Weierstrass functions: continuous but not differentiable anywhere
1–10 of 49 posts
Re: Weierstrass functions: continuous but not differentiable anywhere
#2That's pretty much a defining feature of all (infinitely iterated) fractals [1]
Re: Weierstrass functions: continuous but not differentiable anywhere
#3This function produces an audio signal that sounds like an organ.
Re: Weierstrass functions: continuous but not differentiable anywhere
#4That's pretty much a defining feature of all (infinitely iterated) fractals [1] [1] https://en.wikipedia.org/wiki/Fractal
Cool! I didn't appreciate that. I think its interesting to see that behavior in an everyday Fourier series though.
Re: Weierstrass functions: continuous but not differentiable anywhere
#5The construction of the Weierstrass function (http://en.wikipedia.org/wiki/Weierstrass_function) was new to me, even though I suffered through a semester of baby Rudin.
To me, the canonical example of a continuous but nowhere differentiable function is: any Brownian motion sample path.
Which, like the Weierstrass function, illustrates that functions that look simple may be so weird that our standard intuition does not apply.
Re: Weierstrass functions: continuous but not differentiable anywhere
#6This function produces an audio signal that sounds like an organ. https://gist.github.com/improv32/8414717 http://vocaroo.com/i/s13ZO9QOMULL
All those lovely harmonics ... now dreaming of filter envelopes.
Re: Weierstrass functions: continuous but not differentiable anywhere
#7That's pretty much a defining feature of all (infinitely iterated) fractals [1] [1] https://en.wikipedia.org/wiki/Fractal
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. :)
Re: Weierstrass functions: continuous but not differentiable anywhere
#8Quick and dirty landscape/horizon generator for games?
Re: Weierstrass functions: continuous but not differentiable anywhere
#9Earlier 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. :)
Heh. I hear you. But but on the other hand, a sawtooth is perfectly everyday with an infinite series of coeffs: http://en.wikipedia.org/wiki/Triangle_wave .
Re: Weierstrass functions: continuous but not differentiable anywhere
#10This 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