Thread View: sci.fractals
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2 total messages
Started by Jay R. Hill
Fri, 18 Feb 1994 20:56
DEEPZOOMing
Author: Jay R. Hill
Date: Fri, 18 Feb 1994 20:56
Date: Fri, 18 Feb 1994 20:56
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Tim Wegner <twegner@killerbee.jsc.nasa.gov or twegner@mitre.org for now>
sent me email with these questions. Apparently he was unable to post so
I will.
> At very deep zooms is one increasingly likely to encounter copies
> of structures that exist at shallower magnifications? Is there a
> point to deep zooming? I still believe the answer is yes. Even if
> deep zooming always eventually results in self-similarity (don't
> know if this is true) it is possible that the zoom depth at which
> the self-similarity starts is arbitrarily large. If so, there is
> still a point to deep zooming.
>
> I posted this question in SCI.FRACTALS. I'm not sure it got posted,
> I don't see any answers.
Of course we should DEEPZOOM ;-) but it all depends on what you want
to see. If you get caught in one of those tornado like regions around
a pre-periodic point you will see self-similarity. Just break off to
a branch, try to find a locally largest island MSet and you can find
the whole array of structure shapes there. Each with delicate
filementary structures attached everywhere. These little structures
contain a 'genetic' history of the larger (less zoomed) structures
you were looking at before. Now you can either zoom into one of these
attached filementary structures or dive into the thick of it next to
the large island MSet you are next to. If you do this you can set up
any designs you want by selecting elephant row (the cusp) or its
seahorse cajon (its equivalent to -.75) or even one of its branches (its
-1.75 area is nice), or even one of its pre-periodic point tornados.
Each time you do this - target a locally largest island MSet - you
seem to go through a layer so to speak. After several of these, you
build up an interesting filementary structure history that can be
very nice. I'll post more on this sometime.
It reminds me of the dust jacket descripiton on Poul's SiFi novel 'Tao
Zero' where the ram jet space ship's reverse has failed. "The sound
of a basso gong went thought the ship as they passed through another
galaxy". This is what I have in mind for an awesome mpeg into the
Mandelbrot Set with something like DEEPZOOM or FRACTINT.
Warmly,
Jay "Not tonight honey, my ram jet is still stuck" Hill
PS. If you have svga256.bgi for MSDOS SuperVGA, try this:
--
#include <graphics.h> /* hilljr@jupiter.saic.com (Jay R Hill) */
int huge D(){return 4;}long e=0;main(){float g,s,f,r,i,w=3./1024;int m,l,u,b,c
,d=0;installuserdriver("Svga256",D);initgraph(&d,&b,"");for(;e<393216;){u=(256
*(s=(r=w*(m=e%1024)-2)*r+(i=w*(l=e++/1024)-1.125)*i)-96)*s+32*r<3?1:16+32*r+16
*s<1?2:0;if(u==(s=f=0))do{g=s*s-f*f+r;f=2*s*f+i;s=g;}while(++u<191&&g*g+f*f<4)
;putpixel(m,l,31+u%192);putpixel(m,767-l,31+u%192);}getch();closegraph();}
Re: DEEPZOOMing
Author: Jay R. Hill
Date: Tue, 22 Feb 1994 20:33
Date: Tue, 22 Feb 1994 20:33
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In article <1994Feb18.205648.13940@ast.saic.com>, hilljr@jupiter.saic.com
(Jay R. Hill) writes:
> If you get caught in one of those tornado like regions around
> a pre-periodic point you will see self-similarity.
Follow the sequence of cardioids above the period three bud up
and to the left. The sequence leads to a pre-periodic point in
a spiral path.
> Just break off to a branch, try to find a locally largest island
> MSet and you can find the whole array of structure shapes there.
I have posted 02281554.gif to alt.binaries.pictures.fractals, a
picture of the period 32 Mandelbrotie which is about 10^(-30) long.
The sequence is (for those who might wish to explore)
p Re Im
4 -0.15652016683375506179898665078782 1.03224710892283180167160153243019
5 -0.19804209936425384006706548935925 1.10026953729269853292946466130293
6 -0.21752674703051102736891220156735 1.11445426587329273196477528116824
7 -0.22491595128674005468532625520412 1.11626015745499183500126825424467
8 -0.22733132304700393850020723485875 1.11588359859402312790906438800341
9 -0.22800380192119768440593561256394 1.11546788192985807009468931540234
10 -0.22815164143855000105962532908523 1.11525874731625437467075180497712
11 -0.22816913033455737825914409091174 1.11517762946600472614870779890337
12 -0.22816402388442481254773547314825 1.1151512357871120510828460360341
13 -0.22815914494963267922362645982414 1.11514402267262310759199436509364
14 -0.22815677396428361530981906175257 1.11514249470941736215744240030232
15 -0.22815587332714247866717366366851 1.1151423411958116277129217944865
16 -0.22815558552315644808961321089189 1.11514241004226826652595546152242
17 -0.22815550855263108159568120988684 1.11514246716222284722700817563524
18 -0.22815549289168166313358642159102 1.11514249396830935731555596097925
19 -0.22815549163680663920717494619641 1.11514250394536590772360527215044
20 -0.2281554925322253512909550642675 1.11514250707688756197138804714047
21 -0.22815549319731556856424943014538 1.11514250789577121812967396959899
22 -0.22815549349959990133282720600253 1.11514250805508562105684202780953
23 -0.22815549360990878542334551974513 1.11514250806408643629140304957281
24 -0.22815549364391173034831426363938 1.11514250805273480892297383016298
25 -0.22815549365259618062956104965992 1.115142508045027862555012636284
26 -0.22815549365420261160997315103021 1.11514250804162740475671111807369
27 -0.22815549365424854012252068118215 1.11514250804041014487455664512442
28 -0.22815549365410736972499445466057 1.11514250804004171457292269370484
29 -0.22815549365401844991684724388258 1.11514250803994992821477773170794
30 -0.22815549365398028678543082111145 1.11514250803993389938200454331354
31 -0.22815549365396687998323940973346 1.11514250803993397968632052442503
32 -0.22815549365396289668839451272971 1.11514250803993570912551129479396
> Each with delicate filementary structures attached everywhere.
Try these parameters in DEEPZOOM (For a copy send me email).
DEEPZOOM 65000 -0.22815549365396289668839451272971\
1.11514250803993570912551129479396 64e-30 48e-30 640 480
In this picture the filements are relatively very long, going out
in slowly curving spirals.
> These little structures contain a 'genetic' history of the larger
> (less zoomed) structures you were looking at before.
Each filement has n=p-3 barbs on it where n is the number of
Mandelbroties in the sequence. This is because to get to this point,
we went past n branch points like the first one (branching to the right
ending at +i).
> It reminds me of the dust jacket descripiton on Poul's SiFi novel
> 'Tao Zero'
Make that Poul Anderson.
Warmly,
Jay "Not tonight honey, I'm at the end of the branch" Hill
int m,u,e=0;float g,s,f,r,i;char*_="-moc.cias.retipuj@rjllih=lliH .R yaJ\n";
main(){for(;e<3920;){u=(256*(s=(r=.0325*(m=e%80)-2)*r+(i=.047*(e/80)-1.128)*
i)-96)*s+32*r<3?28:16+32*r+16*s<1?35:0;if(u==(s=f=0))do g=s*s-f*f+r;while((f
=2*s*f+i)*f+(s=g)*g<4&&++u<29);putchar(_[++e>3840&&m<36?35-m:m>78?36:u]);}}
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