**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