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

Interesting

Very interesting question and informative conversation.

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.