Whitney Music Box

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Damn that's beautiful. It's basically a visualization of integers and divisibility. Every time a group of dots crosses the white line, they share a common divisor, which would be the dot on the outside. Getting it to be a visualization of the real numbers (>= 1) would be interesting. Each point takes 1/n time units to go around once, assuming the outer dot (1) takes 1 time unit to go around once.

So what you would have is a space of functions: x = r cos n*t/2pi and y = r sin n*t/2pi where t is time, n is the number, and r is the radius of the circle that n goes around. Whitney's music box makes the intervals more or less equidistant... but man that's not mathematically elegant since you would need an infinitely long line to describe the real numbers. What would be nice is a function that converges to 1 at 0 and 0 at infinity. x = r(n) and y = r(n) should form a nice little spiral. 1/a^n works pretty well though it converges to 0 really freaking fast.

I don't know much Mathematica or whatever graphing package you kids are using these days... but it'd be pretty cool to see this in 3-d (taking time as another dimension.) Each of the functions would plot a spiral through time, but the space of functions would plot an infinite number of spirals, each spiraling at a different rate, with different distances from the origin line (where n = infinity.)

I wish I knew more topology.

http://www.coverpop.com/whitney/index.php?var=v12

Also pretty cool. Obviously, no two dots are ever on the white line at the same time.