The Mathematica code below shows how it was created.
Needs["Miscellaneous`WorldPlot`"]
Wermer[theta_, phi_] :=
Module[{t},
t = If[N[theta] == 0,
phi,
phi Sin[theta] / theta];
N[theta {Sin[t], -Cos[t]}]]
WermerProjection =
(Wermer[Pi(5400 - #1) / 10800, Pi #2 / 10800])& ;
WorldPlot[World,
WorldBackground -> RGBColor[0.340, 1.000, 1.000],
WorldGrid -> None,
WorldFrame -> Thickness[0.004],
WorldProjection -> WermerProjection];
Here is a bigger version
of the plot, and another area-preserving map-projection:
