Coloring Graphics3D as a Function of Spatial Coordinates

Question

I would like to plot a Graphics3D object colored based on the value of some spatial function. For example to plot a triangle colored by the function cf I might start with:

testCorners = {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}};
testTriangle = Triangle[testCorners];
cf [{x_, y_, z_}] = ColorData["NeonColors"][z];

I hope that there is a couple of lines, probably involving Graphics3D that will plot this triangle (or other, more complicated surfaces) colored with the function I have specified (or other more complicated functions of x, y, and z).

I can do fun stuff like color vertices and have Mathematica linearly interpolate over the faces.

Graphics3D[
 Append[PolyhedronData["Dodecahedron", "GraphicsComplex"], 
  VertexColors -> ColorData["NeonColors"] /@ RandomReal[{0, 1}, 20]]]

It may be possible to use a similar method by transforming the surface into a GraphicsComplex object and then setting VertexTextureCoordinates appropriately and supplying a Texture to Graphics3D, but that seems quite complicated...

Answer 1

gc = PolyhedronData["Dodecahedron", "GraphicsComplex"];
minmax = MinMax @ N @ gc[[1, All, -1]]

Graphics3D[Append[gc, 
  VertexColors -> (ColorData[{"NeonColors", minmax}] /@ gc[[1, All, -1]])]]

Alternatively, use your cf with rescaled arguments:

Graphics3D[Append[gc, 
   VertexColors -> (cf /@ Transpose[Rescale /@ Transpose[First @ N @ gc]])]

same picture

Update: "but suppose I had a color function which was radially symmetric like Norm. The dodecahedron would be uniformly colored, even though the centers of each face should have a lower value."

texture = RadialGradientImage["NeonColors"];

Graphics3D[Normal[gc] /. Polygon[x_] :>
  {Texture @ texture,
   Polygon[x, VertexTextureCoordinates -> Rescale[CirclePoints @ Length @ x]]}]

Further examples:

SeedRandom[1]
polyhedra = RandomSample[PolyhedronData["Archimedean"], 12];

Graphics3D[Normal[PolyhedronData[#, "GraphicsComplex"]] /. 
     Polygon[x_] :> {Texture @ texture, 
       Polygon[x, 
        VertexTextureCoordinates -> Rescale[CirclePoints @ Length @ x]]}, 
    Lighting -> "Neutral"] & /@  polyhedra // Multicolumn[#, 4] &