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With[{w = r Exp[I \[Theta]]},
g3 = Im[(1 + w^6)/(w^6 + Sqrt[5] w^3 - 1)] - 1/2;
g1 = -(3/2) Im[(w (1 - w^4))/(w^6 + Sqrt[5] w^3 - 1)];
g2 = -(3/2) Re[(w (1 + w^4))/(w^6 + Sqrt[5] w^3 - 1)];
g = Total[{g1, g2, g3}^2];
ParametricPlot3D[{g1, g2, g3}/g, {r, 0, 1}, {\[Theta], 0, 2 \[Pi]},
PlotTheme -> {"Classic", "ClassicLights"},
PerformanceGoal -> "Quality", PlotPoints -> 50, Boxed -> False,
Axes -> False, ViewPoint -> Below]
]
g3 = Im[(1 + w^6)/(w^6 + Sqrt[5] w^3 - 1)] - 1/2;
g1 = -(3/2) Im[(w (1 - w^4))/(w^6 + Sqrt[5] w^3 - 1)];
g2 = -(3/2) Re[(w (1 + w^4))/(w^6 + Sqrt[5] w^3 - 1)];
g = Total[{g1, g2, g3}^2];
ParametricPlot3D[{g1, g2, g3}/g, {r, 0, 1}, {\[Theta], 0, 2 \[Pi]},
PlotTheme -> {"Classic", "ClassicLights"},
PerformanceGoal -> "Quality", PlotPoints -> 50, Boxed -> False,
Axes -> False, ViewPoint -> Below]
]
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