In reading Michael Trott's Visualization of Riemann Surfaces of Algebraic Functions, he has:
ParametricPlot3D[{r Cos[φ], r Sin[φ], Sqrt[r] Sin[φ/2], SurfaceColor[Hue[φ/(4 π)]]}, {r, 0, 1}, {φ, 0, 4 π}, PlotPoints -> {20, 60}, Boxed -> False, Axes -> False] How would you do this coloring now in Mathematica 9?
Natively in version 9, you can do the following:
ParametricPlot3D[{r Cos[phi], r Sin[phi], Sqrt[r] Sin[phi/2]}, {r, 0, 1}, {phi, 0, 4 Pi}, PlotPoints -> {20, 60}, Boxed -> False, Axes -> False, ColorFunction -> (Hue[#5/(4 Pi)] &), ColorFunctionScaling -> False] 
Alternatively, you can always use the exact code using the V5 emulator:
<<Version5`Graphics` ParametricPlot3D[{r Cos[phi], r Sin[phi], Sqrt[r] Sin[phi/2], SurfaceColor[Hue[phi/(4 Pi)]]}, {r, 0, 1}, {phi, 0, 4 Pi}, PlotPoints -> {20, 60}, Boxed -> False, Axes -> False] 
Of course, the graphics aren't quite as nice. You can go back to the newer style graphics as follows:
<<Version6`Graphics` As of Version 6,
SurfaceColorhas been superseded bySpecularityandGlow.
One could specify the variables explicitly like here :
ParametricPlot3D[{ r Cos[φ], r Sin[φ], Sqrt[r] Sin[φ/2]}, {r, 0, 1}, {φ, 0, 4 π}, ColorFunction -> Function[{x, y, z, r, φ, θ}, {Specularity[#], Glow[#]}& @ Hue[Rescale[φ, {0, 1}]]], PlotPoints -> {20, 60}, Boxed -> False, Axes -> False] 
or adding a more thrilling variation of ColorFunction (singularity when r -> 0) :
ParametricPlot3D[{ r Cos[φ], r Sin[φ], Sqrt[r] Sin[φ/2]}, {r, 0, 1}, {φ, 0, 4 π}, ColorFunction -> Function[{x, y, z, r, φ, θ}, {Specularity[#], Glow[#]}& @ Hue[ Rescale[ φ/r, {0, 1}]]], PlotPoints -> {20, 60}, Boxed -> False, Axes -> False] 
Function[{x, y, z, r, φ}, ...]. $\endgroup$ ...and now, for something that takes a bit of the old, and a bit of the new:
ColoredMakePolygons[vl_List] := Module[{msh = Map[Most, vl, {2}], cols, dims}, cols = Map[First[Cases[Last[#], _?ColorQ, {0, Infinity}]] &, vl, {2}]; cols = Map[If[Head[#] === GrayLevel, #, ColorConvert[#, RGBColor]] &, cols, {2}]; dims = Most[Dimensions[msh]]; GraphicsComplex[Apply[Join, msh], Polygon[Flatten[Apply[ Join[Reverse[#1], #2] &, Partition[Partition[Range[Times @@ dims], Last[dims]], {2, 2}, {1, 1}], {2}], 1]], VertexColors -> Apply[Join, cols]]] /; ArrayDepth[vl] == 3 && Last[Dimensions[vl]] == 4 With[{m = 21, n = 61}, Graphics3D[ColoredMakePolygons[ N @ Table[{r Cos[φ], r Sin[φ], Sqrt[r] Sin[φ/2], Hue[φ/(4 π)]}, {r, 0, 1, 1/(m - 1)}, {φ, 0, 4 π, 4 π/(n - 1)}]], Boxed -> False, Lighting -> "Neutral"]] 
Manipulate[ ParametricPlot3D[ Evaluate@{Re[(1 - α) (r Exp[ I φ])^2 + α (r Exp[I φ])^3], Im[(1 - α) (r Exp[I φ])^2 + α (r Exp[ I φ])^3], r Cos[φ]}, {r, 0, 2}, {φ, -Pi, Pi}, PlotRange -> All, Mesh -> 20, ColorFunction -> (Hue[#5] &), PlotPoints -> 100, MaxRecursion -> 1, BoxRatios -> {1, 1, 1}, PlotRange -> All, Axes -> False, ImageSize -> {435, 435}, Boxed -> False], {{α, 0, "%"}, 0, 1}] 
