Classic Snowflake Domain

Dimension

Power-Law  (Exact value: log(4)/log(3)=1.2618595)

Dirichlet boundary conditions (Res. 243-81-27-9-3-1)

1000 Eigenvalues: Data Graph Extrapolation

Neumann boundary conditions (Res. 243-81-27-9-3-1)

1000 Eigenvalues: Data Graph

Dirichlet boundary conditions (Level 6, Res. 243-81-27-9-3-1)

Eigenvalues: 4500 Graph
2D: 1-25 26-50 51-75 76-100 101-125 126-150 151-175 176-200 201-225 226-250 251-275 276-300 301-325 326-350 351-375 376-400 401-425 426-450 451-475 476-500 501-525 526-550 551-575 576-600 601-625 626-650 651-675 676-700 701-725 726-750 751-775 776-800 801-825 826-850 851-875 876-900 901-925 926-950 951-975 976-1000    Show No.: Go
3D: 1-25 26-50 51-75 76-100 101-125 126-150 151-175 176-200 201-225 226-250 251-275 276-300 301-325 326-350 351-375 376-400 401-425 426-450 451-475 476-500 501-525 526-550 551-575 576-600 601-625 626-650 651-675 676-700 701-725 726-750 751-775 776-800 801-825 826-850 851-875 876-900 901-925 926-950 951-975 976-1000    Show No.: Go
Nodal Lines: 1-25 26-50 51-75 76-100 101-125 126-150 151-175 176-200 201-225 226-250 251-275 276-300 301-325 326-350 351-375 376-400 401-425 426-450 451-475 476-500 501-525 526-550 551-575 576-600 601-625 626-650 651-675 676-700 701-725 726-750 751-775 776-800 801-825 826-850 851-875 876-900 901-925 926-950 951-975 976-1000    Show No.: Go
Graphs: Counting Function Remainder 1 Remainder 2

Neumann boundary conditions (Level 6, Res. 243-81-27-9-3-1)

Eigenvalues: 6000 Graph
2D: 1-25 26-50 51-75 76-100 101-125 126-150 151-175 176-200 201-225 226-250 251-275 276-300 301-325 326-350 351-375 376-400 401-425 426-450 451-475 476-500 501-525 526-550 551-575 576-600 601-625 626-650 651-675 676-700 701-725 726-750 751-775 776-800 801-825 826-850 851-875 876-900 901-925 926-950 951-975 976-1000    Show No.: Go
3D: 1-25 26-50 51-75 76-100 101-125 126-150 151-175 176-200 201-225 226-250 251-275 276-300 301-325 326-350 351-375 376-400 401-425 426-450 451-475 476-500 501-525 526-550 551-575 576-600 601-625 626-650 651-675 676-700 701-725 726-750 751-775 776-800 801-825 826-850 851-875 876-900 901-925 926-950 951-975 976-1000    Show No.: Go
Nodal Lines: 1-25 26-50 51-75 76-100 101-125 126-150 151-175 176-200 201-225 226-250 251-275 276-300 301-325 326-350 351-375 376-400 401-425 426-450 451-475 476-500 501-525 526-550 551-575 576-600 601-625 626-650 651-675 676-700 701-725 726-750 751-775 776-800 801-825 826-850 851-875 876-900 901-925 926-950 951-975 976-1000    Show No.: Go
Graphs: Counting Function Remainder 1 Remainder 2

Dirichlet boundary conditions (Level 5, Res. 81-27-9-3-1)

Eigenvalues: 1000 Graph
2D: 1-25 26-50 51-75 76-100 101-125 126-150 151-175 176-200    Show No.: Go
Energy Contour: 1-20 21-40 41-60 61-80 81-100 101-120 121-140 141-160 161-180 181-200    Show No.: Go
Energy Filled: 1-20 21-40 41-60 61-80 81-100 101-120 121-140 141-160 161-180 181-200    Show No.: Go
Energy Sum of Multiplicities 2: 2+3 4+5 8+9 11+12 15+16 17+18 19+20 21+22 27+28 30+31 32+33 34+35 39+40 41+42 44+45 46+47 49+50 53+54 56+57 59+60 63+64 65+66 67+68 69+70 74+75 78+79 80+81 83+84 85+86 89+90 91+92 94+95 98+99 100+101 104+105 107+108 109+110 113+114 116+117 120+121 122+123 126+127 128+129 131+132 133+134 136+137 139+140 142+143 146+147 149+150 153+154 155+156 157+158 162+163 165+166 167+168 169+170 171+172 176+177 180+181 183+184 185+186 189+190 191+192 193+194 195+196

Neumann boundary conditions (Level 5, Res. 81-27-9-3-1)

Eigenvalues: 1000 Graph
2D: 1-25 26-50 51-75 76-100 101-125 126-150 151-175 176-200    Show No.: Go
Energy Contour: 1-20 21-40 41-60 61-80 81-100 101-120 121-140 141-160 161-180 181-200    Show No.: Go
Energy Filled: 1-20 21-40 41-60 61-80 81-100 101-120 121-140 141-160 161-180 181-200    Show No.: Go
Energy Sum of Multiplicities 2: 2+3 4+5 8+9 10+11 13+14 17+18 21+22 23+24 27+28 29+30 31+32 33+34 40+41 42+43 44+45 46+47 50+51 52+53 57+58 59+60 62+63 64+65 68+69 72+73 75+76 77+78 81+82 83+84 86+87 88+89 91+92 93+94 97+98 99+100 105+106 108+109 111+112 113+114 117+118 119+120 122+123 125+126 128+129 130+131 134+135 136+137 138+139 141+142 146+147 148+149 151+152 153+154 159+160 162+163 165+166 168+169 171+172 174+175 177+178 180+181 182+183 185+186 187+188 190+191 194+195 196+197

Square Snowflake Domain (a = 0.1)

Dimension

Power-Law  (Extrapolation: 1.237456, Exact: 1.15965)

Dirichlet boundary conditions (Res. 100)

1000 Eigenvalues: Data Graph

Neumann boundary conditions (Res. 100)

1000 Eigenvalues: Data Graph

Dirichlet boundary conditions (Level 4, Res. 100)

2D: 1-25 26-50 51-75 76-100 101-125 126-150 151-175 176-200    Show No.: Go
3D: 1-25 26-50 51-75 76-100 101-125 126-150 151-175 176-200    Show No.: Go
Nodal Lines: 1-25 26-50 51-75 76-100 101-125 126-150 151-175 176-200    Show No.: Go
Graphs: Counting Function Remainder 1 Remainder 2

Neumann boundary conditions (Level 4, Res. 100)

2D: 1-25 26-50 51-75 76-100 101-125 126-150 151-175 176-200    Show No.: Go
3D: 1-25 26-50 51-75 76-100 101-125 126-150 151-175 176-200    Show No.: Go
Nodal Lines: 1-25 26-50 51-75 76-100 101-125 126-150 151-175 176-200    Show No.: Go
Graphs: Counting Function Remainder 1 Remainder 2

Square Snowflake Domain (a = 0.2)

Dimension

Power-Law  (Extrapolation: 1.287419, Exact: 1.2811)

Dirichlet boundary conditions (Res. 100)

1000 Eigenvalues: Data Graph
Graphs: Counting Function Remainder 1 Remainder 2

Neumann boundary conditions (Res. 100)

1000 Eigenvalues: Data Graph
Graphs: Counting Function Remainder 1 Remainder 2

Square Snowflake Domain (a = 3. - 2 * sqrt(2))

Dimension

Exact: 1.2465

Dirichlet boundary conditions (Res. 100)

1000 Eigenvalues: Data Graph
Graphs: Counting Function Remainder 1 Remainder 2

Neumann boundary conditions (Res. 100)

1000 Eigenvalues: Data Graph
Graphs: Counting Function Remainder 1 Remainder 2

Contact Information

Robert Strichartz, Cornell University, Department of Mathematics, str@math.cornell.edu
Samuel Wiese, University of Leipzig, Department of Mathematics, sw31hiqa@studserv.uni-leipzig.de