Author Topic: Forms and Fractals  (Read 232 times)

Onyx

Forms and Fractals
« on: June 17, 2017, 02:00:45 pm »
Fractals have a level of "roughness" associated with them. This is evident in the natural world, but I think it's also evident in conscious beings, chimpanzees being a bit coarser in their ability to perceive or communicate than humans despite similar physiology, etc.

The Form of a triangle can't really be a fractal, I don't think, but a fractal can approximate a triangle or at least reveal the concept of triangleness (see image). Fractals and Forms are two different things, but there do seem to be levels of complexity to various natural and unnatural phenomena, ranging from the mundane to the transcendental.
« Last Edit: June 17, 2017, 02:46:51 pm by Onyx »

merytseth

  • Guest
Re: Forms and Fractals
« Reply #1 on: June 17, 2017, 05:28:10 pm »
Fractals have a level of "roughness" associated with them. This is evident in the natural world, but I think it's also evident in conscious beings, chimpanzees being a bit coarser in their ability to perceive or communicate than humans despite similar physiology, etc.

The Form of a triangle can't really be a fractal, I don't think, but a fractal can approximate a triangle or at least reveal the concept of triangleness (see image). Fractals and Forms are two different things, but there do seem to be levels of complexity to various natural and unnatural phenomena, ranging from the mundane to the transcendental.
I know you've done some work with Mandelbrot sets, but have you looked into Weierstrass functions at all?  If you've had a little calculus, you may find it interesting to know these are functions which are continuous everywhere, but differentiable nowhere (which sort of flies in the face of everything you're taught in calc I).  They are related to fractals, I think.  The interesting thing is that the function has rough, jagged detail at every zoom level.
https://en.wikipedia.org/wiki/Weierstrass_function

Xepera maSet

Re: Forms and Fractals
« Reply #2 on: June 17, 2017, 06:40:50 pm »
I'm formally requesting more philosophy of mathematics for Imperishable Star v II.
AKA: Three Scarabs, 1137

"You look up into the night sky - whether as a child or an adult - and if you open yourself honestly, then it is a gateway to mystery, to the unknown."

Onyx

Re: Forms and Fractals
« Reply #3 on: June 18, 2017, 09:49:18 am »
I know you've done some work with Mandelbrot sets, but have you looked into Weierstrass functions at all?

I'm no math expert, is this a correct breakdown of the basic idea?
y =  (1/1) * sin(1 * x) + (1/2) * sin(2 * x) + (1/4) * sin(4 * x) etc.

Would it be possible to reverse the process, say analyze a signal and try to find a Weierstrass function that approximates it?


« Last Edit: June 18, 2017, 11:00:02 am by Onyx »

pi_ramesses

Re: Forms and Fractals
« Reply #4 on: June 21, 2017, 06:06:16 pm »
I recall, in his History of Western Philosophy, that Bertrand Russell had written something to the effect that mathematics remained insecure owing to Weierstrass as well as others. Wouldn't it be accurate to state that, as things currently stand, we are still in the midst of a bit of a mathematical crisis?
Pro omnis dominos viae sinistra, sic itur ad astra
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merytseth

  • Guest
Re: Forms and Fractals
« Reply #5 on: June 22, 2017, 07:10:14 pm »
I'm formally requesting more philosophy of mathematics for Imperishable Star v II.
I will see what I can do :)

merytseth

  • Guest
Re: Forms and Fractals
« Reply #6 on: June 22, 2017, 07:16:37 pm »
I know you've done some work with Mandelbrot sets, but have you looked into Weierstrass functions at all?

I'm no math expert, is this a correct breakdown of the basic idea?
y =  (1/1) * sin(1 * x) + (1/2) * sin(2 * x) + (1/4) * sin(4 * x) etc.

Would it be possible to reverse the process, say analyze a signal and try to find a Weierstrass function that approximates it?
I'll be honest - differentiable nowhere functions aren't something I know a whole lot about other than that they exist, as real analysis isn't really my forte.  I'll look into it, though.
I recall, in his History of Western Philosophy, that Bertrand Russell had written something to the effect that mathematics remained insecure owing to Weierstrass as well as others. Wouldn't it be accurate to state that, as things currently stand, we are still in the midst of a bit of a mathematical crisis?
In that there is still not a single foundation that describes all of mathematics?  I would say so, in that regard - that was something the Russell spent quite a bit of time working on.  

Xepera maSet

Re: Forms and Fractals
« Reply #7 on: October 30, 2017, 03:07:37 pm »
[Admin bump]
AKA: Three Scarabs, 1137

"You look up into the night sky - whether as a child or an adult - and if you open yourself honestly, then it is a gateway to mystery, to the unknown."