Fractals are wonderfully "complex" mathematical equations that self-replicate and often make gorgeous computer-generated pictures and animations. Sometimes adjusting the colors a little on the same equation/setup changes something hideous [IMHO] to something gorgeous, like the below. It doesn't matter if you're mathematically inclined or not, it is more about "fiddling" and art appreciation on the "human scale" IMHO.

Do some searching for fractal galleries and post the ones you like here.

I'll open with a "classic Mandelbrot":


["Julia" is one of the other common fractal sets]

The above Mandelbrot Set is plotted on a quandrant of positive, negative, imaginary and negative-imaginary numbers. A given coodinate is fed into an equation, the answer is refed and refed into the same equation ad-infinitum, and the original coodinate is then colored a certain way depending on what the result does - whether it quickly increases to infinity, or decreases to zero, how fast or slow it does this, etc. The equation itself is deceptively simple.
The mechanism at work is often compared to thatwhich results in the beauty found in nature.

Someone correct me if I'm a little off, I haven't read about fractals for many years.

Excellent Sanders- that's in the 90+% correct category. Here's a tutorial written by a self-proclaimed NON-mathematician (although I claim that myself, too). He is much less "mathematician" than I apparently.

http://www.fractalarts.com/ASF/Tutor1.html

Sanders wins a new [rather intense] Fractal of the Day:


From the page "Fractal Adventures":
http://home.att.net/~Novak.S/

This one ought to keep anyone who has studied Quantum Mechanics pondering for a while...

Woo-hoo!

Some might call that "the hand of God," but I dunno here. I DO KNOW that it is certainly beautiful, and amazing.

dMole

You have hit on one of my favourite topics and one I nearly brought up earlier. I was using an 8-bit 32KB RAM computer in the 1980s to generate these beasts. The program had to run for about 24 hours to complete on a zoomed in area. The numbers of iterations were incredible at times and with a recursive program there is alweays the problem of memory fill if variables are repeatedly declared - one has to work round this in 32KB of which only about half was free for code and variables because of the demanding graphics mode required for colour.

Do you have any, pref' freeware, recommends for software for winblows Xtra Pernickity? *****************

Another area that grabbed my attention was Bio-Morphs - had an 8-bit prog' for that in the late 1980s - that's how I got into Dawkins.

Edit: ***************** Sorry for the stupid question having visited the link and had a browse - I could lose myself for hours on this and Stellarium.

Why do I start thinking Hubble here?

Couple of simple ones to start with from the Mandelbrot set and Mandelbrot Explorer:



and zoomed in:

Hi O892,

Check the links I posted over on the "Broken Science" thread- there are a few fractal music programs over there (I still need to install them and check them out).

Back when I first got into fractals, I was running a [then] "souped up" 80286 "AT" with 1024 KB of RAM, a 40 MB hard disk (most still ran off floppy disk then), DOS 5.0, and a 287 math CoPro. That was a "screamer" back in the day. My drafting teacher was jealous- he had the 2 best drafting machines in the school district, and his only had 30 MB "Winchesters," and we had a 2-3 week rotation to be able to use those CAD machines (1980s).

I was running Fractint off a single 3.5" floppy IIRC, and my fractals ran pretty quickly at 256 color. There was a bifurcation fractal that I loved- it sent me into that whole Robert Frost "Two roads diverged in a wood, " causality, philosophical mode. I can't even recall the name of that one (it might simply be "bifurcation").

EDIT: I am still apalled/disgusted at how much faster the CAD programs and spreadsheets ran under MS-DOS. Of course, I have an admitted strong bias towards LINUX/UNIX myself. Computers designed for the masses, I suppose...

Mostly I just browse the galleries periodically, because of the time factor you mentioned.

Also, so did you just generate the 2 Mandelbrots above that quickly then?

Hi O892,

On the AI/BioMorph thing- I had heard of them vaguely but never got into it. If you need an "AI" laugh, see posts #245 onward at:

http://pilotsfor911truth.org/forum//index....&p=10742845

'Screamer'. You can say that again compared to what I was using - did you cop that 32KB RAM 8-bit computer I mentioned?


Ah! Yes! Fractint was what I tried on the add-in processor card of that Archimedes I mentioned which was running DOS when I started. Even Win 3.1 and 3.11 was a bit bloated for generally available ram.

I still have the little notebook that somebody threw out because it didn't have a hard drive - the original customer had requested the hard drive not be supplied as new - incredible. They had also managed to unship the keyboard screws which were floating about inside. I checked it over, refitted the keyboard and powered it up with some DOS discs I had on floppy (oh! That drive had to be secured too) and checked it out.

Being satisfied I sourced a small HD (smaller form factor than the original which required a bit of extra work sourcing a connector and securing physically) but almost outside the BIOS limit for formating - I had to do a custom one.

It was DOS only but once loaded with Wordstar, Lotus 1-2-3 and a Skymap program it was useful. Very useful for the evening Lotus course I was asked to run at short notice in local college. I prep'd some course notes for that on the Acorn because I could run in a window and do screen grabs as illustrations that way which I then incorporated into notes in a superb Acorn based DTP program Publisher produced by the same company that did the Artworks vector graphic software which morphed into Xara for Windows.

I once found computers quite interesting and exciting well for an ex airframe metal basher, flying controls, hydraulics, pneumatics airframe fuel systems and engines bod (always unkindly called 'grubbers' or 'oilies') they were.


Ah! Those moments of serendipitous bliss inspired by good background music (I prefer classical or even pre-classical baroque Bach and Co., or sometimes 'The Man in Black' or 'The Stones' or 'Shadows') and a wee dram of something and a little turkish for the smoker.



I blame IBM. If they hadn't given that 'person' a chance to use a near ripped-off OS, I layed out the history here:

PC History

then the world would be a better place. When I think of how much an Acorn RISC PC managed with its humble power requirments all I can think of is that under the hood of Winblows is much sloppy coding with probably most of it dedicated to preserving the scource of the 'Crown Jewels'.


No I don't think so. I think the way it worked was a kind of cascade down through the zoom levels with the image at each level being split into a number of cells in a grid. Clicking on a cell takes you to a zoom of that cell which is once again split into a grid of cells, choose a cell and zoom in to another grid and so on. It worked so quick I don't think there was time to generate on the fly.

As for inserting them here, hithertoo I have been using an FTp transfer programme with a dial-up, that's all there was when I started and am quite used to keeping image file sizes down, on the Acorn RISC-PC, that is how my website was assembled and those pics in my sig here added, I decided to bypass that and work through the ftp routine in Exploder 7. Setting that up took longer than the fractals - I had to hunt out my log in details as it worked differently on the Acorn.

Wow, those are some GREAT fractal images - beautiful!!!

And we sort-of almost have been talking about the connection between the math behind these fractals and the mechanisms behind the "beauty" in nature ... but one word is conspicuously missing in this (highly enjoyable) thread ... "feedback". The key mechanism in the fractal-producing equations is feedback - the result keeps getting re-fed into the equation. Nature, or I should say, the way DNA is read in the protein-producing process, is very similar ... "do this over and over until this". DNA is not a blue-print, it's a recipe - it's more like baking a cake than making a building - a set of instructions from which the result can't be visualised - you just follow the recipe and you get a leaf - or a mandelbrot set, or a human baby. And it should be no surprise that the "feedback" mechanism, because it's so effective, and frugal (from an IT perspective) would have been allocated by nature to produce the deep and expansive beauty that we see everywhere.

Hey, who started this thread??? Kudos!

Aha, I think it was a Lyapunov bifurcation fractal, but I can't find an image as yet. When you want the "heavy duty" mathematics- see the Russians- period. I hear that Russian mathematics and novels are directly linked to the length and severity of their winters, somehow.

The Wiki page is actually extremely good on this subject. Give it a look if you're not familiar with these mathematical wonders (it has plenty of pictures).

http://en.wikipedia.org/wiki/Fractal

The original Fractint still appears to be alive and well:

http://www.fractint.org/

It should run quite well under a Win 95/98 DOS window (cough* OF * cough ) EDIT: There appears to be a Win9X version now.

Sanders is clearly requesting some fractal "ferns." I didn't find images yet, but a good Aussie page with generating parameters and small photos is at:
http://www.home.aone.net.au/~byzantium/ferns/fractal.html

EDIT2: The Fractint "plants" tutorial:
http://spanky.triumf.ca/www/fractint/lsys/plants.html

Another good fractal page:
http://sprott.physics.wisc.edu/fractals.htm

And just something pretty:

Sorry about that Sanders but feedback is kinda implied in the term recursive procedures along with variables which I mentioned in my opening post here.

Right - "kinda-implied". But not duely emphasized, at least not for us laymen. I fall by the wayside when it comes to the math, it's over my head ... but I understand the importance of feedback - negative and positive - in fractals, in nature, in our economy, in average American Joe's relationship to his government, in relationships among P4T members, in everything - both positive and negative. Negative feedback tends to equalize things .... positive (sort of a misnomer - actually the evil twin of its fair negative-feedback cousin) feedback is a highly volitle force to be feared .... BETA Video, anyone??? Anyway, positive feedback yeilds the white (or black? - depending on how you color it) part of the Mandlebrot Set. And the housing bubble. And almost everything where human greed interferes in the interest of corrupting the system in order to make a buck. Negative feedback is where we wake up and rise up and throw the bastards in jail.... but as of this posting there is only positive (not very "positive" IMO) feedback going on.

Hey, I've got an idea... [Uh oh- RUN!!! - JK ] Let's re-define the 'positive' feedback as "uppity feedback" or alternately, "propaganda" or "spin" [understood to be "up"ward in this context- gawwd I'm glad my name isn't "Ward." ]

Just an idea...

Thought I would bore you with one more fling for now:



and three derriratives from that one, the first with a Difference Clouds filter and contrast enhanced:



next with a Stained Glass rendering with small cells to be more like a mosaic than the mosaic filter:



and finally an extrusion, crop and contrast enhance, could this represent the Big Bang?:

I've been PM'ing with Omega a bit about the terms 'recursion' and 'feedback' - we're all talking about the same thing, and on the same page. (For the record.)

Wow, where do you guys find these fractals? Those are gorgeous. How is it that a slight instability in a simple mathematical equation can generate such images?? ... it just floors me.

Hi Sanders,

I've provided links to all the galleries that I've been looking at (either here or on the "broken science" thread). I'd wager that several university math and physics departments or professors have some galleries that I haven't found yet.

O829, did you generate the new one (or 4)? I wonder what equation "set" that one is- I don't think I've seen that one before. Are you using Photoshop filters to modify them? I really liked the "Big Bang" one.

I had an old book on fractals that came with a CD and had software and galleries- I really ought to find it.

Looks like the book is out-of-print:
http://www.amazon.com/Fractal-Creations-Bo...s/dp/1878739344

Here's another that looks excellent though:
http://www.amazon.com/Colours-Infinity-Bea...=pd_sim_b_img_3

EDIT2: Apparently David Gilmour did the music for the [non-NTSC] DVD included with the book above that has been on PBS before- Gilmour's proof enough for me.

A search for 'fractal gallery' ought to turn up a lot.

EDIT: I used the quick-n-dirty "self-replicating" and I think we are all saying the same thing (on 3 different continents of course ).

Hi dMole. All the fractal renderings that I have posted up here are from using Mandelbrot Explorer (so it is created from the Mandelbrot set function):

http://www.softlab.ntua.gr/miscellaneous/mandel/

I zoomed in around the fine tracery leading from the lower 'Hudson Bay',

I have not been able to retrace my steps exactly but it went something like in this diagram which gets you to an area where you could find exactly what I have had previously:



Hope that clarifies. Plenty to find variations on a theme along the way there.

Yes I did post process in Photoshop with the filters indicated in the note preceeding each image. Photoshop Element 1 BTW. I would invest in CS3 if it wasn't so bloated with stuff I don't want and which would just hog the machine. I have Paintshop Pro v 8 (last version before that started adding lots of extra unwanteds) for when I need masks or such. I have had Pantshop Pro versions since they were freeware and shareware on magazine cover-discs. I bought v3 for use on Win 3.11, then v 5 for Win 95.

Grr! That HTML 'center' tag gotcha. First try I got the opening tag correct and then spelt it the British way 'centre' in the closing tag, which of course failed!

OK A few more:
These are all done with another program Tiera-Zon as suggested in the galleries at:

http://home.att.net/~Novak.S/

Tiera-Zon can be found at:

http://home.houston.rr.com/fergusonsc/Tierazon-v29/

The first one demonstrates that monochrome does not have to be boring:



Slightly more colourful:



and a detail from the above:




Tiera-Zon is full of variations on a theme the image at left is as produced and the one at right with a Photoshop Difference Clouds filter applied followed by slight lightening and then slightly increased contrast:


I found a Fractal Program Stat's page:

http://home.att.net/~Paul.N.Lee/Most_Used_Counts.html

which provides a handy list of programs with the most popular indicated. How accurate and complete this is I defer to dMole on.

Flarium24 can be found at:

http://www.softlookup.com/display.asp?id=15964

it is a free download but I have not tested it.

If you have money to spend then Ultra Fractal 4 looks good:

http://www.ultrafractal.com/

There are some nice videos around as well. I like this Mandelbrot Julia set morph.
http://www.youtube.com/watch?v=tEKDU28ATLk
and Chaos Visions 1.2
http://www.youtube.com/watch?v=n14M9pl5KeY
Julibrots were probably my favorite exploration back in the early 90s with Fractint, I think., though I did some wild and crazy equation invention experiments as well. Now you can do a lot more in terms of animations, etc.
Here's a nice Julibrot capture. Now imagine that figure as a an explorable three dimensional object, colored based on iteration positions in the Mandelbrot set. Then imagine a Julibrot video animation which changes over "time" (the X axis on the M set). Sort of like the first video linked above, only better. That's where my mind consistently goes when I think of such things...

Here are a couple more videos which at least approach the sort of thing I'm talking about...
Interactive Visualization of Quaternion Julia Sets
Dynamics in the Quaternions

Since the Mandelbrot set can be considered a "map" (if you will) of all the connected Julia sets, basically what I'm visualizing is the evolution of the M set in 4D "space/time" using RGB color values to track how far from the origin (in 3 dimensions) the function has travelled with each iteration. The fourth or time dimension would consist of the non symmetrical axis (X axis) of the M set, so each sequential frame in the video would show a 3D snapshot from a little bit further along the X axis. Not particularly easy to explain in words, but I have a rough picture what it might look like in my head...

I had not intended to add any more fractal diagrams as there is nothing inherently clever about what I am doing. However I generated the one in this post yesterday using Tiera-Zon, no post processing, and it took my breath away. I think that you will agree that it is a stunner:

Wow. You created that?? Awesome.

Here's a purist "old school" fractal- it's even from the Wisconsin Univ. physics dept.

http://sprott.physics.wisc.edu/fractals.htm



Here's an animated one:


http://sprott.physics.wisc.edu/fractals/animated/vclutq.gif

EDIT: TieraZon link at post #21 looks broken O892.

EDIT2: Not so animated anymore. Let's try this:



http://sprott.physics.wisc.edu/fractals/animated/lnqfvo.gif

OK one more- a HIGH VOLTAGE Lichtenberg Figure [/sculpture IMHO]:

http://en.wikipedia.org/wiki/Image:Square1.jpg

As are a few others thrown up by a Google.

However the following site has a link to download the program, I have not tried it so cannot guarantee its authenticity or freedom from nasties. Downloader beware:

http://www.fractalarts.com/ASF/Download.html

I might have a rummage around there myself later.

Edit: Sorry for triple post.

Sheeeeeeeeeeeeeesh! Frigging internet, the link dropped and I thought it had stalled so hit Stop [X] and tried again when the connection looked up only for it to drop out again. This is beginning to pee me off big time!

Waiting for connect.

Go.

Thanks O892.

Going back there took me to Terry Gintz' excellent fractal page. I may need to purchase his software.

http://www.mysticfractal.com/Galleries.html

http://www.mysticfractal.com/Downloads/Login.php

Here's one of his samples:



http://www.mysticfractal.com/fractal_mirage/starpath.jpg

EDIT: The open-source Fractal Forge looks excellent from the screenshots.

http://www.fractovia.org/uberto/

EDIT2: More pages, linked from the Fractal Explorer page:

http://www.eclectasy.com/Fractal-Explorer/index.html

http://www.videocd.org

http://www.videocd.org/thebest/fe202.zip

http://fractalfairy.com

Fractal Landscapes
http://fractalfairy.com/~fexplorer/fl_libr.zip

User Formula Compiler
http://www.fractaldaydreams.com/compiler.zip

http://www.harleyangel.com/~feusers/compiler.zip
-------------
http://www.fractaldaydreams.com

3 of my virus scanners just blessed the above .ZIP files. Fractal Forge doesn't require an install or registry BS- you can just drop the files into a new folder wherever.

I noticed that it had a Mandelbrot named for Turbofan, or vice versa. I merely changed the color palette to "volcano" and here is the result:



It was quite nice when color cycled too, BTW. There is a way to create animations in Fractal Forge, but I didn't dig that deep.

EDIT: The Fractal Forge help file is 32-bit Windows .CHM, which Vista does not support- ask Billg Brother why. The easiest fix is to get a copy of "WinHelp32.exe" from a Win9x/ME/2000/XP computer and drop it into your "main" Windows directory. More info here:

http://www.codejacked.com/viewing-old-help...-windows-vista/

Great stuff dMole.

I figure we have just unlocked an Aladdin's Cave of fractal stuff.

From:

http://fractalfairy.com/

http://wallpapr.fractalfairy.com/wallimages/jwp15.jpg



http://wallpapr.fractalfairy.com/wallimages/jwp07.jpg



And an animation that I really liked (13.5 MB):

http://avi.fractalfairy.com/avis/bird-c-.avi

Here are a couple that I created a while back under Fractal Forge that maturin's recent St. Elmo's Fire post seemed relevant to IMHO (plus I need to finally move them off my desktop). These are "basic" Mandelbrot fractals.

EDIT: The 2nd attachment didn't work. I color cycled the same one above.

Thanx for the fractals, dMole - I really enjoy this stuff.

Here's another for Sanders (since he's into Dragons).

Phoenix Dragon

http://www.mysticfractal.com/2dfractals/PHDRAGON.jpg