[math-fun] Brain-Filling Curves and Gosper splittings
A pointer to a neat picture book featuring very many "space filling" curves: Jeffrey J. Ventrella: Brain-Filling Curves: A Fractal Bestiary, LuLu.com, (2012). The book can be downloaded for free at http://archive.org/download/BrainfillingCurves-AFractalBestiary/BrainFilling... Warning: it is a hefty 235 Megabytes! Two pertinent URLs are http://www.fractalcurves.com/HorrorVacui.html (book as html) http://www.brainfillingcurves.com/ (needs scripting activated) On page 107 (top) there is a curve corresponding to the L-system (and turns of 120 degrees) Start: F Rules: F --> F-F+F+FF-FF + --> + - --> - Here is a rendering http://jjj.de/3frac/R7-x1-curve.pdf What Ventrella does not show is how neatly 3 of these combine into either http://jjj.de/3frac/R7-x1-closed-trihook.pdf or http://jjj.de/3frac/R7-x1-closed-gosper-island.pdf The better known splittings of Gosper's island are http://jjj.de/3frac/gosper-split7.pdf and possibly (3 renderings) http://jjj.de/3frac/R7-2-gosper.pdf http://jjj.de/3frac/R7-2-sty1-gosper.pdf http://jjj.de/3frac/R7-alt-hex-gosper.pdf Here is a bubbly one: http://jjj.de/3frac/R7-bubble-gosper.pdf (cf. http://www.fractalcurves.com/Root7.html ) For the truly bored there are more images under http://jjj.de/3frac/ Best, jj
Duh! Space-filling curves are hard to compress using typical compression algorithms. But they're often quite easy to generate using the Postscript language. I wonder if this book has a Postscript form? At 06:19 AM 11/3/2013, Joerg Arndt wrote:
A pointer to a neat picture book featuring very many "space filling" curves:
Warning: it is a hefty 235 Megabytes!
News: nowadays pdf compresses way better than dvi or ps (at least with those curves!). Actual test: 2.8M -rw-r--r-- 1 jj users 2.8M 2013-11-03 18:08 test.dvi 40K -rw-r--r-- 1 jj users 39K 2013-11-03 18:08 test.pdf 4.1M -rw-r--r-- 1 jj users 4.1M 2013-11-03 18:08 test.ps Let's compress (gzip -9): 96K -rw-r--r-- 1 jj users 95K 2013-11-03 18:08 test.dvi.gz 36K -rw-r--r-- 1 jj users 36K 2013-11-03 18:08 test.pdf.gz 316K -rw-r--r-- 1 jj users 315K 2013-11-03 18:08 test.ps.gz Still sure you want ps? Best, jj P.S.: If anybody does know how to make xdvi render the colors in the (embedded postscript from the) 'picture' environment, I'd be eternally grateful... * Henry Baker <hbaker1@pipeline.com> [Nov 03. 2013 18:01]:
Duh! Space-filling curves are hard to compress using typical compression algorithms.
But they're often quite easy to generate using the Postscript language.
I wonder if this book has a Postscript form?
At 06:19 AM 11/3/2013, Joerg Arndt wrote:
A pointer to a neat picture book featuring very many "space filling" curves:
Warning: it is a hefty 235 Megabytes!
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I'm guessing you are talking about compressing the bitmap, while Henry is talking about writing a program to generate the bitmap. Such a program is probably only a few hundred bytes in PostScript. As an irrelevant aside, I believe all the compression algorithms used by pdf are also available in PostScript. --ms On 03-Nov-13 12:13, Joerg Arndt wrote:
News: nowadays pdf compresses way better than dvi or ps (at least with those curves!).
Actual test:
2.8M -rw-r--r-- 1 jj users 2.8M 2013-11-03 18:08 test.dvi 40K -rw-r--r-- 1 jj users 39K 2013-11-03 18:08 test.pdf 4.1M -rw-r--r-- 1 jj users 4.1M 2013-11-03 18:08 test.ps
Let's compress (gzip -9):
96K -rw-r--r-- 1 jj users 95K 2013-11-03 18:08 test.dvi.gz 36K -rw-r--r-- 1 jj users 36K 2013-11-03 18:08 test.pdf.gz 316K -rw-r--r-- 1 jj users 315K 2013-11-03 18:08 test.ps.gz
Still sure you want ps?
Best, jj
P.S.: If anybody does know how to make xdvi render the colors in the (embedded postscript from the) 'picture' environment, I'd be eternally grateful...
* Henry Baker <hbaker1@pipeline.com> [Nov 03. 2013 18:01]:
Duh! Space-filling curves are hard to compress using typical compression algorithms.
But they're often quite easy to generate using the Postscript language.
I wonder if this book has a Postscript form?
At 06:19 AM 11/3/2013, Joerg Arndt wrote:
A pointer to a neat picture book featuring very many "space filling" curves:
Warning: it is a hefty 235 Megabytes!
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* Mike Speciner <ms@alum.mit.edu> [Nov 04. 2013 07:44]:
I'm guessing you are talking about compressing the bitmap, while Henry is talking about writing a program to generate the bitmap. Such a program is probably only a few hundred bytes in PostScript. As an irrelevant aside, I believe all the compression algorithms used by pdf are also available in PostScript.
--ms
[...]
No bitmaps anywhere here. See http://jjj.de/tmp-tmp/ for the 3 files in question. Reproduce with latex test.tex dvips test.dvi pdflatex test.tex Sure one could write (tiny!) postscript programs to create these pictures, but how many people would be able to do that, leave alone willing to do it? The following (paperfolding dragon) is from http://rosettacode.org/wiki/Dragon_curve --------------------------- %!PS-Adobe-3.0 %%BoundingBox 0 0 300 300 /+ { 90 rotate } def /- {-90 rotate } def /!1 { dup 1 sub dup 0 eq not } def /F { 180 0 rlineto } def /X { !1 { X + Y F + } if pop } def /Y { !1 { - F X - Y } if pop } def /dragon { gsave 270 580 moveto dup 1 sub { 1 2 div sqrt dup scale -45 rotate } repeat F X stroke grestore } def 1 1 16 { dragon showpage } for %%EOF --------------------------- This one gives the "terdragon", in a nice rendering (just cooked it up): --------------------------- %!PS-Adobe-3.0 %%BoundingBox 0 0 300 300 /+ { 60 rotate } def /- {-60 rotate } def /!1 { dup 1 sub dup 0 eq not } def /F { 150 0 rlineto } def /X { !1 { X F + F + X - F - F X } if pop } def /terdragon { gsave 70 280 moveto 90 rotate dup 1 sub { 1 3 div sqrt dup scale -30 rotate } repeat F X stroke grestore } def 1 1 16 { terdragon showpage } for %%EOF --------------------------- Regards, jj Not a P.S., just for my records: generated with stringsubst 4 R--R--R L --RL++LRL++LR--RL R RL++LR--RLR--RL++ + + - - \ | tail -1 | ./bin 6 3 0 > pic-test.tex && make dotex # R9-flake (A Lindenmeyer system and turtle graphics to generate a TeX picture)
participants (3)
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Henry Baker -
Joerg Arndt -
Mike Speciner