Math Is Fun Forum / Fractals in a simple way

bunman00 · 2012-12-14 20:03:16

I am a Gr. 9 student who is doing a project on fractals. I tried looking for explanations on the web but I am unable to understand them. I am very impressed by the discussions going on this forum and hope to receive a simple explanation on fractals and the equations related to it.
Thank you.

bobbym · 2012-12-14 20:21:40

Hi;

Welcome to the forum. The simplest explanation of a fractal comes from Benoit Mandelbrot, the discoverer as he told me and some computer and math types.

Look at figure 1, you see a circle. Notice the top has a nice curvature. Now look at the next drawing where we zoom on the top, it appears much flatter. Finally the third drawing under high magnification the circle looks like a straight line. Fractals are all the curves that do not have that property.

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bunman00 · 2012-12-14 20:29:18

Thanks a tons for the explanation.
I was actually looking for a detailed one although your explanation helped a lot.
Once again thanks.

bobbym · 2012-12-14 20:37:01

Hi;

That is the simplest geometric idea possible. The rest go into topology and iteration, things that are fine for experts but do not really help in the understanding.

Fractals are all the curves that do not have that property.

If you have seen the Mandelbrot fractal then you know that no matter how much you magnify it, it always retains its structure, retains its complexity as does Sierpinski's triangle. Because they do not flatten out like the circle does they are fractals.

The rest of it is algorithmic to me but if you want I can show you how to play the Chaos game as described by Michael Barnsley. It is a great way to introduce fractals.

bunman00 · 2012-12-14 22:42:38

@Bobby, o....ok....thanxx......Any further inputs for my project, I was thinking of making a model and some charts explaining mandelborts in simplified terms.

bobbym · 2012-12-15 00:17:23

Hi;

Did you search the net for some good photos of it?

bob bundy · 2012-12-15 04:36:41

hi bunman00

Welcome to the forum.

Fractal: a shape that looks the same at all magnifications.

My favourite is the Koch Snowflake. (Good one for Christmas decorations)

Stage 1: draw an equilateral triangle.

Stage 2: divide each side in 3 and go along for 1, out in a triangle for 1 step and back for 1 step, then back along the original side for 1 step. Sorry that's a confusing explanation. The pictures below show it lots better.

Stage 3. repeat stage 2 for all the straight edges.

Stage 4: repeat stage 2 for all the straight edges.

...............and so on ......................

The resulting snowflake is bounded easily and so has a finite area. But at each stage the perimeter increases by a factor of 4/3 so the perimeter tends to infinity as you continue through the stages.

Bob

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bunman00 · 2012-12-15 04:57:27

Oh yeah......I di....got some bit of info. If you think of something, pls. Let me know.

bunman00 · 2012-12-15 05:00:14

Bob, hi there....interesting.....will try to follow it, but it is sure confusing. I plan to decorates the back drop of my stand with clouds, Koch snowflakes etc.

bob bundy · 2012-12-15 05:30:14

hi bunman00

I posted in a bit of a rush as I had to go elsewhere.

This new diagram may help you to follow the stages.

It shows what happens to one side of an equilateral triangle as you do the next two stages.

LATER EDIT

I've added another.

Draw an equilateral triangle and by joining the midpoints of the sides divide it into 4 triangles.

(Repeat from here) Colour the middle triangle.

For the other three triangles, join the midpoints and divide into 4 triangles.

Go back to the repeat step.

Bob

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$View Image: kofractal.gif$

bunman00 · 2012-12-15 06:37:18

Thanks a lot for your explanation and by the way can fractals be 2 or 3 dimensional or can it be only
be between 1 and 2.

bob bundy · 2012-12-15 06:47:53

Did you see the edit of my last post? I've added a new picture.

I don't see why a fractal shouldn't be 3D but it would be hard to show it adequately.

Approximately, you could regard the Universe as fractal-like in the sense that galaxies are full of stars and the universe if full of galaxies. Of course the distribution is not regular, but you could probably imagine a universe where it was.

eg, stars in a cube lattice to make a galaxy.

galaxies in a cube lattice to make a super-galaxy.

super-galaxies in a cube lattice to make ............................

Bob

bunman00 · 2012-12-15 07:01:23

Doess anything on earth resemble a ideal fractal which infinitely repeats itself.

bob bundy · 2012-12-15 07:29:53

Good question.

My brother argues that the coastline is an example of a fractal.

If you define the division between the water and the land as the 'tide mark' then this is fractal like as it always looks like a wiggly line at any magnifiaction.

I am not so sure as the line is not uniformly wiggly, but I'm not sure that matters.

As a result the line must be infinitely long in the limit.

Maybe crystals are. When you split a diamond you get smaller diamonds.

Bob

anonimnystefy · 2012-12-15 07:45:19

The fractal I like the most is Menger sponge.

What's interesting about it is that it has a volume of zero!

Last edited by anonimnystefy (2012-12-15 07:45:31)

bob bundy · 2012-12-15 08:40:29

hi Stefy,

That's brilliant! A volume of zero !!!

That could provoke a long thread that beats the various infinity threads.

So if I walk towards it can I just go right through without any of my atoms coming into contact with it ?

Bob

anonimnystefy · 2012-12-15 08:45:40

Not quite...

Bringing the concept of a Menger sponge to real life is not possible, so I am not quite sure how (or if) it would react with atoms and molecules. And I think it wouldn't be a good thing to go there...

bob bundy · 2012-12-15 08:54:36

Oh how disappointing. You take a photo of one and show it to me, and then it turns out it doesn't exist. That's a great camera you've got there.

Come to think of it, I can think of lots of things that don't exist and also have zero volume.

There's a book written by bobbym for example called "My Age and Other Interesting Facts" that doesn't exist. It weighs nothing as well as having zero volume.

Then there's the snowman I made last year. That was made from all the ice I could find when the temperature was 20 degrees C.

And the gin and tonic I make when I've got no tonic. Oh yes, I've got no gin either.

Bob

anonimnystefy · 2012-12-15 09:08:50

The Apollonian gasket is also a nice one, but a bit less three dimensional...

bobbym · 2012-12-15 09:46:48

Hi;

bunman00 wrote:
Doess anything on earth resemble a ideal fractal which infinitely repeats itself.

bob bundy wrote:
My brother argues that the coastline is an example of a fractal.

According to Benoit your brother is correct.

There's a book written by bobbym for example called "My Age and Other Interesting Facts" that doesn't exist. It weighs nothing as well as having zero volume.

Yes and it is equally interesting.

bunman00 · 2012-12-15 16:49:10

Can you explain to me how a cloud, pineapple and fern resemble fractals and how a coastline is a fractal.
Thank you

bunman00 · 2012-12-15 16:58:16

And just to make sure a fractal is an image that has the same shape as the base when zoomed. I got that from library.thinkquest when I googled koch snowflake.

bob bundy · 2012-12-15 17:57:26

hi bunman00

A look at a map shows the coastline as a line that winds about apparently at random. If you view the map at increasing magnifications it still looks like that. You are perhaps expecting that the shape is exactly the same at all magnifications but I don't think that was what was meant when Benoit Mandlebrot first introduced the idea. I don't know the underlying math but you could make a start by looking at

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

I couldn't find much math theory there and it seems you might have to read his book. Maybe you can get it from a library.

As for a fern, take a look at this picture. Each fern leaf is made up of minature ferns which are made up of even more minature ferns .......

$http://1.bp.blogspot.com/-PuB07ZVgHds/TgADEK0skUI/AAAAAAAAAas/LIr0gD8y-gE/s1600/fern+fractal.png$

Bob

bunman00 · 2012-12-15 18:19:15

What is a non-integer dimension.

bobbym · 2012-12-15 19:45:47

Hi bunman00;

You asked for a little bit more.

I was actually looking for a detailed one although your explanation helped a lot.

Okay, you got it. After all what do Barnsley and Mandelbrot know anyway? The essence of fractals begins Isaac Newton and Arthur Cayley. Take a look at the drawing.

When Newton's method is applied to the equation x^3 - 1 = 0, the fractal in the image is created. See the x's those are the roots of equation. Now the red areas are all the initial conditions that converge on that root in the red. Same for the blue and yellow. No one ever imagined such structure in such a simple system until Arthur Cayley first tried to answer the question.

He had already solved it for x^2 - 1 = 0 and assumed that the cubic would only be slightly harder.

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#1 2012-12-14 20:03:16

Fractals in a simple way

#2 2012-12-14 20:21:40

Re: Fractals in a simple way

#3 2012-12-14 20:29:18

Re: Fractals in a simple way

#4 2012-12-14 20:37:01

Re: Fractals in a simple way

#5 2012-12-14 22:42:38

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#6 2012-12-15 00:17:23

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#7 2012-12-15 04:36:41

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#8 2012-12-15 04:57:27

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#9 2012-12-15 05:00:14

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#10 2012-12-15 05:30:14

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#11 2012-12-15 06:37:18

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#12 2012-12-15 06:47:53

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#13 2012-12-15 07:01:23

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#14 2012-12-15 07:29:53

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#16 2012-12-15 08:40:29

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#19 2012-12-15 09:08:50

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#20 2012-12-15 09:46:48

Re: Fractals in a simple way

bunman00 wrote:

bob bundy wrote:

#21 2012-12-15 16:49:10

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#22 2012-12-15 16:58:16

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#23 2012-12-15 17:57:26

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#24 2012-12-15 18:19:15

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#25 2012-12-15 19:45:47

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