A while ago, I made a GIF, which shows a one-parameter family of conformal transformations of the disk which turn it into an equilateral triangle. Of course, the Riemann mapping theorem guarantees the existence of such a conformal map, but it’s given explicitly by the Schwarz-Christoffel mapping (which more generally turns the unit disk or upper half plane into any convex polygon).
Anyway, two earlier GIFs show the images of the concentric circles in the unit disk, whereas the above GIF shows the images of the radial lines as the initial circle is rotated.
My Schwarz-Christoffel function is far too slow, even after compiling, to produce a usable Manipulate, so here I just give the code for generating the above GIF:
Anyway, two earlier GIFs show the images of the concentric circles in the unit disk, whereas the above GIF shows the images of the radial lines as the initial circle is rotated.
My Schwarz-Christoffel function is far too slow, even after compiling, to produce a usable Manipulate, so here I just give the code for generating the above GIF:
ToReal[z_]:={Re[z],Im[z]};scComp=Compile[{{z,_Complex}},NIntegrate[Exp[Total[((1/3.-1)(Log[I#]+Log[(ζ-#)/(I#)]))&/@N[{1,E^((2Iπ)/3),E^(-((2Iπ)/3))}]]],{ζ,0,z}]]
dodecapod=Module[{cols},cols=RGBColor/@{"#EE8374","#54567A"};ParallelTable[Quiet@ParametricPlot[Evaluate@Table[ToReal[I*scComp[rE^(I(θ+t))]],{θ,0.,2π-π/6,π/6}],{r,0.,1.},PlotRange->{{-2,2},{-1.78,2.22}},ImageSize->540,Axes->False,PlotStyle->Directive[CapForm["Round"],Thickness[.008],cols[[1]]],Background->cols[[2]]],{t,0.,π/6-#,#}]&[π/600]];
Export["dodecapod.gif",dodecapod,"DisplayDurations"->{1/25}]