A Convex Surface with Fractal Curvature
Iancu Dima, Rachel Popp, Robert S. Strichartz, Samuel C. Wiese
TL;DR
This paper constructs a fractal-curved convex surface derived from an octahedron, analyzes its spectral properties using finite element methods, and explores the spectrum's behavior.
Contribution
It introduces a novel convex surface with fractal curvature based on the Sierpinski gasket and computes its Laplacian spectrum.
Findings
Computed the bottom of the spectrum of the Laplacian.
Used finite element method on polyhedral approximations.
Speculated on the spectrum's overall behavior.
Abstract
We construct a surface that is obtained from the octahedron by pushing out 4 of the faces so that the curvature is supported in a copy of the Sierpinski gasket in each of them, and is essentially the self similar measure on SG. We then compute the bottom of the spectrum of the associated Laplacian using the finite element method on polyhedral approximations of our surface, and speculate on the behavior of the entire spectrum.
| # | Level 4 | Level 5 | Level 6 | Level 7 | Extrap. |
|---|---|---|---|---|---|
| 0 | 0 (1) | 0 (1) | 0 (1) | 0 (1) | 0.00 |
| 1-3 | 1.37 (3) | 1.37 (3) | 1.37 (3) | 1.37 (3) | 1.37 |
| 4-5 | 3.93 (2) | 3.92 (2) | 3.91 (2) | 3.91 (2) | 3.91 |
| 6-8 | 4.37 (3) | 4.35 (3) | 4.34 (3) | 4.34 (3) | 4.34 |
| 9-11 | 8.38 (3) | 8.3 (3) | 8.28 (3) | 8.27 (3) | 8.26 |
| 12-14 | 8.56 (3) | 8.49 (3) | 8.48 (3) | 8.47 (3) | 8.47 |
| 15 | 9.41 (1) | 9.34 (1) | 9.31 (1) | 9.3 (1) | 9.30 |
| 16 | 13.68 (1) | 13.5 (1) | 13.44 (1) | 13.42 (1) | 13.42 |
| 17-19 | 14.53 (2) | 14.35 (2) | 14.28 (3) | 14.24 (3) | 14.23 |
| 20-21 | 14.62 (3) | 14.36 (3) | 14.3 (2) | 14.28 (2) | 14.27 |
| 22-24 | 15.47 (3) | 15.28 (3) | 15.23 (3) | 15.21 (3) | 15.20 |
| 25-27 | 21.33 (3) | 20.9 (3) | 20.77 (3) | 20.72 (3) | 20.71 |
| 28-30 | 23.14 (3) | 22.54 (3) | 22.33 (3) | 22.25 (3) | 22.21 |
| 31-32 | 23.24 (2) | 22.93 (2) | 22.86 (2) | 22.85 (2) | 22.84 |
| 33-35 | 24.2 (3) | 23.65 (3) | 23.48 (3) | 23.42 (3) | 23.40 |
| 36-37 | 31.14 (2) | 29.95 (2) | 29.56 (2) | 29.41 (2) | 29.35 |
| 38-40 | 31.4 (3) | 30.66 (3) | 30.42 (3) | 30.34 (3) | 30.30 |
| 41-43 | 33.69 (3) | 32.88 (3) | 32.64 (3) | 32.57 (3) | 32.55 |
| 44 | 33.98 (1) | 33.1 (1) | 32.85 (1) | 32.77 (1) | 32.74 |
| 45 | 35.47 (3) | 34.04 (1) | 33.37 (1) | 33.1 (1) | 33.04 |
| 46-48 | 36.02 (1) | 34.4 (3) | 34.12 (3) | 34.03 (3) | 34.00 |
| 49-51 | 42.5 (3) | 40.83 (3) | 40.28 (3) | 40.08 (3) | 39.99 |
| 52-54 | 43.74 (3) | 41.97 (3) | 41.4 (3) | 41.2 (3) | 41.11 |
| 55 | 46.14 (1) | 44.87 (1) | 44.53 (1) | 44.44 (1) | 44.41 |
| 56-58 | 47.63 (3) | 45.97 (3) | 45.32 (3) | 45.09 (3) | 44.96 |
| 59-61 | 48.13 (3) | 46.0 (3) | 45.56 (3) | 45.43 (3) | 45.42 |
| 62-63 | 49.39 (2) | 47.4 (2) | 46.83 (2) | 46.63 (2) | 46.56 |
| 64 | 54.45 (1) | 52.24 (1) | 51.52 (1) | 51.26 (1) | 51.14 |
| 65-66 | 56.42 (2) | 54.0 (2) | 53.24 (2) | 52.97 (2) | 52.86 |
| 67-69 | 59.78 (3) | 56.0 (3) | 54.77 (3) | 54.3 (3) | 54.09 |
| 70-72 | 60.65 (3) | 58.25 (3) | 57.59 (3) | 57.39 (3) | 57.33 |
| 73-74 | 62.74 (2) | 60.31 (2) | 59.67 (2) | 59.5 (2) | 59.44 |
| 75-77 | 64.76 (3) | 61.66 (3) | 60.76 (3) | 60.47 (3) | 60.36 |
| 78-80 | 67.09 (3) | 63.17 (3) | 61.94 (3) | 61.49 (3) | 61.31 |
| 81-83 | 74.27 (3) | 69.87 (3) | 68.4 (3) | 67.87 (3) | 67.62 |
| 84-86 | 76.3 (3) | 71.39 (3) | 69.77 (3) | 69.18 (3) | 68.91 |
| 87-88 | 79.09 (2) | 75.35 (2) | 74.21 (2) | 73.84 (2) | 73.69 |
| 89-91 | 79.66 (3) | 76.28 (3) | 75.46 (3) | 75.29 (3) | 75.22 |
| 92 | 79.76 (1) | 76.64 (1) | 75.77 (1) | 75.56 (1) | 75.47 |
| 93-95 | 81.06 (3) | 77.1 (3) | 76.0 (3) | 75.68 (3) | 75.56 |
| 96-98 | 91.32 (3) | 85.35 (3) | 83.34 (3) | 82.62 (3) | 82.27 |
| 99-101 | 92.9 (3) | 86.03 (3) | 84.23 (3) | 83.7 (3) | 83.53 |
| 102-103 | 95.67 (2) | 87.53 (2) | 84.91 (2) | 83.94 (2) | 83.52 |
| 104 | 96.88 (3) | 88.62 (1) | 85.55 (1) | 84.41 (1) | 83.78 |
| 105-107 | 98.31 (1) | 91.31 (3) | 89.69 (3) | 89.2 (3) | 89.00 |
| 108-110 | 98.48 (3) | 92.63 (3) | 91.11 (3) | 90.72 (3) | 90.58 |
| 111 | 98.52 (1) | 94.12 (1) | 92.85 (2) | 91.65 (1) | 91.58 |
| 112 | 107.97 (1) | 96.25 (1) | 97.62 (3) | 92.49 (1) | 92.05 |
| # | Level 4 | Level 5 | Level 6 | Level 7 | Extrap. |
|---|---|---|---|---|---|
| 113-115 | 108.08 (3) | 99.98 (3) | 99.45 (2) | 96.85 (3) | 96.57 |
| 116-117 | 109.2 (2) | 101.54 (2) | 102.01 (3) | 98.83 (2) | 98.61 |
| 118-120 | 115.67 (3) | 105.42 (3) | 103.18 (3) | 100.8 (3) | 100.22 |
| 121-123 | 116.6 (3) | 106.23 (3) | 107.44 (3) | 102.13 (3) | 101.76 |
| 124-126 | 123.37 (1) | 111.19 (3) | 112.34 (1) | 106.15 (3) | 105.63 |
| 127 | 123.85 (3) | 114.66 (1) | 115.74 (3) | 111.62 (1) | 111.43 |
| 128-130 | 126.96 (3) | 118.98 (3) | 116.93 (2) | 114.69 (3) | 114.07 |
| 131-132 | 129.26 (3) | 119.19 (3) | 117.02 (3) | 115.66 (2) | 115.60 |
| 133-135 | 130.66 (1) | 120.7 (2) | 118.45 (1) | 116.42 (3) | 116.31 |
| 136 | 130.99 (2) | 121.55 (1) | 119.17 (2) | 117.44 (1) | 116.94 |
| 137-138 | 134.71 (2) | 122.05 (2) | 122.73 (3) | 118.26 (2) | 118.11 |
| 139-141 | 135.25 (3) | 125.54 (3) | 124.19 (3) | 121.89 (3) | 121.56 |
| 142-144 | 143.13 (2) | 128.47 (3) | 127.58 (2) | 122.75 (3) | 122.23 |
| 145-146 | 143.36 (3) | 131.36 (2) | 130.9 (3) | 126.36 (2) | 125.81 |
| 147-149 | 157.02 (3) | 136.57 (3) | 139.19 (3) | 128.95 (3) | 128.35 |
| 150-152 | 159.47 (3) | 143.73 (3) | 144.08 (3) | 137.76 (3) | 137.22 |
| 153-155 | 163.81 (3) | 148.98 (3) | 144.56 (3) | 141.95 (3) | 141.07 |
| 156-158 | 166.1 (2) | 150.23 (3) | 148.93 (2) | 142.95 (3) | 141.90 |
| 159-160 | 169.13 (3) | 152.91 (2) | 149.55 (3) | 147.78 (2) | 147.48 |
| 161-163 | 169.22 (3) | 154.06 (3) | 150.52 (3) | 148.11 (3) | 147.54 |
| 164-166 | 170.58 (1) | 156.56 (3) | 151.88 (3) | 148.47 (3) | 147.52 |
| 167-169 | 176.02 (3) | 156.8 (3) | 154.91 (1) | 150.72 (3) | 150.28 |
| 170 | 176.65 (3) | 158.31 (1) | 162.94 (2) | 154.11 (1) | 154.00 |
| 171-172 | 185.4 (3) | 168.55 (3) | 163.57 (3) | 160.66 (2) | 159.67 |
| 173-175 | 186.44 (3) | 170.0 (2) | 165.48 (3) | 162.11 (3) | 161.49 |
| 176 | 192.37 (3) | 171.27 (3) | 166.23 (1) | 163.48 (1) | 162.28 |
| 177-179 | 192.78 (2) | 171.28 (3) | 166.93 (3) | 163.8 (3) | 163.21 |
| 180-182 | 193.05 (1) | 173.2 (1) | 167.66 (1) | 165.77 (3) | 165.61 |
| 183 | 196.51 (1) | 174.11 (1) | 169.02 (3) | 166.12 (1) | 165.36 |
| 184-186 | 198.24 (3) | 176.18 (3) | 173.13 (3) | 166.73 (3) | 165.61 |
| 187-189 | 198.73 (3) | 179.28 (3) | 174.77 (2) | 171.33 (3) | 170.44 |
| 190-191 | 201.58 (2) | 181.95 (2) | 180.61 (2) | 172.04 (2) | 170.53 |
| 192-193 | 217.16 (3) | 188.64 (2) | 185.07 (3) | 178.51 (2) | 177.62 |
| 194-196 | 221.2 (2) | 192.53 (3) | 187.06 (1) | 182.85 (3) | 181.87 |
| 197 | 225.43 (3) | 197.73 (3) | 189.17 (3) | 183.24 (1) | 181.14 |
| 198-200 | 226.05 (3) | 197.93 (1) | 190.58 (3) | 186.46 (3) | 185.27 |
| 201-203 | 226.14 (1) | 198.84 (3) | 194.71 (1) | 187.93 (3) | 186.83 |
| 204 | 235.29 (1) | 201.4 (1) | 196.43 (1) | 192.62 (1) | 192.46 |
| 205 | 241.4 (3) | 207.81 (1) | 199.31 (3) | 192.86 (1) | 190.92 |
| 206-208 | 244.76 (1) | 209.82 (3) | 207.73 (3) | 195.86 (3) | 194.13 |
| 209-211 | 248.45 (3) | 219.27 (2) | 208.8 (2) | 203.62 (3) | 201.47 |
| 212-213 | 249.14 (3) | 219.3 (3) | 212.48 (3) | 205.39 (2) | 203.43 |
| 214-216 | 249.79 (2) | 220.7 (3) | 213.16 (3) | 208.68 (3) | 207.65 |
| 217 | 254.04 (1) | 223.52 (1) | 213.48 (1) | 210.22 (1) | 208.48 |
| 218-220 | 255.38 (3) | 223.54 (3) | 214.29 (3) | 210.97 (3) | 209.41 |
| 221-223 | 257.26 (3) | 224.89 (3) | 216.61 (2) | 211.32 (3) | 209.67 |
| 224-225 | 259.79 (3) | 225.39 (2) | 220.61 (3) | 214.18 (2) | 213.43 |
| 226-228 | 261.87 (2) | 229.5 (3) | 221.13 (3) | 217.85 (3) | 216.76 |
| # | Level 4 | Level 5 | Level 6 | Level 7 | Extrap. |
|---|---|---|---|---|---|
| 229-231 | 264.86 (3) | 230.81 (3) | 222.42 (1) | 218.03 (3) | 216.94 |
| 232 | 272.86 (1) | 233.29 (1) | 230.83 (2) | 219.63 (1) | 218.49 |
| 233-234 | 274.65 (2) | 242.37 (2) | 232.62 (3) | 226.85 (2) | 224.58 |
| 235-237 | 281.02 (3) | 246.9 (3) | 235.64 (3) | 227.85 (3) | 225.39 |
| 238-240 | 294.81 (2) | 247.35 (3) | 237.53 (2) | 232.05 (3) | 231.18 |
| 241-242 | 295.55 (3) | 251.22 (2) | 242.32 (3) | 233.58 (2) | 231.71 |
| 243-245 | 303.96 (3) | 257.87 (3) | 246.39 (3) | 236.83 (3) | 234.13 |
| 246-248 | 304.15 (3) | 261.82 (3) | 251.53 (3) | 241.68 (3) | 238.64 |
| 249-251 | 305.81 (3) | 266.33 (3) | 257.87 (3) | 246.83 (3) | 243.71 |
| 252-254 | 310.2 (1) | 270.84 (3) | 262.7 (3) | 253.11 (3) | 250.95 |
| 255-257 | 315.55 (3) | 276.12 (3) | 267.68 (2) | 258.49 (3) | 256.18 |
| 258-259 | 317.26 (2) | 280.48 (2) | 268.27 (3) | 263.42 (2) | 261.59 |
| 260-262 | 321.5 (3) | 281.96 (1) | 270.92 (3) | 264.13 (3) | 261.71 |
| 263-265 | 326.86 (3) | 282.89 (3) | 273.58 (1) | 265.73 (3) | 264.53 |
| 266-268 | 327.19 (3) | 285.66 (3) | 273.72 (3) | 268.41 (3) | 266.90 |
| 269-270 | 336.55 (3) | 287.59 (3) | 274.02 (2) | 269.02 (2) | 267.47 |
| 271-273 | 341.09 (2) | 290.22 (2) | 274.57 (3) | 270.65 (3) | 268.31 |
| 274 | 344.79 (3) | 292.16 (3) | 283.35 (1) | 271.51 (1) | 268.74 |
| 275 | 345.82 (3) | 302.76 (3) | 287.54 (3) | 276.56 (1) | 273.96 |
| 276-278 | 353.11 (1) | 303.45 (1) | 288.09 (3) | 282.66 (3) | 279.81 |
| 279-281 | 360.97 (3) | 303.78 (3) | 295.23 (3) | 284.2 (3) | 282.55 |
| 282-284 | 360.99 (2) | 311.17 (3) | 295.96 (1) | 289.99 (3) | 287.58 |
| 285 | 363.24 (3) | 312.4 (1) | 296.93 (3) | 291.09 (1) | 288.59 |
| 286-287 | 366.82 (1) | 315.92 (3) | 297.99 (2) | 291.65 (2) | 287.48 |
| 288-290 | 370.24 (3) | 317.18 (2) | 301.67 (2) | 291.78 (3) | 288.43 |
| 291-292 | 371.34 (3) | 322.13 (2) | 304.11 (3) | 295.19 (2) | 291.42 |
| 293-295 | 376.36 (2) | 325.33 (3) | 310.04 (3) | 297.44 (3) | 294.41 |
| 296 | 382.34 (1) | 328.38 (3) | 310.61 (3) | 302.76 (1) | 299.38 |
| 297-299 | 383.35 (3) | 330.1 (3) | 311.07 (1) | 303.7 (3) | 300.61 |
| 300-302 | 388.63 (3) | 335.19 (1) | 315.3 (1) | 305.67 (3) | 302.57 |
| 303 | 392.67 (3) | 336.88 (1) | 316.4 (2) | 308.67 (1) | 303.89 |
| 304-305 | 394.0 (3) | 338.1 (2) | 321.72 (3) | 309.82 (2) | 305.24 |
| 306-308 | 395.84 (2) | 341.85 (1) | 325.79 (1) | 316.28 (3) | 312.01 |
| 309 | 396.62 (1) | 341.87 (3) | 328.41 (3) | 321.57 (1) | 319.54 |
| 310-311 | 416.57 (3) | 346.51 (3) | 328.87 (2) | 321.95 (2) | 318.15 |
| 312-314 | 416.99 (3) | 351.96 (2) | 335.88 (3) | 322.38 (3) | 320.85 |
| 315-317 | 422.45 (2) | 360.9 (3) | 338.16 (3) | 328.08 (3) | 324.14 |
| 318-320 | 424.11 (1) | 361.41 (3) | 341.41 (3) | 330.71 (3) | 325.64 |
| 321-323 | 435.62 (2) | 366.5 (2) | 345.84 (1) | 333.3 (3) | 328.33 |
| 324 | 435.77 (3) | 367.16 (3) | 347.11 (2) | 336.65 (1) | 333.41 |
| 325 | 445.38 (3) | 374.47 (1) | 350.8 (1) | 340.76 (1) | 338.34 |
| 326-327 | 447.57 (3) | 376.51 (3) | 352.45 (3) | 341.27 (2) | 335.74 |
| 328-330 | 449.1 (1) | 379.87 (3) | 357.95 (3) | 345.02 (3) | 340.66 |
| 331-333 | 450.18 (1) | 382.46 (3) | 360.47 (1) | 351.64 (3) | 348.31 |
| 334 | 452.25 (3) | 384.72 (1) | 360.5 (3) | 352.39 (1) | 349.10 |
| 335-337 | 465.24 (1) | 387.13 (1) | 364.91 (3) | 354.74 (3) | 351.62 |
| 338-340 | 468.7 (3) | 396.03 (3) | 366.64 (3) | 355.08 (3) | 349.72 |
| 341-343 | 469.18 (3) | 399.26 (2) | 372.59 (2) | 357.03 (3) | 352.18 |
| # | Level 4 | Level 5 | Level 6 | Level 7 | Extrap. |
|---|---|---|---|---|---|
| 344-345 | 475.76 (2) | 399.32 (3) | 374.47 (3) | 364.75 (2) | 359.96 |
| 346-348 | 481.12 (3) | 400.07 (3) | 376.35 (3) | 366.73 (3) | 363.02 |
| 349-351 | 483.09 (2) | 405.95 (3) | 379.65 (3) | 367.49 (3) | 361.10 |
| 352-354 | 489.5 (3) | 407.43 (1) | 379.67 (1) | 370.32 (3) | 365.52 |
| 355 | 493.52 (1) | 410.16 (3) | 390.25 (2) | 372.37 (1) | 367.35 |
| 356-357 | 499.79 (3) | 422.95 (2) | 393.67 (3) | 380.68 (2) | 374.58 |
| 358-360 | 509.36 (3) | 424.44 (3) | 395.82 (3) | 383.73 (3) | 377.65 |
| 361-363 | 510.71 (2) | 430.47 (3) | 396.96 (3) | 384.32 (3) | 378.77 |
| 364-366 | 511.2 (3) | 433.33 (3) | 404.26 (3) | 386.75 (3) | 381.10 |
| 367-369 | 521.12 (3) | 435.62 (2) | 405.4 (2) | 392.64 (3) | 385.94 |
| 370-371 | 521.55 (2) | 440.62 (2) | 407.96 (2) | 395.13 (2) | 388.87 |
| 372-373 | 523.18 (3) | 441.97 (3) | 414.76 (1) | 398.75 (2) | 391.79 |
| 374 | 526.68 (1) | 444.89 (1) | 416.44 (3) | 404.86 (1) | 398.66 |
| 375-377 | 533.4 (3) | 448.71 (3) | 418.93 (3) | 405.36 (3) | 398.59 |
| 378-380 | 536.16 (3) | 458.1 (3) | 425.65 (3) | 407.83 (3) | 402.84 |
| 381-383 | 539.51 (3) | 459.87 (1) | 428.42 (3) | 414.1 (3) | 408.64 |
| 384-386 | 541.85 (1) | 462.15 (3) | 430.28 (1) | 419.44 (3) | 412.61 |
| 387-388 | 547.75 (3) | 463.65 (3) | 432.0 (2) | 421.21 (2) | 413.75 |
| 389 | 549.18 (2) | 467.98 (3) | 433.65 (3) | 422.46 (1) | 415.84 |
| 390-392 | 553.87 (3) | 470.79 (2) | 437.44 (3) | 423.35 (3) | 416.14 |
| 393-395 | 559.46 (3) | 472.57 (3) | 444.03 (3) | 426.76 (3) | 419.63 |
| 396-398 | 567.23 (3) | 480.13 (3) | 446.33 (3) | 432.79 (3) | 425.99 |
| 399-401 | 574.44 (1) | 480.56 (1) | 449.68 (1) | 434.37 (3) | 427.32 |
| 402-404 | 577.38 (3) | 483.45 (3) | 452.66 (2) | 439.61 (3) | 433.19 |
| 405-407 | 588.63 (2) | 491.07 (3) | 453.22 (3) | 443.02 (3) | 435.24 |
| 408-410 | 590.79 (3) | 493.44 (2) | 456.9 (3) | 445.54 (3) | 437.45 |
| 411-413 | 594.91 (3) | 494.94 (3) | 459.47 (3) | 447.86 (3) | 442.29 |
| 414-415 | 606.12 (1) | 495.35 (2) | 460.8 (2) | 450.21 (2) | 444.31 |
| 416-418 | 609.45 (3) | 497.8 (3) | 475.84 (1) | 461.66 (3) | 454.48 |
| 419-421 | 612.36 (2) | 513.83 (1) | 476.74 (3) | 463.69 (3) | 455.97 |
| 422 | 614.61 (3) | 515.77 (3) | 476.82 (3) | 463.91 (1) | 453.68 |
| 423 | 623.89 (2) | 522.22 (1) | 478.85 (1) | 464.65 (1) | 454.90 |
| 424-425 | 641.18 (3) | 524.61 (3) | 481.9 (2) | 465.9 (2) | 458.41 |
| 426-427 | 641.7 (3) | 530.62 (2) | 488.24 (2) | 474.77 (2) | 466.83 |
| 428-430 | 650.4 (1) | 536.23 (2) | 488.48 (3) | 474.9 (3) | 465.56 |
| 431 | 653.88 (3) | 536.69 (3) | 493.48 (1) | 477.32 (1) | 470.41 |
| 432-434 | 657.49 (1) | 544.86 (3) | 498.91 (3) | 486.25 (3) | 476.44 |
| 435-437 | 659.13 (2) | 545.65 (3) | 502.99 (3) | 488.8 (3) | 482.00 |
| 438-440 | 676.14 (3) | 547.0 (1) | 506.87 (3) | 494.09 (3) | 486.49 |
| 441-443 | 679.73 (2) | 551.04 (3) | 508.39 (2) | 494.58 (3) | 487.07 |
| 444-445 | 681.0 (3) | 552.17 (2) | 508.86 (3) | 495.16 (2) | 486.99 |
| 446-448 | 698.25 (3) | 557.04 (3) | 510.66 (3) | 496.13 (3) | 487.99 |
| 449-451 | 702.54 (1) | 559.13 (3) | 512.14 (3) | 498.7 (3) | 488.65 |
| 452 | 702.84 (3) | 563.66 (3) | 521.19 (1) | 504.46 (1) | 498.38 |
| 453-454 | 708.14 (1) | 566.38 (1) | 523.5 (2) | 509.05 (2) | 502.00 |
| 455-457 | 715.85 (3) | 570.25 (2) | 527.58 (1) | 515.94 (3) | 507.25 |
| 458 | 719.75 (3) | 577.52 (3) | 529.2 (3) | 517.35 (1) | 509.18 |
| 459-461 | 722.16 (3) | 578.34 (1) | 536.85 (3) | 521.81 (3) | 514.89 |
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A Convex Surface with Fractal Curvature
Iancu Dima
Department of Mathematics, Ithaca College, 953 Danby Road, Ithaca, NY 14850, USA
,
Rachel Popp
The Graduate Center, CUNY, 180 Queens Gate, 365 Fifth Avenue, New York, NY 10016, USA
,
Robert S. Strichartz
Department of Mathematics, Cornell University, Malott Hall, Ithaca, NY 14853, USA
and
Samuel C. Wiese
Department of Mathematics, Universität Leipzig, Augustusplatz 10, 04109 Leipzig, Germany
(Date: July 8, 2019.)
Abstract.
We construct a surface that is obtained from the octahedron by pushing out 4 of the faces so that the curvature is supported in a copy of the Sierpinski gasket in each of them, and is essentially the self similar measure on SG. We then compute the bottom of the spectrum of the associated Laplacian using the finite element method on polyhedral approximations of our surface, and speculate on the behavior of the entire spectrum.
Key words and phrases:
Laplacian, eigenvalues, eigenfunctions, polyhedra, curvature
2000 Mathematics Subject Classification:
35P05
ID was supported by the National Science Foundation through the Research Experience for Undergraduates (REU) Program, Grant DMS-1156350.
RP was supported in part by the National Science Foundation, Grant DMS-1162045
RSS was supported in part by the National Science Foundation, Grant DMS-1162045.
SCW was supported by the Foundation of German Business (SDW)
1. Introduction.
A convex surface in 3-space may be regarded as an Alexandrov space, and so it has a well-defined curvature (as a measure) and a well-defined Laplacian. In this paper we construct an example of such a surface whose curvature is a fractal measure related to the Sierpinski Gasket (SG). For an introduction see the first chapter of [4]. Our construction produces the surface as a limit of convex polyhedra . The curvature of each is a discrete measure supported on its vertices, and these discrete measures will converge to the fractal measure on . Each has a Laplacian, and the limit is the Laplacian on . We compute the spectra of each of the Laplacians on and thereby obtain approximations to the spectrum of the Laplacian on .
The first approximating polyhedron (with 6 vertices) is just a regular octahedron. Half of the faces will remain flat in the construction, but with folding. Of course folding does not introduce curvature. The other four alternating faces will have a SG inscribed in them, and then the “upside down triangles” in the SG picture will be pushed out, step-by-step, as increases. When just a single central triangle will be pushed out of each face. The result will be a polyhedron with 18 vertices, the original 6 together with 3 new vertices along the sides of the 4 faces (The flat faces will fold to accomodate the eruptions in the other curved faces). If we choose the angles correctly, the original vertices will have their curvature reduced from to , while the new vertices will have curvature . Note that by the Gauss-Bonet Theorem the total curvature measure on will then be the sum of the discrete measures on the 4 faces with weight on each of the 6 vertices on the face (the original vertices lie in exactly 2 of these faces). Thus, we have the first discrete approximation to a standard self-similar measure with total mass on the SG pictured on these faces.
The same idea of pushing out successively smaller triangles to create from produces polyhedra whose curvature gives better and better approximations to the self-similar SG measures. The details are given in Section 2. In Section 3 we describe the finite element method we use to approximate the spectra of each . In Section 4 we discuss the numerical results of these computations.
2. Geometry
2.1. Construction of angles
Each of the four curved faces in is subdivided into triangles of the SG construction labeled by words of length , together with “upside down” equilateral triangles that will be flat. So let denote the triangle associated with the word , namely where is the full face, and let denote the angles at the vertices for . We will choose
[TABLE]
where are integers to be described below.
Theorem 2.1**.**
Suppose we can find values of that are symmetric under permutations of and satisfy the following conditions:
[TABLE]
Then the curvature of is at every vertex.
Proof.
Condition (2.2) shows that the angle sum for each triangle is . Now consider a vertex of the original octahedron. Then it has two angles of each from the flat faces and two angles of at the curved faces by (2.4) and its symmetric versions. Thus the total angle sum is and so the curvature there is . Next consider a vertex interior to one of the faces. Then it has one angle from the equilateral triangle that touches at that point, and one angle from the edge of a triangle, and then by (2.3) the remaining two angles add to , so again the curvature is . Here we use the observation that the last two angles have the form and . ∎
Our goal is to construct the constant satisfying the conditions of the theorem. By the symmetry it suffices to do this for words with the initial , so . We show a solution for and , filling in the -value in each angle in base-3 notation (Fig. 1).
The conditions are easily verified by inspection, and since and , condition (2.3) holds at the bottom vertices for (similarly since for ) when we join up with the symmetric lower thirds of the face. We observe that the -values on each of the two lower thirds for are identical to the whole set of values. This will be true in general when we describe an inductive construction from level to level , so we only need to describe the -values on the top third. We also note that all -values are divisible by , and the only base-3 digits are [math] and .
If we we examine the values of along the right boundary edge, that is each or , we see a decrease from top to bottom. For we have , , , and . If we interpret as a base-2 number then these increase [math], , , and if we consider these decrease , , , . Note that if we add a final [math] and interpret as a base-3 number we get the correct -values. In our first algorithm we extend this to the case of general .
Algorithm 2.2**.**
For any integer base-2 written as
[TABLE]
This is rereading the base- representation as base- and adding the digit [math] at the end. Then set
[TABLE]
if is a word of length consisting of digits [math] or only, and
[TABLE]
Next we describe the inductive algorithm that computes -values on level given the values on level .
Algorithm 2.3**.**
By symmetry we may assume , so with . We consider three cases:
- (1)
consists of digits [math] and only. Then we apply Algorithm 2.2 to set . We then set
[TABLE]
so that (2.3) holds at the vertex, and
[TABLE]
so that (2.2) holds on the triangle. Similarly, if consists of digits [math] and only, we define by symmetry. 2. (2)
If , then set
[TABLE]
and similarly . Note that this gives that bottom two thirds at level equal to the whole of level . 3. (3)
If and both digits and occur in , then set
[TABLE]
Theorem 2.4**.**
The -values constructed by Algorithms 2.2 and 2.3 satisfy the conditions of Theorem 2.1.
Proof.
Since we know this holds for we may proceed by induction. Every triangle at level in cases and has identical -values to a triangle at level , so (2.2) holds. For triangles in case we obtain (2.2) by adding (2.8), (2.9), and (2.10). Also (2.4) holds by Algorithm 2.2.
It remains to verify (2.3). For vertices in the interior of the lower thirds this follows by the induction hypothesis. For the vertex where they intersect this is the condition , but it is easy to see by induction that both values are zero. For vertices along the left and right edges (2.3) is an immediate consequence of (2.8) and (2.9).
If is a triangle along the right edge we will verify (2.3) at the bottom left vertex. Here we consider two separate cases. If ends in the digit then , and so (2.10) means . If we write then by local symmetry (easily established by induction) so (2.3) holds. On the other hand, if ends in the digit , then (2.6) implies while the -value on the adjacent is by local symmetry. Finally, for vertices further in the interior, condition (2.3) follows by induction. ∎
2.2. Computing the sidelengths
After we compute the -values inductively, we get all the angles from (2.1) and set up a system of equations using the law of sines, symmetry, and norming the distance between the points associated to the angles and to be , where , . The system will be uniquely solved for all the sidelengths. We also compute the area of every subtriangle that will be needed for the FEM.
2.3. Assembling the polyhedra
We use the sidelengths and angles to display the net of a level polyhedron, that will be triangulated using [3]. Figure 2 shows the low-level triangulation of one SG face.
To put the 4 SG faces and 4 flat faces together, we will need make identifications ”inside” the SG-faces to stich up the small slits and also identify the sides of SG-faces and flat faces. Figure 3 shows the complete mesh with and without identifications, where two vertices are identified when they are connected by a blue line. This is done in a way that the 22 left faces are rotated to the right and put ”on top” of the 22 faces on the right to form the polyhedron.
We numerically fold up the net using [1] to display the polyhedra in 3D. Figure 4 shows level , where for only the edges belonging to the flat faces are shown. Note that for folding occurs on the inner flat faces too, but we won’t show them since they don’t contribute to the curvature. Although there is zero curvature on the non-SG (flat) faces, the vertices added to the SG-faces cause the non-SG faces to fold. This creates trapazoids which must be triangulated for the FEM.
The solid has 3 symmetry axes through two opposing tops (by ) and 4 symmetry axes through the middles of two opposing faces (one flat, one curved) by and . It’s symmetry group is with order 24. We represent it as
[TABLE]
and write the generators as matrices in in a coordinate systems where the center of the solid is at the origin and the coordinate axes run through the vertices where 4 faces meet. Choose
[TABLE]
then corresponds to a reflection through the z-axis and fixes an opposing pair of vertices and cyclically permutes the others. The group has 2 irreducible representations of degree 1, 1 of degree 2, and 2 of degree 3.
3. FEM
To implement the finite element method, we need a Gram matrix and an energy matrix which are sparse matrices filled with contributions from areas of triangles for each of the vertices. We compute the beginning of the spectrum for different levels and increasing subdivisions.
3.1. Eigenvalues
We approximate the spectrum of the Laplacian on the surface by extrapolating over the spectrum of lower level polyhedra. The Laplacian on the surface is just the usual two dimensional Laplacian on the planar realization, such that values and normal derivatives along the identified edges match. By the spectrum we mean a study of both the eigenvalues and and eigenfunctions satisfying
[TABLE]
It is known that the eigenvalues form an increasing sequence tending to infinity, and satisfying the Weyl asymptotic law
[TABLE]
where is the area of the surface. In Figures 5-7 we show the graphs of the eigenvalue counting function , the difference
[TABLE]
and the average of the difference
[TABLE]
based on our numerical computations.
The multiplicities at the beginning of the spectrum of level are shown in Figure 8. They are distributed as follows: of eigenvalues have multiplicity 1, have multiplicity 2 and have multiplicity 3. This corresponds to the squares of dimensions of irreducible representations (1, 2 and 3) of the symmetry group .
We notice a pattern of multiplicities around consecutive multiplicity 1 occurances (3-3-1-1-3) starting at positions 9, 38, 105, 198… and found that for the corresponding pair of eigenfunctions the individual faces are either invariant under all symmetries or just under one (skew)symmetric reflection.
We extrapolate exponentially using levels 4 to 7. Figure 9 shows the beginning of the spectrum for the first levels.
Tables 1-4 show the beginning of the spectrum for level 1 to 7 without subdivisions. The numbering in the left column refers to Level 7. Note that below the double-horizontal line on the next page the multiplicities of level and do not match any more.
3.2. Eigenfunctions
We display the eigenfunctions as described above in Fig. 3, and study the symmetries at the two selected tops ( and ) locally, and their relation to each other. The eigenfunctions may be sorted into symmetry types according to the irreducible representations of the symmetry group of the polyhedron. We will show examples of the eigenfunction symmetry types for level and . Note, that the types of symmetries that occur in the eigenfunctions correspond to the irreducible representations of the symmetry group of the polyhedron, , which has two 1-dim. representations, one 2-dim. representation and two 3-dim. representations.
There are two types of eigenfunctions of multiplicity 1, since there are two 1-dim. irreducible representations of . One type is symmetrical with regard to both axes (Fig. 10), the other has horizontal symmetry and is skew-symmetrical vertically or the other way around (Fig. 11). This is equivalent to saying that one type has rotational symmetry and the other has rotational skew-symmetry by with regard to the origin.
There is one type of pairs of eigenfunctions from a 2-dimensional eigenspace. Both eigenfunctions from such a pair have horizontal symmetry. Individually, they seem to have rotational symmetry by between the tops. One eigenfunction in the 2-dimensional eigenspace is obtained from the other by rotation of and inverting on half. Two pairs for level and are shown in Fig. 12 and Fig. 13.
There are two types of eigenfunctions from an eigenspace of dimension 3. Eigenfunctions from the first type seem to have rotational symmetry by between the tops. One of them has horizontal symmetry, and vertical symmetry with respect to the vertical axes through the tops. The other two have horizontal skew-symmetry through the horizontal axes of one of the tops and vertical symmetry through the other (Fig. 14). For the second type: one has horizontal and vertical skew-symmetry, one has vertical skew-symmetry and rotational skew-symmetry around the tops and the last one has vertical symmetry and rotational skew-symmetry around the tops (Fig. 15).
More data and eigenfunctions can be found at [6].
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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- 2[2] T. Shioya, Geometric Analysis on Alexandrov Spaces , Sugaku Expositions, 24 (2011), 145 – 167
- 3[3] J. R. Shewchuk, Delaunay Refinement Algorithms for Triangular Mesh Generation , Comp Geom Theor Appl, 22 (2002), 21 – 74
- 4[4] R. S. Strichartz, Differential equations on fractals: A tutorial , Princeton University Press, Princeton, New Jersey (2006)
- 5[5] R. S. Strichartz, Average error for spectral asymptotics on surfaces , Commun Pure Appl Anal, 15 (2016), 9 – 39
- 6[6] R. S. Strichartz, S. C. Wiese, A convex surface with fractal curvature , http://pi.math.cornell.edu/~sw 972/fractal_curvature.html (updated July 8, 2019)
- 7[7] S. Winter, M. Zähle, Fractal curvature measures of self-similar sets , Adv Geom, 13(2) (2010), 229 – 244