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This is an attempt to make a case with some mounting ears, but all just mathematically. You can see that it was not much of a success. The Mathematica function (RegionPlot3D) does not handle sharp edges elegantly. Though to be fair, the GNU Octave function isosurface does not either. I think that is because they are both working from a 3D grid.

\frac{1}{{(x^{2} + y^{2} + 0.25 {(z - 1.)}^{2})}^{0.5}} + \frac{1}{{({(x - 0.943)}^{2} + y^{2} + 0.25 {(z + 0.333)}^{2})}^{0.5}} + \frac{1}{{({(x + 0.471)}^{2} + {(y - 0.816)}^{2} + 0.25 {(z + 0.333)}^{2})}^{0.5}} + \frac{1}{{({(x + 0.471)}^{2} + {(y + 0.816)}^{2} + 0.25 {(z + 0.333)}^{2})}^{0.5}} > 4.8

Octave Code:

1; # Prevent Octave from thinking that this is a function
   # though one is defined here
   
  function w = f(x2,y2,z2,c,r,e) 
   x  = (x2-c(1))/r(1);
   y  = (y2-c(2))/r(2);
   z  = (z2-c(3))/r(3);
   # function at origin must be <0, and >0 far enough away.  w=0 defines the surface
   th = atan2(x,y);
   r = sqrt(x.^2+y.^2);
   phi = th + (r+r.^2)*0.05;
   w = ((x/70).^4+(y/70).^4+(z/30).^4).^(-8.0);
   w=w+((1/4)*((x+y+40+134)/(4*1.4))./(1.0+((((x+67)/4).^2+((y+67)/4).^2).^0.5-2).^2))./(1+((z-20)/20).^2);
   w=w+((1/4)*((x-y+40+134)/(4*1.4))./(1.0+((((x+67)/4).^2+((y-67)/4).^2).^0.5-2).^2))./(1+((z-20)/20).^2);
   w=w+((1/4)*((-x+y+40+134)/(4*1.4))./(1.0+((((x-67)/4).^2+((y+67)/4).^2).^0.5-2).^2))./(1+((z-20)/20).^2);
   w=w+((1/4)*((-x-y+40+134)/(4*1.4))./(1.0+((((x-67)/4).^2+((y-67)/4).^2).^0.5-2).^2))./(1+((z-20)/20).^2);
   w=w-1;

   endfunction;

GNU Octave

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