How can I make my own fractals?

NemoReally

around the one and a half second mark in Mathcad 15 on a Dell Dimension 521 (Athlon 54 dual-core 2.5 GHz with 4GiB RAM running Windows 7 32-bit) - the lighter the image, the quicker the escape. Around one second for an n=64 standard Mandelbrot over the same x and y ranges.

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piercebeatz

That's interesting that yours rendered much more quickly than Bill's... Do you think that it is because your program is faster?

I'm considering investing in a math program solely for creating original fractals like this one (although good graphing abilities and other features wouldn't hurt). Should I get the Mathematica student edition? Any other recommendations?

Thanks

NemoReally

Depends what you mean by 'program'. I'm using Mathcad, a completely different application. I think the Mathcad program I showed is equivalent to the Mathematica one which implies that Mathcad is faster, but it may be that the Mathematica Table function is slower than an equivalent Mathematica For program.

If you're going to solely create fractals, then I'd recommend a specialized fractal application. If you want a generalized application then it would be helpful to know what else you would want to do. Mathematica SE seems like a very good choice. However, I use Mathcad because it's a good general purpose mathematical application that allows the user to use near-standard maths notation to write equations and display/graph results on the same 'piece of paper'. The image I posted is exactly as the Mathcad worksheet looks, including the fractal plot (*).

Have a look at http://www.ptc.com/product/mathcad/ - you can download an evaluation copy and the user forum http://communities.ptc.com/community/mathcad is a pretty good place to ask for advice.

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(*) The gray dotted line that goes across the plot is where the page boundary would be if I'd printed the worksheet as is - I'd move stuff around to avoid this if I was actually going to print it. The image below shows a variant of the program in which I'd shuffled the 'regions' further up the page. It also illustrates another feature of Mathcad, that is the ability to write programs across the page as well as down the page. Mathcad evaluates left-to-right then top-to-bottom. Consequently, the second function f supersedes the first definition resulting in a Mandelbrot like figure (note the power of z is 2.1 rather than 2, hence the bident rather than the single prong along the negative x-axis).

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piercebeatz

Do you know of any specialized fractal programs? Also, is it possible to color the fractals that you render in Mathcad? Do you know if this is possible for Mathematica?

Bill Simpson

I assumed from the wording of the posts before I got into this that the original poster was a beginner, didn't know a lot about this and just wanted to get started. What I originally provided in Mathematica was intended to be as simple as possible so that a beginner might be able to understand it. I carefuly and intentionally put nothing more in that than absolutely necessary. There are things that could be added that would speed it up, but with additional complexity to have to understand.

If the original question had been "I have years of experience, how can I make this go fast?" then my answer might have been very different.

Take a look at
http://mathematica.stackexchange.com...enerating-code
and see how this speeds up the process by orders of magnitude. I can understand some of that, but I have not yet been able to see how to incorporate your fractal function. Those folks are usually way beyond my skill level.

To answer your question about color, change my original simple code thus

ArrayPlot[Table[n = 0; z = a + I*b; r = z; While[Norm[r] < 3. && n < 20, r = r^z; n++]; n, {a, -3., 3., .01}, {b, -3., 3., .01}], ColorFunction ->Hue, ColorFunctionScaling -> True]

While I really hesitate to describe in "market terms", everything comes at some price. If you want to change the way a result looks you have to pay some price to accomplish that, if you want to make something faster you have to pay some price to accomplish that. The price may be small or not.

As a general rule of thumb I expect that if figuring out how to get something sort of working takes x then figuring out how to get it really exactly working precisely correctly takes between two and ten x more, getting a graphic result display that sort of shows what is going on takes another x, getting the graphics correct to show what is going on takes two to five x more and getting the graphics to be exactly precisely they way that you want them in every detail takes five to twenty x more and possibly even infinity times x more for some people.

I don't want to diminish your motivation or excitement. However you might keep this in mind. Every math operation, +,-,*,/,^ creates a bit more error. If you only do a small number of those operations then you only get a small amount of additional error. If you do lots and lots and lots of those operations you get more and more and more error. Fractal calculations do few operations in some places. Those places are almost always a single simple color. Fractal calculations do vast vast numbers of operations in some places. Those places are almost always where the complicated patterns and colors appear. Thus I propose that it is possible that almost everything about fractals is little or nothing more than a graphical display of floating point calculation error. There is a lot more that could be written to support this, but this is probably long enough already and saying anything against the myth of fractals is perhaps a lost cause anyway.

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NemoReally

a. I'm afraid I don't know of any specialized fractal programs - Googling should give you a list.
b. Yes, see image below. It makes use of a short function mono2colr (not displayed) that indexes the count into a colour map. Note that I've changed the function to raise z to the cube root of a for this example - gives it a slightly Escher look, IMO.
c. Yes, Bill Simpson has already answered this one.

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piercebeatz

Bill, like I said, I don't know anything about computer science, so you were correct about your assumptions. When I get the chance, I'm going to invest in the Mathematica student edition and continue to play around with these fractals (and maybe one day write an award winning essay!).