Well, I'm already confused. The domain is complex numbers where the real part is >= 0? I think it would be helpful to make that clear from the get-go. When I see "Cartesian grid", I think R^2, not C.

My prof even said to me "If you'd pay attention, you'd understand this!". I was the only one who was brave enough to ask questions about it!

https://en.wikipedia.org/wiki/Smith_chart#/media/File:Animat...

https://www.researchgate.net/publication/3427377_Escher's_ar...

"A conceptual relation between Circle Limit IV, an artistic creation by M.C. Escher, and Smith Chart, geographical aid for microwave engineering created by P.H. Smith, was established. The basis of Escher's art and Smith chart can both be traced back to invariance of the cross ration of four complex numbers under Möbius transformation on the domain of complex numbers. The Smith chart can be used as an aid for constructing Escher-like drawings that display periodic mosaic patterns and at the same time convey the perception of infinite progression within a unit circle."

f(z) = (z − 1)/(z + 1)

The Smith chart is useful to electronic engineers because a given VSWR (the thing you try to minimise to get a good antenna) becomes a circle about the center of the Smith chart

VSWR= (1 + | Γ |)/(1 - | Γ |)

So to make your antenna better, get the plot closer to the center. Whether it is above the line or below tells you whether the antenna is inductive or capacitive and hence which kind of loading to add.

https://www.youtube.com/watch?v=0z1fIsUNhO4

Also this one: https://www.ti.com/lit/an/swra046a/swra046a.pdf