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Personally I think the cool thing about these(and also fractals, which display this as iterations go to infinity) is that they have infinite lengths in a finite(though obviously infinitely subdividable) span.
Earlier quoted context omitted.
Cool! I didn't appreciate that. I think its interesting to see that behavior in an everyday Fourier series though.
I'm not sure that I'd call something with an infinite number of non-zero Fourier coefficients an "everyday" Fourier series. :)
This is why images get blurry when resized up! For some reason better algorithms never seem to become popular.
This function produces an audio signal that sounds like an organ. https://gist.github.com/improv32/8414717 http://vocaroo.com/i/s13ZO9QOMULL
I am quite surprised it does not sound like white noise. Anyway, I played the output of that script through some filters, and this sounds nice: play -c 2 weierstrass.wav gain -12 loudness chorus 0.5 0.9 50 0.4 0.25 2 -t 60 0.32 0.4 2.3 -t 40 0.3 0.3 1.3 -s reverb gain 12
This will sound quite interesting, but it's a really bad approximation of the Weierstrass function! http://www.wolframalpha.com/input/?i=sin%28x%29+%2B+0.9*sin%...
You can modify it slightly and get a function that has no integral, though it still has a well defined area under its curve (do the limit after the subtraction of the two otherwise-would-diverge sums).
>The function has detail at every level, so zooming in on a piece of the curve does not show it getting progressively closer and closer to a straight line. Rather between any two points no matter how close, the function will not be monotone.
Quick and dirty landscape/horizon generator for games?
See here for more: http://freespace.virgin.net/hugo.elias/models/m_perlin.htm
Earlier quoted context omitted.
Cool! I didn't appreciate that. I think its interesting to see that behavior in an everyday Fourier series though.
I'm not sure that I'd call something with an infinite number of non-zero Fourier coefficients an "everyday" Fourier series. :)
You can modify it slightly and get a function that has no integral, though it still has a well defined area under its curve (do the limit after the subtraction of the two otherwise-would-diverge sums).
Not sure what you mean by this, since your comment has no context -- there are no limits or sums in the link. However, even the humble Riemann integral would be able to find the area under the graph of a bounded function on a compact interval which is continuous on a set of full Lebesgue measure. Any other function only has a "well-defined area" for some seriously stunted notion of area; you can integrate more functi…
f(x) = sum[n = 0 to infinity]((1/2)^n sin((1/20)^n pi x))
f(x) = sum[n = 0 to infinity](10^n / pi cos((1/20)^n pi x))
Which diverges because the input to the cosine function limits to 0 as n -> infinity, so the cos limits to 1, and we get 10+100+1000+10000...
But if you integrate from two points and delay doing the limit until after then you get
F(x1, x2)
= sum[n = 0 to infinity](10^n / pi (cos((1/20)^n pi x2) - cos((1/20)^n pi x1)))
= sum[n = 0 to infinity](10^n / pi (2*sin(pi/2 (1/20)^n (x1-x2))*sin(pi/2 (1/20)^n (x1+x2)))
~= sum[n = 0 to infinity](10^n / pi (2*(pi/2 (1/20)^n (x1-x2))*(pi/2 (1/20)^n (x1+x2)))
= sum[n = 0 to infinity]((1/40)^n pi/2 *(x1-x2)(x1+x2))
Apologies for any math mistakes. This was all off the cuff. I wouldn't be surprised if some of the more general integration or summation strategies can also handle this case.
I was finding it hard to understand why it wasn't differentiable (guess my maths is getting a bit rusty) until I read the below[0] which helped it make sense to me: >The function has detail at every level, so zooming in on a piece of the curve does not show it getting progressively closer and closer to a straight line. Rather between any two points no matter how close, the function will not be monotone. [0] http://en…