Dror Bar-Natan: Talks: Fields-0911:

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

Hilbert's 13th Problem

Pensieve Header: Some pictures for the Kolmogorov-Arnol'd solution of Hilbert's 13th problem.

<< C:\drorbn\AcademicPensieve\2009-11\Hilbert13th-Program.m

phi2:= Phi[phi0, 2, 0.3, 2/3];
phi26:= Phi[phi2, 12, 0, 0.95];
Plot[{ phi0[x], phi2[x], phi26[x]}, {x,0,1},
PlotPoints -> 479
]

phi5:= Phi[ϕ];
Timing[
Cascade5=Rasterize[
Plot3D[phi5[x]+λ*phi5[y], {x,0,1}, {y,0,1},
PlotPoints -> 51, Mesh -> 14, Exclusions -> None, ViewPoint -> {-2,-2,1}, ImageSize ->800 , AxesLabel ->Automatic, NormalsFunction -> None
]
]
]

phi2:= Phi[ϕ, Subdivisions -> 2, Slope -> 0.4, FlatFraction -> 2/3];
phi26:= Phi[ϕ, ϕ0 -> phi2, Subdivisions -> 12, FlatFraction -> 0.95];
Plot[{ x, phi2[x], phi26[x]}, {x,0,1},
PlotPoints -> 479
]

Xs=Union@@Phi[{X0, X1, X3}, ϕ0 -> phi2, Subdivisions -> 12, FlatFraction -> 0.95];
Ys=Union@@Phi[{Y0, Y1, Y3}, ϕ0 -> phi2, Subdivisions -> 12, FlatFraction -> 0.95];
l=Length[Ys];
Cascade26=Rasterize[
ListPlot3D[
Join @@ Table[{Xs[[i]], Xs[[j]], Ys[[i]]+λ*Ys[[j]]}, {i, l}, {j,l}],
InterpolationOrder ->1, Mesh -> 14, ViewPoint -> {-2,-2,1}, ImageSize ->800 , AxesLabel ->Automatic, NormalsFunction -> None, ColorFunction -> Hue, ColorFunctionScaling -> False
]
]

Xs

Ys

Xs=Union@@Phi[{X0, X1, X3}];
Ys=Union@@Phi[{Y0, Y1, Y3}];
l=Length[Ys];
Cascade=Rasterize[
ListPlot3D[
Join @@ Table[{Xs[[i]], Xs[[j]], Ys[[i]]+λ*Ys[[j]]}, {i, l}, {j,l}],
InterpolationOrder ->1, Mesh -> 14, ViewPoint -> {-2,-2,1}, ImageSize ->800, NormalsFunction -> None, ColorFunction -> Hue, ColorFunctionScaling -> False
]
]

Xs

Ys

phi26