriangleTriangulate function. For an illustration, see the attached image.
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