Sort[
{
Area[#]/(4Pi)*CountryData["World", "Area"] / CountryData[#,"Area"],
#
}& /@ AllCountries
]

CountryColour["Canada"]

CountryShape["Monaco"]

Length[CountryShape["Canada"]]

Total[Length[CountryShape[#]]& /@ AllCountries]

(# -> CountryColour[#]-CountryColour["Russia"])& /@ AllCountries

Simplify[
#,
{Element[ϕ, Reals], Element[θ, Reals]}
]& /@ S1Act[-Arg[First[sigma[LonLat[ϕ, θ]]]], sigma[LonLat[ϕ, θ]]] /.{
Arg[(E^(I ϕ)) (Sqrt[1+Sin[θ]])] -> ϕ
}

Simplify[
#,
{Element[ϕ, Reals], Element[θ, Reals], Element[α, Reals]}
]& /@ pi[S1Act[α, sigma[LonLat[ϕ, θ]]]]

False && MakeImage["G20DiskMap", Graphics[{
EdgeForm[Black],Circle[{0,0},1], Circle[{0,0},1],
{
Hue[CountryColour[#]],
CountryShape[#] 
/. p_LonLat :> (s=sigma[p][[1]]; s={Re[s], Im[s]}; s)
/. {CountryShape  -> List, Region[pts___] :> Polygon[{pts}]}
}& /@ 
G20
}]]

False && MakeImage["WhirlCanada", Graphics3D[{
Whirl["Canada"],
grid
}]]

False && MakeImage["WhirlFirst10", Graphics3D[Whirl /@ Take[AllCountries, 10]]]

False && MakeImage["WhirlAusCan", Graphics3D[Whirl /@ {"Australia", "Canada"}]]

False && MakeImage["WhirlG20", Graphics3D[Whirl /@ G20]]