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Fractals
#1
We had a thread about these things on the old forum - and, I guess it's time for me to re-post it now :P

Fractals are infinitely-detailed geometric objects which are "self-similar": that is, they exhibit the same basic patterns no matter how far you zoom in on them. Sometimes, they're used to model features of the natural world (for example, trees and coastlines both have fractal-like properties); other times, they're just admired for their own beauty. 

These fractals can be wonderfully complex and intricate; however, they sometimes arise from very simple processes. I'll illustrate with an example called the Sierpinski carpet, first discovered by Polish mathematician Wacław Sierpiński in 1916:

Step 1 - Start out with a black square (this one has size 81x81): 

[Image: O1sudwd.png]

Step 2 - Cut out the middle, from one-third of the way across to two-thirds of the way across:

[Image: Lhpg4YS.png]

What we have now resembles a square doughnut, composed of eight black squares (which are smaller versions of the black square we had in step 1). Think of it like a Rubik's Cube with the centre removed, leaving just the eight outer cubes. 

Step 3 - Cut out the middle of each of the eight smaller squares, from one-third of the way across to two-thirds of the way across: 

[Image: zV68XUK.png]

Now, we basically have a ring of eight square doughnuts, which are mini versions of the ones we had in step 2. 

Step 4 - Repeat Step 3 on each of the eight mini 'square doughnuts': 

[Image: 2LPo1iu.png]

Now, we have eight mini versions of the ring in step 3 (each consisting of eight tiny 'square doughnuts').

Step 5 - Repeat Step 4 on each of the eight mini rings: 

[Image: Z4VV9aU.png]

Step 6 - Keep going like this forever. I can't actually go any further (because I've run out of pixels) - but, you get the idea. 

Notice how, at every step of our construction, we have eight smaller versions of whatever we had at the previous step. This is what "self-similarity" means: if we could keep this process up forever, then we would end up with a fractal. 

So, do you know anything else about fractals that you'd like to share?
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#2
I had this fractal creator ages ago.

Julia type:
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Mandelbrot type:
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#3
Not really, plank length.
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#4
(07-11-2018, 01:51 AM)Dapocalypse Wrote: Not really, plank length.

True, that stops any genuine fractals from existing in the real world. I only said coastlines were fractal-like (in reality, you'd hit problems once you get down to the atomic level). 

Still, this doesn't stop theoretical fractals from existing, and being a useful way for mathematicians to model certain real-world objects!
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#5
Okay, this story is a few weeks old, but it is freaky :O

Apparently, scientists have created the world's smallest fractals - out of electrons:

https://www.sciencealert.com/check-out-t...-electrons

Not sure how this can be applied in the real world, but I really hope someone thinks of something!
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#6
I've seen them before. They're cool.
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#7
I did attempt to build one in Minecraft once.

Sadly, you can't really go any smaller than the size of a regular block (at least, not without mods), so the only alternative is to make it really big. Of course, I don't have the patience for that!
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#8
Here's an interesting one. Scientists have managed to build a laser that produces light in a fractal pattern:

https://phys.org/news/2019-01-team-fractal-lasers.html

This was first theorised back in 1998; however, it took them a little over two decades to build a laser that could actually do it. Now, they think they can make one that produces 3D fractals - not just 2D!
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#9
My mind’s playing tricks on me but it looks 3D to me.
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