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Comment by ermir | Hacker News Reader

I assume these waves are not continuous but made out of many points and then just simulated over time, like a game engine. But since fractals have infinite detail, does that mean that for every number of points we choose for the wave we'll get a radically different wave pattern?

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I gave this some thought and came to the conclusion that a continuous wave hitting a fractal would not have a defined reflection, since the fractal does not have a traditional derivative it would also not have a tangent at a point.

No, below a certain resolution (depending on the wavelength) the waves just see the average shape (which is smooth has a derivative). Think about this in real live imperfections in the atomic structure of a mirror don't influence the reflected light rays, or the shape of waterwaves reflected of a concrete wall is not influenced by the roughness of that concrete.

you are right, I've recently done some Points Billiard in Koch Fractals video which tries to do classical billiard in Koch Fractal which is exactly what you write about averaging etc. (I had to average the normals around hitting point as the normal vector was not defined here), see the video here: https://youtu.be/v9KS03hiSU8

This depends on the wavelength. Essentially a wave hitting an object that has features that are smaller than roughly the wavelength will see those features as an average. So we therefore don't need to simulate the infinite resolution fractal. This is somewhat more complicated in the video because the waves are pulses, i.e. they contain many frequencies.

The video actually illustrates this effect when simulating the "narrow" ring waves and the "wide" ones. Those two contain different frequency components and thus see more or less of the fractals.

My initial thought was that was two particles that are infinitesimally close at the start of the simulation will have radically different reflections and paths when they hit the fractal, since the fractal has no local similarity, therefore the choice of the number of particles to use in the simulation will greatly affect the shape of the reflected wave. However by the same logic that the fractal has no local similarity we can also conclude that because the fractal has no derivative, and therefore no tangent, and therefore no way to calculate the reflection angle for any point at all.

Generally it is better to think of waves as waves, not as being composed of particles as they behave quite differently, for example particles can't interfere (the dark bright patterns we see). I assume the simulation to create the video simulates the wave equations and the are not composed of "particles".

precisely, this is solution to wave equation (I found this thread as suggested by youtube - you people here gave me quite a lot of traffic recently ;)

Would that have then something in common with butterfly effect?

these waves are produced by finite difference (spatially finite resolution) approximation to contunuous wave equation so in some sense you are right but on the other hand - these are NOT particles at all

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