On this page several configurations of a traingulation consisting of 18 triangles. It exhibits various degencrecaise that cannot be explained by the existing theory.
The most symmetric configuration consists of a regular hexagon inside a regular hexagon with all the triangles being congruent and equilateral, as illustrated in the nearby figure.
For each configuration, the following table contains the following information:
Conf. | Desc. | binary | V 0 | V 1 | V 2 | V 3 | V 4 | V 5 | V 6 | CT Vtcs |
1 | Description | Binary | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 7 |
2 | Description | Binary | 3 | 3 | 3 | 4 | 3 | 3 | 3 | 6 |
3 | Description | Binary | 3 | 3 | 3 | 4 | 4 | 3 | 3 | 5 |
4 | Description | Binary | 3 | 3 | 4 | 5 | 4 | 3 | 3 | 4 |
5 | Description | Binary | 3 | 4 | 5 | 5 | 4 | 3 | 3 | 3 |
6 | Description | Binary | 3 | 4 | 5 | 5 | 4 | 3 | 4 | 2 |
7 | Description | Binary | 3 | 4 | 5 | 5 | 4 | 4 | 5 | 1 |
8 | Description | Binary | 3 | 4 | 5 | 5 | 5 | 5 | 5 | 1 |
9 | Description | Binary | 3 | 5 | 5 | 5 | 5 | 5 | 5 | 1 |
10 | Description | Binary | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 0 |
11 | Description | Binary | 6 | 5 | 5 | 5 | 5 | 5 | 5 | 0 |
The following table gives dimension of various spline spaces on the above described configurations. For each case, the true dimension is computed and compared to Schumaker's lower bound (for that particular configuration). When the dimension and the lower bound agree, their common value is listed. If they disagree, the dimension is listed, followed by a slash, and the discrepancy. Thus, for example, the dimension on the completelysymmetric split (configuration C1) when r=1 and d=2 is 12 and the lower bound is 9.
m and n are the dimension of the linear system that must be analyzed. They are included to illustrate the capacity of the MDS code.
For clarity, r and d are listed on a yellow background, and the matrix dimensions on a purple background. A cyan background indicates that the dimension of the spline space actually equals the dimension of the space of polynomials of degree d. Thus in that case all splines are in fact polynomials. A green background indicates that there are nontrivial splines, and the true dimension exceeds Schumaker's lower bound.
At present there is no theory that explains the cases highlighted in green.
The dimension of spline spaces completely understood (in the
sense that we know a minimal determining set) in the case
that d>3r+1. However, for the range of r
considered here, discrepancies only occur for d<2r+1.
It is an open question whether discrepancies occur for
larger of values of d, either on the Morgan-Scott
split for values of r not considered here, or on
other configurations.
r | d | C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | C10 | C11 | m | n |
1 | 2 | 12 / 3 | 11 / 2 | 10 / 1 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 48 | 43 |
1 | 3 | 33 | 33 | 33 | 33 | 33 | 33 | 33 | 33 | 33 | 33 | 33 | 72 | 91 |
2 | 3 | 16 / 3 | 15 / 3 | 14 / 3 | 12 / 2 | 11 / 2 | 10 / 2 | 10 / 3 | 10 / 3 | 10 / 3 | 10 / 4 | 10 / 4 | 120 | 91 |
2 | 4 | 34 / 3 | 32 / 2 | 30 / 1 | 28 | 27 | 26 | 25 | 25 | 25 | 24 | 24 | 168 | 157 |
2 | 5 | 67 | 66 | 65 | 64 | 63 | 62 | 61 | 61 | 61 | 60 | 60 | 216 | 241 |
3 | 4 | 21 / 3 | 20 / 3 | 19 / 3 | 17 / 3 | 16 / 4 | 15 / 4 | 15 / 6 | 15 / 8 | 15 / 9 | 15 / 11 | 15 / 11 | 216 | 157 |
3 | 5 | 39 / 9 | 36 / 7 | 33 / 5 | 29 / 3 | 26 / 2 | 24 / 1 | 23 / 2 | 21 / 2 | 21 / 3 | 21 / 5 | 21 / 5 | 288 | 241 |
3 | 6 | 64 / 4 | 61 / 2 | 59 / 1 | 56 | 54 | 53 | 51 | 49 | 48 | 46 | 46 | 360 | 343 |
3 | 7 | 108 | 107 | 106 | 104 | 102 | 101 | 99 | 97 | 96 | 94 | 94 | 432 | 463 |
4 | 5 | 27 / 3 | 26 / 3 | 25 / 3 | 23 / 3 | 22 / 4 | 21 / 4 | 21 / 6 | 21 / 8 | 21 / 9 | 21 / 11 | 21 / 12 | 336 | 241 |
4 | 6 | 46 / 9 | 43 / 8 | 40 / 7 | 35 / 5 | 31 / 4 | 28 / 3 | 28 / 6 | 28 / 8 | 28 / 9 | 28 / 12 | 28 / 13 | 432 | 343 |
4 | 7 | 72 / 11 | 67 / 8 | 62 / 5 | 55 / 1 | 51 | 49 | 46 | 44 | 43 | 40 | 39 | 528 | 463 |
4 | 8 | 107 / 4 | 103 / 2 | 100 / 1 | 96 | 93 | 91 | 88 | 86 | 85 | 82 | 81 | 624 | 601 |
4 | 9 | 163 | 161 | 159 | 156 | 153 | 151 | 148 | 146 | 145 | 142 | 141 | 720 | 757 |
5 | 6 | 34 / 3 | 33 / 3 | 32 / 3 | 30 / 3 | 29 / 4 | 28 / 4 | 28 / 6 | 28 / 8 | 28 / 9 | 28 / 11 | 28 / 12 | 480 | 343 |
5 | 7 | 54 / 9 | 51 / 9 | 48 / 9 | 42 / 7 | 39 / 8 | 36 / 8 | 36 / 12 | 36 / 14 | 36 / 15 | 36 / 19 | 36 / 20 | 600 | 463 |
5 | 8 | 81 / 18 | 75 / 15 | 69 / 12 | 60 / 7 | 53 / 4 | 48 / 2 | 47 / 5 | 45 / 5 | 45 / 6 | 45 / 10 | 45 / 11 | 720 | 601 |
5 | 9 | 115 / 16 | 107 / 11 | 99 / 6 | 89 | 85 | 82 | 78 | 76 | 75 | 71 | 70 | 840 | 757 |
5 | 10 | 159 / 6 | 153 / 3 | 148 / 1 | 143 | 139 | 136 | 132 | 130 | 129 | 125 | 124 | 960 | 931 |
5 | 11 | 225 | 222 | 219 | 215 | 211 | 208 | 204 | 202 | 201 | 197 | 196 | 1080 | 1123 |
6 | 7 | 42 / 3 | 41 / 3 | 40 / 3 | 38 / 3 | 37 / 4 | 36 / 4 | 36 / 6 | 36 / 8 | 36 / 9 | 36 / 11 | 36 / 12 | 648 | 463 |
6 | 8 | 63 / 9 | 60 / 9 | 57 / 9 | 51 / 8 | 48 / 10 | 45 / 10 | 45 / 15 | 45 / 19 | 45 / 21 | 45 / 26 | 45 / 27 | 792 | 601 |
6 | 9 | 91 / 18 | 85 / 16 | 79 / 14 | 69 / 10 | 61 / 8 | 55 / 6 | 55 / 12 | 55 / 16 | 55 / 18 | 55 / 24 | 55 / 25 | 936 | 757 |
6 | 10 | 126 / 23 | 117 / 18 | 108 / 13 | 95 / 6 | 84 / 1 | 79 | 75 / 2 | 69 | 67 | 66 / 5 | 66 / 6 | 1080 | 931 |
6 | 11 | 169 / 18 | 158 / 11 | 147 / 4 | 137 | 131 | 127 | 121 | 117 | 115 | 109 | 108 | 1224 | 1123 |
6 | 12 | 222 / 5 | 215 / 2 | 210 / 1 | 203 | 197 | 193 | 187 | 183 | 181 | 175 | 174 | 1368 | 1333 |
6 | 13 | 301 | 297 | 293 | 287 | 281 | 277 | 271 | 267 | 265 | 259 | 258 | 1512 | 1561 |
7 | 8 | 51 / 3 | 50 / 3 | 49 / 3 | 47 / 3 | 46 / 4 | 45 / 4 | 45 / 6 | 45 / 8 | 45 / 9 | 45 / 11 | 45 / 12 | 840 | 601 |
7 | 9 | 73 / 9 | 70 / 9 | 67 / 9 | 61 / 9 | 58 / 12 | 55 / 12 | 55 / 18 | 55 / 24 | 55 / 27 | 55 / 33 | 55 / 34 | 1008 | 757 |
7 | 10 | 102 / 18 | 96 / 17 | 90 / 16 | 79 / 13 | 72 / 14 | 66 / 13 | 66 / 21 | 66 / 27 | 66 / 30 | 66 / 38 | 66 / 39 | 1176 | 931 |
7 | 11 | 138 / 30 | 128 / 25 | 118 / 20 | 103 / 13 | 90 / 8 | 81 / 4 | 80 / 11 | 78 / 15 | 78 / 18 | 78 / 26 | 78 / 27 | 1344 | 1123 |
7 | 12 | 181 / 31 | 168 / 23 | 155 / 15 | 137 / 5 | 124 | 119 | 111 | 105 | 102 | 94 | 93 | 1512 | 1333 |
7 | 13 | 231 / 21 | 217 / 12 | 204 / 4 | 192 | 184 | 179 | 171 | 165 | 162 | 154 | 153 | 1680 | 1561 |
7 | 14 | 293 / 5 | 285 / 2 | 279 / 1 | 270 | 262 | 257 | 249 | 243 | 240 | 232 | 231 | 1848 | 1807 |
7 | 15 | 384 | 379 | 374 | 366 | 358 | 353 | 345 | 339 | 336 | 328 | 327 | 2016 | 2071 |
8 | 9 | 61 / 3 | 60 / 3 | 59 / 3 | 57 / 3 | 56 / 4 | 55 / 4 | 55 / 6 | 55 / 8 | 55 / 9 | 55 / 11 | 55 / 12 | 1056 | 757 |
8 | 10 | 84 / 9 | 81 / 9 | 78 / 9 | 72 / 9 | 69 / 12 | 66 / 12 | 66 / 18 | 66 / 24 | 66 / 27 | 66 / 33 | 66 / 35 | 1248 | 931 |
8 | 11 | 114 / 18 | 108 / 18 | 102 / 18 | 90 / 15 | 84 / 18 | 78 / 18 | 78 / 27 | 78 / 33 | 78 / 36 | 78 / 45 | 78 / 47 | 1440 | 1123 |
8 | 12 | 151 / 30 | 141 / 27 | 131 / 24 | 114 / 17 | 101 / 14 | 91 / 11 | 91 / 21 | 91 / 27 | 91 / 30 | 91 / 40 | 91 / 42 | 1632 | 1333 |
8 | 13 | 195 / 38 | 181 / 31 | 167 / 24 | 146 / 13 | 128 / 5 | 118 / 2 | 114 / 8 | 106 / 6 | 105 / 8 | 105 / 18 | 105 / 20 | 1824 | 1561 |
8 | 14 | 246 / 35 | 229 / 25 | 212 / 15 | 189 / 2 | 177 | 170 | 160 | 154 | 151 | 141 | 139 | 2016 | 1807 |
8 | 15 | 305 / 22 | 288 / 12 | 273 / 4 | 259 | 249 | 242 | 232 | 226 | 223 | 213 | 211 | 2208 | 2071 |
8 | 16 | 378 / 5 | 368 / 2 | 360 / 1 | 349 | 339 | 332 | 322 | 316 | 313 | 303 | 301 | 2400 | 2353 |
8 | 17 | 481 | 474 | 467 | 457 | 447 | 440 | 430 | 424 | 421 | 411 | 409 | 2592 | 2653 |
9 | 10 | 72 / 3 | 71 / 3 | 70 / 3 | 68 / 3 | 67 / 4 | 66 / 4 | 66 / 6 | 66 / 8 | 66 / 9 | 66 / 11 | 66 / 12 | 1296 | 931 |
9 | 11 | 96 / 9 | 93 / 9 | 90 / 9 | 84 / 9 | 81 / 12 | 78 / 12 | 78 / 18 | 78 / 24 | 78 / 27 | 78 / 33 | 78 / 36 | 1512 | 1123 |
9 | 12 | 127 / 18 | 121 / 18 | 115 / 18 | 103 / 16 | 97 / 20 | 91 / 20 | 91 / 30 | 91 / 38 | 91 / 42 | 91 / 52 | 91 / 55 | 1728 | 1333 |
9 | 13 | 165 / 30 | 155 / 28 | 145 / 26 | 127 / 20 | 115 / 20 | 105 / 18 | 105 / 30 | 105 / 38 | 105 / 42 | 105 / 54 | 105 / 57 | 1944 | 1561 |
9 | 14 | 210 / 45 | 195 / 38 | 180 / 31 | 157 / 20 | 137 / 12 | 123 / 6 | 122 / 17 | 120 / 23 | 120 / 27 | 120 / 39 | 120 / 42 | 2160 | 1807 |
9 | 15 | 262 / 49 | 243 / 38 | 224 / 27 | 197 / 12 | 175 / 2 | 165 | 156 / 3 | 145 | 141 | 136 / 7 | 136 / 10 | 2376 | 2071 |
9 | 16 | 321 / 42 | 299 / 28 | 277 / 14 | 251 | 239 | 231 | 219 | 211 | 207 | 195 | 192 | 2592 | 2353 |
9 | 17 | 388 / 25 | 368 / 13 | 351 / 4 | 335 | 323 | 315 | 303 | 295 | 291 | 279 | 276 | 2808 | 2653 |
9 | 18 | 470 / 5 | 459 / 2 | 450 / 1 | 437 | 425 | 417 | 405 | 397 | 393 | 381 | 378 | 3024 | 2971 |
9 | 19 | 585 | 577 | 569 | 557 | 545 | 537 | 525 | 517 | 513 | 501 | 498 | 3240 | 3307 |
[08-Apr-1999]