Pensieve Header: Embedding RP^2 in R^4, following Hillman.

{x,y,z}={Cos[t]Cos[s], Cos[t]Sin[s], Sin[t]};
f={x^2-y^2, x y, y z, x z};

Rasterize[
{c, h1,h2,h3}=f;
ParametricPlot3D[
{h1,h2,h3}, {t, 0, Pi/2}, {s,-Pi,Pi},
ColorFunction -> (Hue[c/2 /. {t-> #4, s-> #5}]&),
ColorFunctionScaling -> False, ViewPoint -> {2,2,2}
],RasterSize -> 1024
]

Rasterize[
{h1,h2,h3, c}=f;
ParametricPlot3D[
{h1,h2,h3}, {t, 0, Pi/2}, {s,-Pi,Pi},
ColorFunction -> (Hue[c/2 /. {t-> #4, s-> #5}]&),
ColorFunctionScaling -> False
],RasterSize -> 1024
]

c

g = {
{1,0,0,0},
{0,1,0,0},
{0,0,0,1},
{0,0,1,0}
}.f;
g=f;
True || (g=Table[2Random[]-1,{4},{4}].f);