High-Precision Deep Zoom

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Secant Method

The secant method is an iterative technique for numerically solving equations that is very similar to Newton's method. For details, see the Technical page on this. It converges more slowly to the solution than Newton's method does. That may sound undesirable, and when used for its intended purpose (solving equations) it is. It does have the advantage of not requiring knowledge of the derivative of the function that you're trying to solve.

The main benefit of this method from the standpoint of fractal art is that it makes much more interesting pictures than Newton's method or the more sophisticated variations.

Two videos have been made based on the secant method. SecantAnimation1 uses the cosine function, and SecantAnimation2 uses (z+1)(z-1)(z+2)(z-2).

Cube Roots of Unity
Secant Cubic Fractal

One of many possible ways of approaching the solution to z3-1=0 with the secant method. This initializes the first guess to 0 and the second guess to the image point. The image is centered at (0,0) and spans a height of 11 vertically in the complex number plane (44/3 horizontally).

Secant Cubic Fractal
Secant Cubic Fractal
Secant Cubic Fractal

This is the center of the previous image magnified.

The top image shows the unaltered iteration count data. Counts are obviously integers, so this image has flat bands. The middle image is the same as the top but has exponential smoothing applied.

The bottom image shows a magnified portion of this fractal. As with most fractals in this large group, the pattern repeats infinitely many times as we magnify the fractal more and more. Still, this is a very nice formula.

Roots of Cosine Function
Secant Cubic Fractal Secant Cubic Fractal Secant Cubic Fractal Secant Cubic Fractal

Here we apply the secant method to the cosine function.

This fractal was used in SecantAnimation1.

The top image shows an overview of this amazingly complex fractal. It is centered at (0,0), and each of the points surrounded by a violet ring is a multiple of 2 pi. If you look very closely you will notice that they are all different, unlike most fractals in this family. This means there is great potential for deep zooming.

The second image is a magnification to a size of 1 vertically.

The third and fourth images are progressively deeper zooms off to the side, around -12 pi. Note in the third image that all the detail is different in each period. Look at how complex this thing is! Exploring all this is going to be a lot of fun.