Envisioning Julia sets

The Julia set is approximated by the black dots in the image below. The set is generated by a function. The arrows highlight the working of the function. The dark red arrow represents an input point as a direction and distance from the center of the plot. We first double it's angle relative to the right-hand axis and then take the square of the distance from the center, called the magnitude of the point. Then we add a constant value to get the function's output.

click the images for an interactive version

Repeating this function will lead us all across the Julia set on an erratic course. The green arrows below show where repeated application of the function will get us. Note how this creates a self-similarity between features of different sizes in the set.

The Julia set is infinitely deep, but the computer displays finitely-sized pixels. If we zoom in on part of the display we will see the pixelation of the display. But we can then recalculate the approximation of the set and will see more of the same structure more finely than before.

What we have looked at so far is but one Julia set. If we change the direction and length of the blue arrow that is added as part of the function, we get an entirely different shape. Below you can see two further examples of Julia sets.

But the basic properties of Julia sets we explored above are still valid.

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