Peter Alfeld Department of Mathematics College of Science University of Utah The Bernstein Bézier Form Home Page Examples Spline Spaces and Minimal Determining Sets User's Guide Residual Arithmetic Triangulations How does it work? Bibliography

Dimensions on a Hexagonal Configuration

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:

• Number of the configuration
• A link to the MDS code generated description of the configuration and an image,
• A link to a binary file that can be read by the MDS code,
• For each of the interior vertices, the number of distinct slopes assumed by the edges emanating from those vertices.
• The total number of interior vertices for which this number is 3 (i.e., the minimum. Such vertices are sometimes called Clough-Tocher Vertices
 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

Dimensions

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]