Mandelbrot's Clock

Calculus Level 4

Mandelbrot has a \(24\)-hour digital clock with a light above it.

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

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