Mandelbrot's Clock

Calculus Level 4

Mandelbrot has a 2424-hour digital clock with a light above it.

Every minute, a computer in the clock calculates a complex number c=(16h2)+(115m2)i,c = \big(\frac{1}{6}h - 2\big) + \big(\frac{1}{15}m - 2\big)i, where hh is the hour and mm is the minute. If the orbit of znz_n for zn+1=zn2+cz_{n + 1} = z_n^2 + c with z0=0z_0 = 0 tends to infinity, the light turns green, but if the orbit is bounded, the light turns red.

For example, at 1818:3030, c=(16182)+(115302)i=1,c = \big(\frac{1}{6}18 - 2\big) + \big(\frac{1}{15}30 - 2\big)i = 1, and this gives the sequence 0,1,2,5,26,,0, 1, 2, 5, 26, \dots, which tends to infinity, so the light on the clock would be green. But at 66:3030, c=(1662)+(115302)i=1c = \big(\frac{1}{6}6 - 2\big) + \big(\frac{1}{15}30 - 2\big)i = -1, and this gives the sequence 0,1,0,1,0,,0, -1, 0, -1, 0, \dots, which is bounded, so the light on the clock would be red.

If you were to look at the clock at a random time during the day, would the light above the clock be more likely to be green or red?


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