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This is the second part of a series of articles about fractals, aiming to introduce the mathematical side of fractals to the DA community, who are familiar with the artistic aspects of them. The information given will be very basic and won't require anything beyond basic mathematical knowledge.

Note: Even though the articles are only loosely connected to each other, they are, nonetheless, parts of a series and it is recommended that you read them in the order that they were published for a more complete understanding.


<a href = news.deviantart.com/article/34…>Part 1: What Are Fractals? : An Introduction
Part 2: The History of Fractals - 1 : Fractals before Gaston Julia
Part 3: The History of Fractals - 2 : Julia and Mandelbrot Sets
Part 4: Fractals and Computers : IFS and Escape-Time Fractals
Part 5: Fractals and Art : Mathematics Meet Art

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The History Of Fractals, Pt 1

This article, especially the second part where I will introduce some early and simple fractals and fractal sets found by certain people, will require a little more than just basic mathematical knowledge (i.e. some calculus). I will try to simplify as much as possible, but I’m not a teacher and have not exactly mastered the mathematics of fractals, so it might sound a little too complicated, so be warned :). If you have any questions about the parts that I couldn’t explain well, please do ask and I’ll gladly try to answer them…

Fractals of Nature

As mentioned in the first part, fractals are the nature’s geometry, so it’s only appropriate to start this article with a few examples of fractals from nature.  One classical example is a tree:

Tree by fireman55 ---> Branch by twomixjunkie ---> Veins by SlinkyJynx

As you can see, from roots to branches to the tiny veins, the tree is a natural fractal that shows the same branching pattern at all scales. Another popular example from Kingdom Plantea is the fern:

:thumb6889875: (early growth period) :thumb67256616: (fully grown fern)

As a matter of fact, whenever you’re thinking of a branched structure in nature (veins, neurons, streaks of lightning etc.) you’re thinking of a fractal due to the natural, random distribution. Two relatively different examples of natural fractals are;
Snowflakes  snowflake by felixw,  and the Romanesco  :thumb46183955: (that’s an edible fractal! :hungry: )

Of course, the fractals of nature are limited when it comes to infinity, since infinity is only a theoretical concept and doesn’t exist in practice, so the fractal structure breaks down at a certain point (the atomic scale if not before). As a matter of fact, the computed fractals are never really infinite either, since theoretically, it would take an infinite amount of time to calculate and display infinity. But I digress…

The Weierstrass Function

Weierstrass Graph by esintu
The Weierstrass Function was one of the first fractal discoveries, even though Mr. Weierstrass didn’t know it then. This function presents a challenge to the idea that every continuous function is differentiable except on a set of isolated points, a basic rule of differentiability. As can be seen in the graph, the function is self similar; the corners, when zoomed in, have a pattern similar to that of the overall function.  No matter the magnification, the function never becomes “smooth” and is made up of “corners”, which means that it’s not differentiable at any point, a good reason for why some mathematicians would refer to the group of Weierstrass Functions as “ Pathological functions”.

For those who are interested, the function is
Weierstrass Function by esintu
where “a” is any constant. The function was originally defined for a=2, but it has the same properties for any “a”. If you have a graphing calculator, you can easily try this. Just manually write the summation up to k=7 or something and try zooming around the graph of the function. I tried it, real fun ;)

The Cantor Set

Cantor Set by esintu

The Cantor Set is one of the most simplistic fractals, introduced by Georg Cantor in 1883. It is formed by removing the middle one third of a straight line and repeating the process for the resulting lines for infinite times (of course, as mentioned before, this is only the theoretical Cantor Set due to infinite iterations being impossible).

For those who are interested, stretch your minds and think about what the phrase “the Cantor Set consists of holes” means and how you can prove it. If you can’t figure it out, go check the comments for an explanation ;)

In the previous article, I had mentioned that the Cantor Set has a dimension of 0.63. Here’s a simplified way to get there;

Fractal dimension = log(n) / log (1/l) where n is the number and l the length of the lines acquired by the first iteration. (a simplified version of the original formula – you can’t calculate the dimensions of more complicated fractal forms as easily (i.e. the Mandelbrot Set))

For the Cantor set: n=2, l=1/3. Therefore:
log(2) / log (3) = 0.63...

The Koch Curve

Koch Curve by esintu

The Koch Curve is actually very similar to the Cantor Set. The only difference is that, to form the Koch Curve, after removing the middle third of a straight line an equilateral “tent” is added in its place. This is, obviously, repeated infinitely to form a perfect Koch Curve.

Now you can go figure out what the dimension of the Koch Curve is! ;) (check the comments for the right answer)

The Sierpinski Triangle

Sierpinkski by esintu
To form a Sierpinski Triangle, the midpoints of the three sides of a triangle are connected to form a new triangle in the middle, which is then removed. The step is repeated infinitely. The Sierpinski is used in fractal art much more often than the aforementioned, simpler fractal forms. Just browse for “Sierpinski” in the fractal art gallery to see some lovely uses of this rather simple form…

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Useful Links:

<a href=www.public.asu.edu/~starlite/S…> A thorough analysis of fractal structures found at the grand canyon.
<a href=www.fourmilab.ch/images/Romane…>
An interesting article about the fractal structure of the Romanesco

More links will come with each article, so keep an eye on this links section ;)
Add a Comment:
 
:iconjurrivortex:
Jurrivortex Featured By Owner Mar 14, 2008  Hobbyist Digital Artist
human brain is also a very interesting example of a natural fractal structure.
Reply
:icondaisuke-paster:
Daisuke-Paster Featured By Owner Mar 13, 2008
I can't believe this is news, sorry to say this but anyone that read this article and didnt know what the hell he was talking about, even though everyone should have known, are really retarded and stupid.
Reply
:icongossamer-light:
gossamer-light Featured By Owner Feb 26, 2008   Digital Artist
:giggle: what a plum on the tree today! XD I liked this article muchly!
Reply
:iconideviant:
IDeviant Featured By Owner Feb 25, 2008  Hobbyist Digital Artist
Nicely pitched :clap: I hadn't heard of the Weierstrass Function before. What, though, is k? Even MathWorld fails to mention its identity.

BTW, links not working for me :(
Reply
:iconesintu:
esintu Featured By Owner Feb 25, 2008  Hobbyist Digital Artist
oh, the k is defined in the notation of the formula, notice the signs below and above the sigma sign; it's any integer from 1 to infinity, and the function is the sum of that phrase for all values of k. :)

links aren't working for anyone, a mistake on my part, sorry :hmm:
Reply
:iconideviant:
IDeviant Featured By Owner Feb 26, 2008  Hobbyist Digital Artist
Duh! *slaps own wrist*
Reply
:iconsya:
Sya Featured By Owner Feb 24, 2008  Professional General Artist
:clap: Great article!
Reply
:iconesintu:
esintu Featured By Owner Feb 24, 2008  Hobbyist Digital Artist
thanks! :)
Reply
:iconerror732:
Error732 Featured By Owner Feb 24, 2008
Finally, some math!

I'm excited for Mandelbrot's entrance (next article, I'd bet?). Though, I suppose, to get that far, you'll have to explain complex numbers, which, means Gauss and Euler . . . and, later, quaternions! Well, maybe I'll leave the story to you.

"infinity is only a theoretical concept and doesn’t exist in practice."

I disagree. How long does it take to fall to the center of a black hole? How far does a gravitational field extend?

:thumbsup: But I approve, anyhow.
Reply
:icondiscarn8:
Discarn8 Featured By Owner Feb 25, 2008  Hobbyist Digital Artist
"How long does it take to fall to the center of a black hole?" - Undefined, as a black hole has an event horizon. Everything within that event horizon is unable to be measured, and is therefore, for any practical scenario, out of bounds.

"How far does a gravitational field extend"? Well, we don't know. *grin* If there is a quanta of gravity, there is a finite distance beyond which the charge is no longer large enough to generate a quanta....

/me takes off his science-geek hat...
Reply
:iconerror732:
Error732 Featured By Owner Feb 25, 2008
By the Einstenian model, as you approach the center of the black hole, time dilates to the point where, though, from your perspective, you're accelerating almost instantaneously, the world would see you slow to a near halt. I say "see" despite said event horizon. I don't see any reason to assume passing beyond this horizon would alter gravitational law, however.

Fine then. "What does the force of gravity approach as two masses approach each other?" There's a plethora of examples; the specific choice is immaterial.

Takes off? Whyever would you do that? =P
Reply
:icondiscarn8:
Discarn8 Featured By Owner Feb 25, 2008  Hobbyist Digital Artist
Due to the math which wraps up what we understand to be an event horizon (altho' we've never observed one in the wild...) there's a clear disconuity. Divide by zero error, reboot this branch of surreality - new laws may or may not apply. *grin*

What does the force of gravity approach as two masses approach each other? F = (G * M(1) * M(2)) / d^2 where F is the force due to gravity, M(1) is the mass of object 1, M(2) the mass of object 2, d is the least-path distance between the two masses, and G is the so-called "Gravitational Force Constant", approximately 6.7E-11 Newtons meter2 / kilogram2 (in metric units).

*doublecheck* Um, nope, no infinity in there, as all known massive objects have nontrivial radii, and therefore d can never reach zero....

*impish grin*

As to why take the sci-geek chapeaux off? Because sometimes other headwear is more appropriate! *bigger grin*
Reply
:iconerror732:
Error732 Featured By Owner Feb 29, 2008
Hmm. I've more reading to do on black holes.

Why does it have to be a massive object? Even two elementary particles of epsilon radii will approach infinite force, albeit slowly. I can always choose an epsilon or delta to suit my needs.
Reply
:icondiscarn8:
Discarn8 Featured By Owner Feb 29, 2008  Hobbyist Digital Artist
Hawking radiation. The small black holes evaporate with a bang!
Reply
:iconesintu:
esintu Featured By Owner Feb 24, 2008  Hobbyist Digital Artist
yea, you're right about infinity, in a way.. I was thinking about more familiar issues when I said that, you know, the nature that we see and experience every day but yea, it was a too-generalized sentence, I agree.. I don't have too much insight about space, so that's an issue too :D thanks for pointing that out..

both Julia and Mandelbrot will be in the next article, you're right. I wasn't planning to go into too much detail about quaternations since I'm aiming to make these articles as simple as possible in order not to scare people off of maths, but I might do some research and include them if I understand them enough to be able to simplify them :)
Reply
:iconerror732:
Error732 Featured By Owner Feb 29, 2008
I'd be interested in an article on math in art in general--not just fractals. Everything from the direct (hypotrichoid designs) to the more subtle (golden ratios in portraiture).
Reply
:iconesintu:
esintu Featured By Owner Mar 1, 2008  Hobbyist Digital Artist
well, you should hope a mathematician sees that comment then :D I'm not all that good at maths, and I've learnt all this stuff with research since I was interested in what exactly I was doing with fractals..
I might do further research and go into less fractal-related topics too though, once I'm done with these articles... :)
Reply
:iconerror732:
Error732 Featured By Owner Mar 1, 2008
I've a different problem; I'm majoring in math, but I don't know if I know enough about the art. :P
Reply
:iconesintu:
esintu Featured By Owner Mar 1, 2008  Hobbyist Digital Artist
oh I see.. well, I can't really promise, but I'll try and summon up an article once this series is done :)
Reply
:iconerror732:
Error732 Featured By Owner Mar 1, 2008
I wouldn't ask you to! I really ought to do the research and write it myself.
Reply
:iconfractalmonster:
FractalMonster Featured By Owner Feb 24, 2008
Glad you continue this series :clap: Think you must have very little time as you are a student ;)
Reply
:iconesintu:
esintu Featured By Owner Feb 24, 2008  Hobbyist Digital Artist
indeed, time is certainly an issue.. glad you like the series :)
Reply
:iconfractalmonster:
FractalMonster Featured By Owner Feb 25, 2008
Of cause :D
Reply
:icongyrojet:
Gyrojet Featured By Owner Feb 24, 2008
Nice of you to enlighten people about what they are.
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:iconesintu:
esintu Featured By Owner Feb 24, 2008  Hobbyist Digital Artist
:)
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:icongyrojet:
Gyrojet Featured By Owner Feb 25, 2008
For mathematics! :iconscienced:
Reply
:iconslobo777:
slobo777 Featured By Owner Feb 24, 2008  Hobbyist Digital Artist
Looks like you've done a lot of research, and I enjoyed your article.

Technically the Cantor Set consists of the "dust" of infinite 0-dimensional points left when you take away the "holes". The holes aren't in the set, though they are in the shape of course, so it's just a nit-picking detail, sorry :nerd:
Reply
:iconesintu:
esintu Featured By Owner Feb 24, 2008  Hobbyist Digital Artist
well yea, you're right about the holes and dust actually, but it was just a phrase that I found in one of the books I was looking through, so I just took it as I saw it.. thanks for pointing it out though :)
Reply
:iconsophquest:
Sophquest Featured By Owner Feb 24, 2008   Digital Artist
:love: An excellent article! :w00t: Well done! :glomp:
Reply
:iconesintu:
esintu Featured By Owner Feb 24, 2008  Hobbyist Digital Artist
thanks! :)
Reply
:iconesintu:
esintu Featured By Owner Feb 24, 2008  Hobbyist Digital Artist
The Answers:
1) The Cantor Set consists of holes.. Being as it is a hard concept to get the hang of, it's easy to figure out by simple logic. Think of the function M(n) as the number of lines, L(n) as the lenght of each line and T(n) as the total length of the Cantor Set, n being the number of iterations. M(n) equals 2^n, L(n) equals 3^(-n) (if the length of the original line is 1), and T(n), being length*number of lines, equals (2/3)^n. As the number of iterations approaches infinity and therefore the perfect Cantor Set, the total length of the set approaches 0. So a perfect Cantor Set is made up of holes since it exists but doesn't have any measurable lenght.

2) The dimension of the Koch Curve is; log(4)/log(3), which is equal to 1.26..
Reply
:iconlyc:
lyc Featured By Owner Feb 24, 2008
it's great to finally see part 2! the romanesco pic is really good, you've done some great research for pix on da :)

btw, did you have a backup copy of your work, or is this redone from scratch?
Reply
:iconesintu:
esintu Featured By Owner Feb 24, 2008  Hobbyist Digital Artist
heh thanks.. I finally found the first draft that I had printed out and just went from there.. I added the nature part just yesterday though, that wasn't included in the written project but only in the presentation that I did in class..
as for the pics, there were a few better photos that I found via google, but the News doesn't allow embedding pics from outside DA :shakefist:

I really hope that it won't be another 6 months before I post the third part :XD:
Reply
:iconblueeyesfairy:
BlueEyesFairy Featured By Owner Feb 24, 2008
:clap: :clap: just studying them for school :aww:
Reply
:iconesintu:
esintu Featured By Owner Feb 24, 2008  Hobbyist Digital Artist
:D
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