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Eagle FeathersLilac ExponentThe Mandelbrot Set
Lilac Exponent
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Description: This is an unusual visualization of a Mandelbrot zoom, for which I cannot take full credit. I chose the zoom location and color scheme (one of my multifrequency ones) but someone named David Madore came up with the technique used here.

That technique is as follows: invert the complex plane about the zoom target (here the center of a minibrot shallowly into high-precision arithmetic -- 22 decimals) and then use polar coordinates as screen coordinates, and moreover the y axis is the logarithm of the modulus, rather than the modulus.

This is analogous to a Mercator projection of the Riemann sphere, with infinity at the north pole and the target point at the south. Horizontal lines in the image represent circles of constant radius (hence the repetition along the x axis) about the target point. Higher precision math is needed farther south.

The math to convert screen coordinates to Mandelbrot coordinates is exceptionally simple. First map screen coordinates into the complex plane. Then multiply by i and take the complex exponential -- z -> eiz. Now invert at zero (take the reciprocal and complex conjugate, in either order) and add the target point (with normal complex number addition).

Since this mapping is conformal, spirals don't get squished or skewed.

The result is what you see here: infinity is infinitely far up, the target point is infinitely far down, and everything else is in between, with higher magnifications toward the bottom.  In principle, everything you'd see in a zoom movie from the full set down to the target minibrot is in this single still image. The black blob at the top is the bulk of the M-set; the large black blob a short way south of it is a shallow minibrot in a weed-tendril. The small minibrot farther down with an enlarged "nose" pointing south is in that one's seahorse valley. The fat minibrot just below that one is the previous one's main spike minibrot. And the black area at the bottom is the target minibrot, which is a smaller spike minibrot and is down in the elephant valley area of the larger spike minibrot.

Now, here's the really freaky thing. In theory, the most efficient way to compute a zoom movie is to compute an image like this (but higher resolution!) and then create the movie frames from it by mapping vertical "windows" of one horizontally-repeating strip into the image space through the inverse mapping. The desired input resolution is: horizontal resolution equal to the circumference, in pixels, of a circle about the frame center and passing through the frame corners; vertical resolution equal to the radius in pixels of that same circle, times a factor for oversampling, times the number of zooms from one pixel radius to that circle's radius needed to get from whole-set view to target view.

In fact, any movie frame centered on the target point and within a certain magnification range can be computed, in principle, using the data in an image like this of sufficient resolution; with any given magnification and (via a plain old translation along the x axis of the input data applied before anything else) rotation.

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Keywords: Mandelbrot Seahorse Valley 
Posted by: Pauldelbrot September 15, 2011, 12:43:21 AM

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December 18, 2012, 08:43:48 PM
Nice but where is the main cardiod in this image ?

Now, here's the really freaky thing. In theory, the most efficient way to compute a zoom movie is to compute an image like this (but higher resolution!) and then create the movie frames [...]

Well do you have the mathematical proof ?  embarrass

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