Triadic Neumann Eigenfunctions

The following images are contour plots of the first 24 eigenfunctions (ordered by eigenvalue) of the negative Laplacian on a 'triadic Koch snowflake' with zero Neumann boundary conditions (normal derivatives). They represent the harmonics, ordered by frequency of vibration, of a surface whose boundary is a level-5 approximation to a fractal curve with fractal dimension log(4)/log(3) -- about 1.26. Note the 60-degree, 120-degree, and/or 180-degree symmetries and anti-symmetries. The yellow regions, representing nodal lines, are defined by contour values in the range [-.01,.01]. The other regions are associated with contour intervals of length .125. The corresponding eigenvalues are shown in parentheses


Function 1 (11.917)
Function 2 (11.917)
Function 3 (23.272)
Function 4 (23.272)
Function 5 (27.735)
Function 6 (52.534)
Function 7 (86.172)
Function 8 (86.172)
Function 9 (112.95)
Function 10 (112.95)
Function 11 (119.55)
Function 12 (141.72)
Function 13 (141.72)
Function 14 (141.97)
Function 15 (148.62)
Function 16 (153.20)
Function 17 (153.20)
Function 18 (185.97)
Function 19 (200.21)
Function 20 (210.55)
Function 21 (210.55)
Function 22 (221.65)
Function 23 (221.65)
Function 24 (231.21)
Square Dirichlet Eigenfunctions

The following images are contour plots of the first 50 eigenfunctions (ordered by eigenvalue) of the negative Laplacian on a 'square Koch snowflake' with zero Dirichlet boundary conditions. They represent the harmonics, ordered by frequency of vibration, of a drumhead whose boundary is a level-4 approximation to a fractal curve with fractal dimension 1.5. Note the 90-degree and/or 180-degree symmetries and anti-symmetries. The yellow regions, representing nodal lines, are defined by contour values in the range [-.01,.01]. The other regions are associated with contour intervals of length .125.


Function 1
Function 2
Function 3
Function 4
Function 5
Function 6
Function 7
Function 8
Function 9
Function 10
Function 11
Function 12
Function 13
Function 14
Function 15
Function 16
Function 17
Function 18
Function 19
Function 20
Function 21
Function 22
Function 23
Function 24
Function 25
Function 26
Function 27
Function 28
Function 29
Function 30
Function 31
Function 32
Function 33
Function 34
Function 35
Function 36
Function 37
Function 38
Function 39
Function 40
Function 41
Function 42
Function 43
Function 44
Function 45
Function 46
Function 47
Function 48
Function 49
Function 50
Square Neumann Eigenfunctions

The following images are contour plots of the first 49 eigenfunctions (ordered by eigenvalue) of the negative Laplacian on a 'square Koch snowflake' with zero Neumann boundary conditions (normal derivatives). They represent the harmonics, ordered by frequency of vibration, of a surface whose boundary is a level-4 approximation to a fractal curve with fractal dimension 1.5. Note the 90-degree and/or 180-degree symmetries and anti-symmetries. The yellow regions, representing nodal lines, are defined by contour values in the range [-.01,.01]. The other regions are associated with contour intervals of length .125.


Function 1
Function 2
Function 3
Function 4
Function 5
Function 6
Function 7
Function 8
Function 9
Function 10
Function 11
Function 12
Function 13
Function 14
Function 15
Function 16
Function 17
Function 18
Function 19
Function 20
Function 21
Function 22
Function 23
Function 24
Function 25
Function 26
Function 27
Function 28
Function 29
Function 30
Function 31
Function 32
Function 33
Function 34
Function 35
Function 36
Function 37
Function 38
Function 39
Function 40
Function 41
Function 42
Function 43
Function 44
Function 45
Function 46
Function 47
Function 48
Function 49
Evolution of Magnetic Field in Superconducting Film

The following movie simulates the evolution of the induced magnetic field in a superconducting film dimensioned 30 by 30 (in units of penetration depth) with no external field applied. The Ginzburg- Landau parameter is \kappa = Sqrt(0.5), and the initial configuration consists of a pair of degree-2 vortices and a pair of anti-vortices.


Movie