Dror Bar-Natan: Talks: Fields-0911:

Dror Bar-Natan: Academic Pensieve: 2009-11:

Hilbert's 13th Problem

Pensieve Header: Hilbert's 13th problem - the heavey computations.

SetDirectory[
$MachineName /. {
"dror-x61" -> "C:/drorbn/AcademicPensieve/2009-11"
} /. $MachineName -> "."
];
<< Hilbert13th-Program.m

phi4:= Phi[phi0, 4, 0.15, 0.5];
g=G[f, phi4];
Plot[g[z], {z, 0, 1+λ}]

Step1Trenches=Rasterize[Plot3D[g[phi4[x]+λ*phi4[y]], {x,0,1}, {y,0,1},
Mesh -> 15, AxesLabel ->None, PlotPoints -> 500, ViewPoint -> {-2,-2,2}, Axes ->None, Boxed -> False
]]

Export[
"Step1Trenches.png",
ImageCrop[Step1Trenches]
]

phi1= Phi[Identity, 2, 0.3, 2/3];
phi2= Phi[phi1, 12, 0.0001, 0.6];
phi3= Phi[ϕ, ϕ0 -> phi1, Subdivisions -> 12, Slope -> 0.0001, FlatFraction -> 0.6];
g1=G[f, phi1];
g2=G[f, phi2];

Plot[{phi1[x], phi2[x]}, {x,0,1}]

Plot[
g1[z], {z,0,1+λ},
PlotPoints-> 200
]

Plot[
g2[z], {z,0,1+λ},
PlotPoints-> 5000
]

Rasterize[
Plot3D[phi2[x]+λ*phi2[y], {x,0,1}, {y,0,1},
PlotPoints -> 119, Mesh -> 11,ViewPoint -> {-2,-2,1}, Axes ->None, Boxed -> False,  NormalsFunction -> None, ColorFunction -> (Hue[g2[#3]]&)
]
]

Rasterize[
DensityPlot[
g2[phi2[x]+λ*phi2[y]], {x,0,1}, {y,0,1},
PlotPoints -> 119,Mesh -> 11
]
]

Timing[
Step2CascadeWithG2=Rasterize[
Plot3D[phi2[x]+λ*phi2[y], {x,0,1}, {y,0,1},
PlotPoints -> 1199, Mesh -> 11,ViewPoint -> {-2,-2,1}, NormalsFunction -> None, ColorFunction -> (Hue[g2[#3]]&), ImageSize -> 1600, Axes ->None, Boxed -> False
], ImageSize -> 1600, RasterSize -> 1600
]
]

Export[
"Step2CascadeWithG2.png",
Step2CascadeWithG2
]

Do[
phi2[i]= Phi[Identity, 2, 0.02, 0.95, i/5];
g2[i]=G[f, phi2[i]],
{i,0,4}
];
Do[
phi3[i]= Phi[Identity, 3, 0.01, 0.95, i/5];
g3[i]=G[f, phi3[i]],
{i,0,4}
]

Step3Trenches=Rasterize[
Plot3D[
(1/5)*Sum[g2[k][phi2[k][x]+λ*phi2[k][y]],  {k ,0, 4}],
{x,0,1}, {y,0,1}, PlotPoints -> 500, Axes ->None, Boxed -> False, ViewPoint -> {-2, -2, 1}
]
]

Export[
"Step3Trenches.png",
Step3Trenches
]