Fractals - any real uses?

 

I love fractals, the question is are they useful for anything?

 

I knowthey preduct certain shapes in nature but do they have real uses or are they just pretty patterns?

 

Cheers,

 

Ryan Jones

I think that there certainly are a lot of uses behind fractals. For instance, one of the most infamous fractals out there is the Lorenz attractor. It's quite nice to look at, but more importantly, the model behind it simulates convection currents in fluids.

 

Admittedly, stuff like the Mandelbrot set doesn't have such a direct application and to a certain extent, fractals are just interesting mathematical objects. But, the systems behind the pretty pictures often have a very direct application in some physical situation.

 

So I suppose an answer to your question is both yes and no; it depends on whether you're interested in the system behind the fractal or not

So I suppose an answer to your question is both yes and no; it depends on whether you're interested in the system behind the fractal or not

 

I see

 

Arn't fractals chaeotic systems? If they rethen that would explain their applications in some areas such liquid dynamics as you stated above

 

Cheers,

 

Ryan Jones

West et al. (1997) proposed that the 3/4 power law for metabolic rates was the consequences of fractal space-filling networks. We're not sure their argument is mathematically correct, of even that the 3/4 power law exist, but still...

 

West, G.B., Brown, J.H. and Enquist, B.J. 1997. A General Model for the Origin of Allometric Scaling Laws in Biology. Science 276, 122-126.

West et al. (1997) proposed that the 3/4 power law for metabolic rates was the consequences of fractal space-filling networks. We're not sure their argument is mathematically correct' date=' of even that the 3/4 power law exist, but still...

 

West, G.B., Brown, J.H. and Enquist, B.J. 1997. A General Model for the Origin of Allometric Scaling Laws in Biology. [i']Science[/i] 276, 122-126.

 

Sounds interesting... Never heared that one before.

 

Cheers,

 

Ryan Jones

There's been considerable research into the use of fractals for image compression

 

See http://inls.ucsd.edu/~fisher/Fractals/

There's been considerable research into the use of fractals for image compression

 

See http://inls.ucsd.edu/~fisher/Fractals/

 

Thats certainly an interesting use, I'd think fractals could be used in encrption too because of their essentially random natue. Anyone agree with that?

 

Cheers,

 

Ryan Jones

Some artists use fractals in thier work. But fractals have technological applications too. For example antenna engineers use lots of small antennas n either place them randomly or regularly spaced to make antenna arrays. These fractal anteenas power important applications, ex: signal intelligence, RFID, telematics, military, aerospace, wireless, cell phones.

Here are some pictures of fractal anteenas:

http://www.antlab.ee.ucla.edu/~johng/art/fractal_collage.jpg

Some artists use fractals in thier work. But fractals have technological applications too. For example antenna engineers use lots of small antennas n either place them randomly or regularly spaced to make antenna arrays. These fractal anteenas power important applications' date=' ex: signal intelligence, RFID, telematics, military, aerospace, wireless, cell phones.

Here are some pictures of fractal anteenas:

http://www.antlab.ee.ucla.edu/~johng/art/fractal_collage.jpg

[/color']

 

I guess that makes sence because fractals have a large surface area... You could makea really big antena with a HUGE suface area

 

Cheers,

 

Ryan Jones

I guess that makes sence because fractals have a large surface area... You could makea really big antena with a HUGE suface area

but the anteena's that r used in cellphones need to be really small' date=' so that's why use fractal anteenas which hav fractal geometry cuz they make it possible to place a considerably long wire in a very small area.[/color']

that's really cool. It'd be cool if it was collapsable too, when you slide it in ; )

but the anteena's that r used in cellphones need to be really small, so that's why use fractal anteenas which hav fractal geometry cuz they make it possible to place a considerably long wire in a very small area.

 

True, very true... I can think of a few uses for them but as Dave said they seem to be more of a mathematical curiosity...

 

Cheers,

 

Ryan Jones

Fractal imagery was explored extensively by the abstract expressionist painter Jackson Pollock, known for his ability to create formless composition:

 

http://www.physics.hku.hk/~tboyce/ap/topics/pollock.html

Do any of you guys watch the mathematic solving cop TV show, "Numb3rs"? A lot of the solving solutions was using all the information such as locations, addresses, numbers, anything to find the bad guys. The mathematican would use fractal mathematics to solve the distriubution of the datas, and find the most confirmed positions.

 

Sorry if I don't make any sense.

 

 

Also, more importantly, it's a definition of beauty of art. (math is just behind the images).

Well, we have some interesting uses here - I knew they were used in art as they are brilliant to look at - never knew they could be used to solve distribution problems though.

 

Thanks all!

 

Ryan Jones

I once attended a presentation at SUNY Stony Brook where a professor presented how fractals represented the effectiveness of Newton's approximations when searching for zeros on the complex plane of a given polynomial.

That's another quite famous example - in fact last year I wrote a program in Fortran to do that for one of my modules. It's also very simple to explain.

 

If you've done the equivalent of AS-level mathematics, then you'll probably know the Newton-Raphson method for approximating roots to a function. Given some starting point x₀, we calculate better approximations by using the formula:

 

[math]x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}[/math]

 

and apply this iteratively a specific number of times. What you probably don't realise is that this can be extended to functions defined over the complex plane; that is [imath]f:\mathbb{C} \to \mathbb{C}[/imath]. Throwing complex analysis aside, it works very effectively, but the interesting thing is your choice of starting point. If you map out which colour gets mapped to which root on some domain contained inside the complex plane, you'll get an image very similar to this.

 

This approximates all roots to the equation [imath]f(z) = z^5 - 1[/imath]. The darker the colour, the more iterations it takes to get close to the root.