Hidden wonders of the Pixel formula

I stumbled upon an amazing combination of variables: the standard Pixel formula with Gradient for family Outside coloring (sam.ucl) and the Inverse fractal plane mapping. How does such a transformation take place from a bunch of squares formed by the gradient to a beautiful rounded geometric pattern? I am baffled and pleasantly surprised!

pixelInverseFractalPlane {
fractal:
  title="pixel inverse fractal plane" width=600 height=600 layers=2
  credits="Kathy;1/20/2020" antialiasing=yes
layer:
  caption="Layer 2" opacity=100
mapping:
  center=0.00/0.00 magn=1.00112 transforms=1
transform:
  filename="Standard.uxf" entry="Inverse" p_radius=1.0 p_center=0/0
  p_usescreen=no
formula:
  maxiter=25000 filename="Standard.ufm" entry="Pixel"
inside:
  transfer=none
outside:
  density=1.049392 transfer=linear filename="sam.ucl"
  entry="FamilyGradient" p_mode=Squares p_tilem=0/0 p_sizesq=1.18
  p_rottile=0.0
gradient:
  linked=yes smooth=yes rotation=-47 index=36 color=14356239 index=152
  color=16382198 index=216 color=46591 index=369 color=1310828
  index=370 color=1966080 index=-9 color=11403778
opacity:
  smooth=yes rotation=-47 index=36 opacity=255 index=152 opacity=255
  index=216 opacity=255 index=369 opacity=255 index=370 opacity=255
  index=-9 opacity=255
layer:
  caption="Layer 1" opacity=100 visible=no
mapping:
  center=0.00/0.00 magn=1.00112 transforms=1
transform:
  enabled=no filename="Standard.uxf" entry="Inverse" p_radius=1.0
  p_center=0/0 p_usescreen=no
formula:
  maxiter=25000 filename="Standard.ufm" entry="Pixel"
inside:
  transfer=none
outside:
  density=1.049392 transfer=linear filename="sam.ucl"
  entry="FamilyGradient" p_mode=Squares p_tilem=0/0 p_sizesq=1.18
  p_rottile=0.0
gradient:
  linked=yes smooth=yes rotation=-47 index=36 color=14356239 index=152
  color=16382198 index=216 color=46591 index=369 color=1310828
  index=370 color=1966080 index=-9 color=11403778
opacity:
  smooth=yes rotation=-47 index=36 opacity=255 index=152 opacity=255
  index=216 opacity=255 index=369 opacity=255 index=370 opacity=255
  index=-9 opacity=255
}

The Inverse mapping performs an operation known as "circular inversion"; do a web search on the term to get lots of technical information. But you can think of it as turning the plane inside-out. Doing this maps (infinite) lines to circles that touch the origin (or whatever the center of the inversion is).

In your example, the black square centered at the origin maps to the black outline around the outside of the second image. The four thick blue lines that touch that square extend to infinity, so map to the four largest circles. The four squares adjacent to the center one map to the four boat like shapes inside those circles. The four squares diagonally adjacent to the center one map to the smaller shield like shapes between those. The center square itself maps to the outside of the black outline; most of it is outside the image.

Thank you so much for this, Rick. I didn't know that the inverse fractal plane took the form of circular patterns. There is a wealth of information out there like you said on the topic, much of which is too technical for me. I do however value your feedback and now I have a better understanding of the outcome!