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Some time ago I stumbled across this page:

http://www.flickr.com/photos/arenamontanus/sets/72157615740829949/

I found the idea super cool! These pictures show zooms into points of the standard Mandelbrot set. Except that x and y of the image do not correspond to real and imaginary parts of C but x corresponds to the logarithmic distance from the zoompoint and y corresponds to the angular direction around the zoom point.

I love how the images can contain many equally sized minibrots and how the structure varies through the whole image. What I don't like is that the aspect ratio of the images is by definition superwide, one can not generate cool wallpapers or posters from that. Also, zooming into the center of a spiral generates superboring ///////// patterns.

Does anyone have an idea on how to generate images that are equally cool but without these problems? Maybe one could find a transformation that enlarges detail and minibrots, a Homunculus picture of the Mandelbrot set so to say:

http://1.bp.blogspot.com/-DUcGIqmz0mA/Tdyot4jXDcI/AAAAAAAABfE/25rbP7BR-q8/s1600/Homunculus%2Bdrawing.jpg

I also saw this post in the forum:

http://www.fractalforums.com/mandelbrot-and-julia-set/on-the-superposition-of-mandel-like-sets/

and it seems that the z*log(1+z)+c version enlarges minibrots:
http://hiato.uctleg.net/mandelol/mandelol1.png

Sadly the continuity is broken. But maybe a transformation during iteration is not the worst idea for this goal.

At least, one could apply some fisheye morphing at the root locations. If noone has a better idea, I will maybe try that.

For the mercator projections: Sadly no, the aspect ratio is fixed for those. That is because the x-axis is given by the zoom factor and the y-axis is always fixed by the fact that one full circle covers 360 degrees, never more or less. So the deeper you zoom the longer (along the x-axis) you image becomes while the y-axis remains the same.

But I like the idea of looking for a projection that covers a greater range of shapes and scales and/or enlarges the satellites. I have experimented using an inversion of the set at a circle of radius 1. Points close to the center of the circle will be enlarged and projected far away from the origin. But this does not really help to show more satellites, instead it creates a large image of the features close to the inversion center and everything else is rather small.

Here are some examples I made to show what I mean:
http://rychveldir.deviantart.com/art/Sun-Vortex-265633801
http://rychveldir.deviantart.com/art/Web-213251523
http://rychveldir.deviantart.com/art/Seahorse-Flame-139353303

I can post the formula if you want. It has been a while since I programmed it and I'd have to look it up.

Here's an idea. How about taking a logarithmic spiral strip out of the set, using the radial position as the x coordinate? The aspect ratio could be controlled by changing the base of the logarithm along with the vertical resolution of the image.