How do the mathematics of fractals work?

[questionposter]

I read a few articles, and I already know how fractals themselves work, but I don't get how I'd write a formula for them or make them on paper. Basically all I know about them mathematically is that in a Julia set it just repeats a function over and over again, and a Mandelbrot set uses imaginary numbers, and the initial points don't escape to infinity.

Edited November 10, 2011 by questionposter

[the tree]

Strictly speaking, the Mandelbrot set doesn't have to be generated using imaginary numbers, you can get the same set using geometrical terms but it's messier and there is really no point. But it's pretty unclear what you're asking for. What is it that you'd actually like to know/understand?

[questionposter]

Strictly speaking, the Mandelbrot set doesn't have to be generated using imaginary numbers, you can get the same set using geometrical terms but it's messier and there is really no point. But it's pretty unclear what you're asking for. What is it that you'd actually like to know/understand?

 

Is there some way I can punch something into a graphing calculator and have it generate something?

[questionposter]

Ok, well, what's an equation for a fractal? Can I just plug in any equation and repeat it a certain way to make a fractal?

[Bignose]

http://www.amazon.com/Introducing-Fractals-Graphic-Nigel-Lesmoir-Gordon/dp/1848310870/ looks like what you want

[questionposter]

http://www.amazon.co.../dp/1848310870/ looks like what you want

 

I was thinking more of actual information to view.

[the tree]

Is there some way I can punch something into a graphing calculator and have it generate something?

Not really, you need very specific programs to plot fractals.

 

If you think of your oridinary plot of say [imath]y=f(x)[/imath], then you're highlighting a subset of [imath]\mathbb{R}^2[/imath], specifically [imath]\{ \left( x,y \right) \in \mathbb{R}^2 \, | \, f(x)=y \}[/imath], which is fairly easy to generate - it'll roughly be something like:

for x in range
y=f(x)
Image[x,y](color)=black
loop

Even for a calculator trying to plot [imath]f(x,y)=0[/imath] the process will amount to (but be more efficient than):

for x in xrange
for y in yrange
 if f(x,y)=0
  Image[x,y](color)=black
 endif
loop
loop

But checking to see if a number in the Mandlebrot set takes a lot more than just some arithmetic function, in short, it's just not what graphing calculators are for.

 

The Wikipedia article on the Mandlebrot set has a pseudocode implementation for plotting a Mandlebrot, but it's basically very roughly checking each number one by one.

[DrRocket]

I read a few articles, and I already know how fractals themselves work, but I don't get how I'd write a formula for them or make them on paper. Basically all I know about them mathematically is that in a Julia set it just repeats a function over and over again, and a Mandelbrot set uses imaginary numbers, and the initial points don't escape to infinity.

 

Then you realize that a fractal is a topological subslpace of [math]\mathbb R^n[/math] for which the Hausdorff dimensiion is strictly greater than the topological dimension. A fractal is not necessariy determined by a "formula". There are various ways of constructing fractals, usually involving some sort of limiting process.

 

Wiki is not always the best reference.

 

Try Fractal Geometry, Mathematical Foundations and Applcations by Falconer.

[md65536]

I read a few articles, and I already know how fractals themselves work, but I don't get how I'd write a formula for them or make them on paper. Basically all I know about them mathematically is that in a Julia set it just repeats a function over and over again, and a Mandelbrot set uses imaginary numbers, and the initial points don't escape to infinity.

 

What's been discussed here so far involves iterating over a grid of pixels and using a function to evaluate a color for each pixel.

 

To draw fractals on paper you'd probably want to do it a different way.

If you take a shape or a line segment, and modify it in some way to create additional (smaller) shapes or line segments, and then modify each of those segments in a similar way, and keep going until the details are small enough, you should end up with a fractal.

 

Some easy fractals to start with:

 

Mountain:

Start with two horizontal points at either end of the page.

For each set of two adjacent points, find the midpoint and then raise it or lower it randomly by some factor that is proportional to the horizontal distance between the points (so that the random offset gets smaller as the points get closer together), and add a point there.

When the points are close enough to not be worth subdividing further, connect them with a line.

 

Tree:

Start with one vertical branch from top to bottom of the page.

For each branch, add some specific formulation of child branches. Eg. split each branch into thirds, and add a "twig" 1/3rd along the branch, pointing 60deg to the left, with length 2/3rd of the branch, and add another twig 2/3rd along the branch pointing to the right with length 1/3rd of the branch. This particular formula will give you a lopsided looking tree.

Repeat for all new twigs/branches that are big enough.

 

If you do this and look at what you've drawn, you'll see that each branch on its own looks exactly like the full tree, only smaller.

 

 

Lame-looking snowflake:

Draw an equilateral triangle.

For each line segment, divide it into 3 sections and draw an equilateral triangle jutting out of the middle section. Repeat.

 

 

 

These are just 3 simple examples off the top of my head. There are probably much cooler, interesting ones out there, with instructions. It's not hard to come up with your own variations. If you look at simpler computer-generated fractals, you may be able to detect a pattern that can be used to draw each smaller iteration, based on a bigger iteration.

[Schrödinger's hat]

Serpinski triangle:

Draw a triangle.

In the middle of this, draw an upside down triangle, dividing it into four triangles

Everywhere you see an upright triangle, draw an upside down triangle in it until you get bored

 

Fern fractal's are also interesting: http://en.wikipedia....i/Barnsley_fern

They are based on copying the full image into itself in four different ways

Edited December 9, 2011 by Schrödinger's hat

[Xittenn]

Maybe look into the Gamma Function and Riemann Zeta Function, these seem to have properties that are considered 'fractal.' I've never bothered to study fractals, but I have noticed these while doing reading.

 

A quick google for Gamma Function Fractals gave me this guys blog, he seems to have some interest.

[DrRocket]

Maybe look into the Gamma Function and Riemann Zeta Function, these seem to have properties that are considered 'fractal.' I've never bothered to study fractals, but I have noticed these while doing reading.

 

A quick google for Gamma Function Fractals gave me this guys blog, he seems to have some interest.

 

A fractal is a topological space, not a function. By whatever definition you please, whatever "fractal" means to you it is not something that is at all smooth.

 

The gamma and zeta functions are analytic on their domeains and meromorphic in the complex plane. That makes them very smooth indeed.

[Xittenn]

A fractal is a topological space, not a function. By whatever definition you please, whatever "fractal" means to you it is not something that is at all smooth.

 

The gamma and zeta functions are analytic on their domeains and meromorphic in the complex plane. That makes them very smooth indeed.

 

 

Thank you for pointing that out to me. If you could maybe expand on your earlier statements with regard to "topological subslpace of [math] R^n [/math] for which the Hausdorff dimensiion is strictly greater than the topological dimension" - DrRocket O.O

Edited December 10, 2011 by Xittenn

[DrRocket]

Thank you for pointing that out to me. If you could maybe expand on your earlier statements with regard to "topological subslpace of [math] R^n [/math] for which the Hausdorff dimensiion is strictly greater than the topological dimension" - DrRocket O.O

 

 

That requires a rather lengthy discussion of the various concepts of dimension in topology, which goes well beyond the algebraic notion that applies to vectors spaces such as [math]\mathbb R^n[/math]. Hausdorff dimension is based on measure theory, which itself is a significant topic.

 

The book Fractal Geometry, Mathematical Foundations and Applcations by Falconer gives a nice treatment in a single source. While not directly concerned with fractals (it predates the work of Mandelbrot bit quite a bit), you might also like Dimension Theory by Hurewicz and Wallman which is the classic treatment of dimension in topology.

 

The point is that to really understand fractals requires quite a bit more than just the pop-sci treatments that are common. It is a subject that has received quite a bit of hype in popularizations, because of the pretty pictures. It is more than just the result of iterated functions (and less than what the hype would lead you to believe). You will have to decide for yourself if it is worth the effort.