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The simplest 3d mandelbrot idea, in my opinion, is to begin with a vector in 3 space, written in spherical coordinates <x,y,z> = <r cos theta sin phi, r sin theta sin phi, r cos phi> and square it three times, one in the xy plane, one in the xz plane and one in the yz plane, then displace it by <c_1, c_2, c_3>. That is, treat each pair of components to a squaring, one after the other. It keeps the idea of squaring and displacing, in line with the usual 2d mandebrot formula, z_n+1=z_n^2+c, The result is <r^4 cos 2 theta sin 2 phi, r^4 sin 2 theta sin 2 phi, r^4 cos 2 phi> +<c_1, c_2, c_3>, very close to the simplest bulb update formula, but with a 4th power radius. If we wanted to assure that the vector was only squared in distance, we could double pairs of angles and take r^sqrt(2) each time to get <r^2 cos 2 theta sin 2 phi, r^2 sin 2 theta sin 2 phi, r^2 cos 2 phi> +<c_1, c_2, c_3>. Alternatively, perhaps a squaring *and* displacing in each of three planes successively, or an average of a squaring and displacing of each pair of the three components of the vector.

But what most people seem to want is a bulby tendrily thing, like the 2 d version. But there are some interesting things going on in the  second power mandelbulb <r^2 cos 2 theta sin 2 phi, r^2 sin 2 theta sin 2 phi, r^2 cos 2 phi> +<c_1, c_2, c_3>. If you, ahem, do a "proctological examination" by looking up the x axis from the right, you will see that there is an infinite set of holes in holes in holes, corresponding in a nice way to the idea of a thickened tendril, with branches upon branches upon branches.

have seen some similar structures, but, could you eventually post a picture?