Suppose there's a firecracker at every grid point and they all go off at time t = 0. The
noise propagates out from each grid point in a circle moving at the speed of sound, so the white (resp. black) regions consist of all points that have heard an odd (resp. even) # of
2/n
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If we change our perspective a bit, we can see that at time t, the number of
that an observer at a point p has heard is equal to the number of grid points within distance v*t (where v is the speed of sound). (image from http://mathworld.wolfram.com/GausssCircleProblem.html …) 3/npic.twitter.com/dJNDGp3Arf
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Translating back to the black & white images, we see that we can determine the number of times a given point has switched colors by counting grid points inside a circle, and for large circle radii, this is approximately π*r^2. 4/5
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(The Gauss circle problem https://en.wikipedia.org/wiki/Gauss_circle_problem … asks for a bound on that error.) Now let's turn to the question of whether the fraction of black and white pixels goes to 1/2 for large radius r. 5/6
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I can translate this into questions about the even/odd parity of the number of grid points within distance r of each point of the square, but I don't know enough number theory to be able to tell whether this follows from known results...
Anyone wanna take it from here? 6/6Show this thread -
Here's a thread with some similar movies and a link to some more discussion (in French) by
@gro_tsen:https://twitter.com/gro_tsen/status/1120692989846872064 …Show this thread
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with an acoustic reinterpretation: 1/n