Julia Set

The Julia Set is named after the French mathematician Gatson Julia who investigated their properties in 1915, culminating in a paper published in 1918.

Julia was interested in the iterative properties of the more general expression:

z4 + z3/(z-1) + z2/(z3 + 4z2 + 5) + c.

However, now the Julia Set is associated with those points z0 = x + iy on the complex plane for which the series of form zn+1 = zn2 + c, does not tend to infinity.

How are the images of the Julia Set generated?

You may have seen some really beautiful images of the Julia Set:

Computing the Julia Set is quite straightforward using the brute force method approach. One must simply assign each pixel a number in the complex plane, which is then the starting point of the series. The series is iterated for each starting point and two colours are assigned for the two cases which can arise: the series diverges to infinity (usually white) or it does not (usually black).

Below is the Julia Set for f(z) = z2 – 0.75. Note that the other colours in the image indicate how soon the iterates left towards infinity (going from red, yellow, green, blue and magenta in decreasing order of speed).

For almost every c, the Julia Set generates a fractal.

Sources: 1 | 2 | 3

M x

 

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3 comments

  1. That’s funny… I always thought the mathematician that discovered the Julia set named it after his girlfriend :D. I did enjoy your blog post – even if I regret loosing the lovely imaginative story…

    Liked by 1 person

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