Hilbert's 13th Problem

Pensieve Header: Hilbert's 13th problem - Step 2 to level 3.

<< "../../2009-11/Hilbert13th-Program.m"

ϕ1:= Phi[Identity, 2, 0.3, 2/3];
ϕ2:= Phi[ϕ1, 8, (0.3^2), 2/3];
ϕ3:= Phi[ϕ2, 32, (0.3^3), 2/3];
Step2phis = Plot[{ x, ϕ1[x], ϕ2[x], ϕ3[x]}, {x,0,1},
PlotPoints -> 1279, ColorFunction -> Automatic, AxesStyle -> Thick, PlotStyle -> {Directive[Red,Thick], Directive[Green,Thick], Directive[Blue,Thick], Directive[Black]}
]

Timing[
L3Step2Cascade=Rasterize[
Plot3D[ϕ3[x]+λ*ϕ3[y], {x,0,1}, {y,0,1},
PlotPoints -> 1279, Mesh -> 95,ViewPoint -> {-2,-2,1}, NormalsFunction -> None, Boxed -> False, Axes -> None, ImageSize -> 2400
], ImageSize -> 2400, RasterSize->2400
]
]

Export[
"L3Step2Cascade.png",
ImageCrop[L3Step2Cascade]
]

phi1:= Phi[Identity, 2, 0.3, 2/3];
phi2:= Phi[phi1, 12, 0, 0.8];
phi3:= Phi[ϕ, ϕ0 -> phi1, Subdivisions -> 12, Slope -> 0, FillFactor -> 0.8];
g1=G[f, phi1];
g2=G[f, phi2];

Step2G1=Plot[
g1[z], {z,0,1+λ},
AxesStyle -> Thick, PlotStyle -> Thick
]

Export[
"Step2G1.png",
ImageCrop[Step2G1]
]

Step2G2 = Plot[
g2[z], {z,0,1+λ},
PlotPoints-> 5000, AxesStyle -> Thick, PlotStyle -> Thick
]

Export[
"Step2G2.png",
ImageCrop[Step2G2]
]

Timing[
Step2CascadeWithG1=Rasterize[
Plot3D[phi2[x]+λ*phi2[y], {x,0,1}, {y,0,1},
PlotPoints -> 301, Mesh -> 23,ViewPoint -> {-2,-2,1}, NormalsFunction -> None, ColorFunction -> (Hue[g1[#3]]&), Boxed -> False, Axes -> None
]
]
]

Export[
"Step2CascadeWithG1.png",
ImageCrop[Step2CascadeWithG1]
]

Timing[
Step2DensityWithG1=Rasterize[
DensityPlot[
g1[phi2[x]+λ*phi2[y]], {x,0,1}, {y,0,1},
PlotPoints -> 301, Frame -> False
]
]
]

Export[
"Step2DensityWithG1.png",
ImageCrop[Step2DensityWithG1]
]

Timing[
Step2DensityWithG2=Rasterize[
DensityPlot[
g2[phi2[x]+λ*phi2[y]], {x,0,1}, {y,0,1},
PlotPoints -> 359, Mesh -> 11, Frame -> False
]
]
]

Export[
"Step2DensityWithG2.png",
Step2DensityWithG2
]