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On some thread somewhere somebody mentioned something like this. This formula is similar to the normal Mandelbulb, except that the latitude coordinate lines converge toward the poles as fast as the longitude lines converge, making the longitude & latitude transformations together a conformal map. (Notice how the latitude & longitude lines continue to form squares even near the poles, rather than long skinny rectangles as with the normal latitude spacing.) If you include r however, the coordinate system is not conformal. In this system, the equator is at phi = 0 and the poles are at phi = +-infinity. To square a point I squared the radius and doubled theta and phi as usual.


Unfortunately, the result isn't fractal except in the xy-plane, but I'm posting it anyway because it looks interesting, like an inflated Mandelbrot set. Here's degree 7:


Here's degree 2:

I used a max iteration count of 64, but I was also bailing out based on a distance estimation, which probably removed some of the details. Also notice on the degree 7 version how the "seahorse valleys" look like they protrude into the balloon so they get covered up and you can't see the full depth of the valleys. This might be happening on the degree 2 version too. Also I don't think the distance estimation formula I'm using works very well with these fractals.