What is the deepest fractal?

[AluminumStudios]

Hi,

I've been interested in fractals for a while now. I've done a little light reading, wrote a C program a few years ago to draw mandelbrots, and played with the DOS Fractint.

I understands the basics of them, and how arbitrary precision math routines let you do deep zooming, where the data values become EXTREMELY small, much smaller than a 64 bit or 80 bit value that is native to a CPU can hold ...

My question is what is the deepest fractal that has ever been drawn/calculated? I read that Fractint (a DOS fractal program still in development, currently at version 20) can basically zoom into a fractal so far that "you can look at an area of a fractal the size of your screen, on a fractal that is the size of the orbit of Jupiter around the Sun."

Has more advanced software been written? Have crazy deep fractals ever been caluculated on super-computers? Is there a good 32 bit fractal generation program for Windows that has the same deep zooming abilities as Fractint (which due to it's 16bit nature is a pain to use in the background on Windows 2000)

Thanks in advance.

 

[Shalmanese]

fractals can be zoomed in to any arbitrary LOD. They rely on formulas not bitmaps. you just have to dynamically rescale your variables every so often.

 

[AluminumStudios]

I understand what fractals are .. but I was asking what the deepest fractal zoom done was. The bignum library (something I believe Fractint uses) and CPU power have their limits, meaning there has to be a practical limit as to how deep a fractal can be zoomed on current computers.

I was also curious if there was any deep zooming software better than Fractint.

 

[Shalmanese]

It depends on what speed you want to go at. if your zooming in at 10x per frame, then obviously you can go very fast but you miss out on a lot of detail in between.

 

[wurmyhi]

I would hazard that the "deepest" fractal is the Universe...


...but then again, I might have read Rudy Rucker's "Spacetime Doughnuts" one too many times...

 

[Smilin]

I thought I saw some software one time that would actually do infinite zooming somehow but it didn't do it very fast at all (couple minutes per screen methinks).

I wish I could think of the name, I'm sorry.

Checkout this program called XAOS though. It runs well in windows in near-realtime and has direct X support and whatnot. It doesn't zoom too deep and it's more of a toy than a real tool but man does it make some cool eyecandy!!! (Think color cycling, continous rotation and motion-blur zooming)


Fractals are dope.

 

[Placer14]

Does anyone want to share their love/passion of fractals with someone interested in expanding their knowledge?

(ie.: What are fractals?)

 

[Haircut]

Originally posted by: Placer14
(ie.: What are fractals?)

Think of two magnets on a piece of paper, with a metal pendulum hanging above them. When you release the pendulum it will move about until it comes to rest over one of the magnets.

Now imagine drawing a line on the piece of paper that shows the transitional lines where letting go of the pendulum one side of the line it will go to one magnet and on the other side of the line it will go to the other magnet.
This is a chaotic system, i.e. changing the position of the pendulum very slightly can cause it to go to the other magnet. The pattern of the transitional lines is effectively a fractal.

To get the patterns that you see in the Mandelbrot set a pretty simple equation is used and the Mandelbrot set is just a pattern in complex space showing which starting values for this equation tend towards infinity.

 

[AluminumStudios]

(ie.: What are fractals?)

Placer14 - fractals are more or less graphs of equations that use imaginary numbers. They produce increadibly detailed, intricate patterns that closely model many systems in the real world such as crystals, leaves growing on tress, magnetic patterns, etc.

I'm sure you'll recognize one as soon as you see it. THe most well known is a "Mandelbrot" fractal named after a mathemetician. Here's a URL I found real quick with some fractal images - http://www.mehmib.freeserve.co.uk/

THe neat thing about fractals is that you can "zoom" in on them. As you do this they reveal an infinite amount of detail. But as you zoom in, the points become smaller and smaller (more and more decimal places) and take much more CPU power to calculate.

My interest largly lies in the mathematical computation of them.

 

[Moohooya]

Assuming most people use built in numeric types, the VC++ long double is an 80 bit double, 1 sign bit, 15 bit for the exponent and 64 for the mantissa. This gives you increments as small as 1/(2^(2^14) * 2^64) = 4.5e-4952
So I'd assume that many people have gone this small as no special math libraries are required. How many people have gone smaller, someone I'm sure.

Assuming a fractal from 1 to -1 displayed on a screen 15" across, zoomed all the way in, the fractal would be 2.6e4944 km across. Given a lightyear is 9.4544e12 km, we are talking 2.8e4931 light years across. I've no clue how large the galazy is, but I'd guess if your screen was the size of a quark, the fractal would still be larger then the galaxy.

Think I'm kidding? Check out Powers of 10 they only have 39 or so powers and we have a whopping 4952

Moohoo

 

[Hayabusa Rider]

Originally posted by: Moohooya
Assuming most people use built in numeric types, the VC++ long double is an 80 bit double, 1 sign bit, 15 bit for the exponent and 64 for the mantissa. This gives you increments as small as 1/(2^(2^14) * 2^64) = 4.5e-4952 So I'd assume that many people have gone this small as no special math libraries are required. How many people have gone smaller, someone I'm sure. Assuming a fractal from 1 to -1 displayed on a screen 15" across, zoomed all the way in, the fractal would be 2.6e4944 km across. Given a lightyear is 9.4544e12 km, we are talking 2.8e4931 light years across. I've no clue how large the galazy is, but I'd guess if your screen was the size of a quark, the fractal would still be larger then the galaxy. Think I'm kidding? Check out Powers of 10 they only have 39 or so powers and we have a whopping 4952 Moohoo

How long would this calculation take to perform?

 

[Era]

Originally posted by: Moohooya
Assuming most people use built in numeric types, the VC++ long double is an 80 bit double, 1 sign bit, 15 bit for the exponent and 64 for the mantissa. This gives you increments as small as 1/(2^(2^14) * 2^64) = 4.5e-4952
So I'd assume that many people have gone this small as no special math libraries are required. How many people have gone smaller, someone I'm sure.

Assuming a fractal from 1 to -1 displayed on a screen 15" across, zoomed all the way in, the fractal would be 2.6e4944 km across. Given a lightyear is 9.4544e12 km, we are talking 2.8e4931 light years across. I've no clue how large the galazy is, but I'd guess if your screen was the size of a quark, the fractal would still be larger then the galaxy.

Think I'm kidding? Check out Powers of 10 they only have 39 or so powers and we have a whopping 4952

Moohoo

C+Z(n)^2=Z(n+1) . Repeat that enough many times to reach an exponent of -4952. You'll have almost 5000 leading zeros followed by 19 meaningfull numbers. Now what would the cumulative error be?
IMO the picture would look more like uncle Joe then Mandelbrot set.

 

[Moohooya]

How long would this calculation take to perform?

No longer than any other long double calculation. My machine can do a billion of these in around 6.5 seconds.

C+Z(n)^2=Z(n+1) . Repeat that enough many times to reach an exponent of -4952. You'll have almost 5000 leading zeros followed by 19 meaningfull numbers. Now what would the cumulative error be?

A fractal is an image of how fast or slow a set of points when transformed by a function move towards some limit. It does not matter how many leading or following zeros there are. Admitadly, we do need some precision during these calculations or else our results will be wrong, so perhaps an exponent of only -4946 is feisable. However the precision with an exponent of-4938 is the same as that with an exponent of 1.

IMO the picture would look more like uncle Joe then Mandelbrot set.

Since there are an infinate number of fractals, each being infinate in size and infinate resolution, I'm sure you'll see Uncle Joe in there more than a few times!

 

[Shalmanese]

Actually, even though a fractal is infinite and non-repeating, it does mean that every permutation exists within it.

Consider the decimal expansion .101101110111101111101111110.....
Even though this sequence is infinite and non-repeating, the number 2 will never appear anywhere.

 

[zogg]

Whats an uncle Joe?

 

[Shalmanese]

Uncle Joe

 

[Moohooya]

Consider the decimal expansion .101101110111101111101111110.....
Even though this sequence is infinite and non-repeating, the number 2 will never appear anywhere

Well you'll only get values that are valid. A fractal is a pictorial representation of a mathematical function, and hence a picture of Uncle Joe is valid, not that this means you will see Uncle Joe.

I'm yet to be convinced that you won't get Uncle Joe in a fractal. While you won't get the infinate sequence of 1010 etc in your sequence above, you would get it in an infinate number of infinate sequences.

Imagine you take Uncle Joe and resize the image to 8x8 in 8 colours. There are only 2048 possible images. You will see Uncle Joe in a fractal, frequently. If the image is 256x256 in 16 bit colour, you only have 65536**65536 possible images. Sure that is alot, but finate.

I'll have to think about this some more. Any math gurus following this thread?

 

[Shalmanese]

Fractals, although never repeating, will have certian limitations placed upon them. While I am sure you could come up with SOME fractals that showed an uncle Joe, I am pretty sure that fractals like the Mandelbrot have inherent limitations that would forbit it from exhibiting something like an Uncle Joe (I guess it would also depend on how you are representing the pattern space)

 

[Moohooya]

I admit that the chance of finding Uncle Joe in the Mandlebrot is pretty slim to none. Ferns, waves, sea horses, yep got all of those.

I bought this book on Fractals 18 or so years ago (perhaps called the Chaos of Fractals.) There are some pictures in there that look like Dr Seuss's faces! Long, rough edges, weird colours with stripes, big eyes etc.

 

[Smilin]

My fractals, looky:




http://home.indy.rr.com/smilin

 

[Joemonkey]

Heh, sorry this is not really "high tech" but I do in fact have a tattoo of a (modified) fractal pattern...