if
\({(x,y,z)\to \frac{(x^2+y^2+z^2)((x^2-y^2)(x^2-z^2),2xy(x^2-z^2),2xz(x^2-y^2))}{\sqrt{x^4((x^2+y^2+z^2)^2-14y^2z^2)+y^2z^2(2x^2(y^2+z^2)+y^2z^2)}}+c}\)
then
\({(x,y,0)\to {(x^2-y^2,2xy,0)}+c}\)
\({(x,0,z)\to {(x^2-z^2,0,2xz)}+c}\)
and
\({|\frac{(x^2+y^2+z^2)((x^2-y^2)(x^2-z^2),2xy(x^2-z^2),2xz(x^2-y^2))}{\sqrt{x^4((x^2+y^2+z^2)^2-14y^2z^2)+y^2z^2(2x^2(y^2+z^2)+y^2z^2)}}| = |x^2+y^2+z^2|}\)
so this formula extends the Mandelbrot set from 2 perpendicular planes into 3 dimensions
additionally
\({(x,y,y)\to \frac{(x^2+2y^2)((x^2-y^2),2xy,2xy)}{(x+y)\sqrt{x^4+6x^2y^2+y^4}}+c}\)