Clicking on the button labeled cells on the Control panel brings up the Cell Control Panel. This page describes that panel.
A cell is a configuration with one interior vertex such that all tetrahedra share that vertex. Cells are interesting in their own right, but they are particularly important because understanding the dimension of spline spaces on cells for all values of d would make it possible to determine the dimension of spline spaces on general configurations for d sufficiently large.
The cell package produces abstract configurations which are lists of 4-tuples representing tetrahedra. The elements of the four tuple represent points (usually on the sphere), and a particular assignment of coordinates to these points define a configuration, a particular realization of that particular abstract configuration.
An (abstract) configuration is equivalent to an (abstract) triangulation of the sphere. Associated with each vertex of that triangulation is its degree. We also associate with each vertex the sequence of degrees of its neighboring vertices traversed in order. Such degree sequences are equivalent if they are identical after possibly changing the starting vertex and reversing the direction of the sequence.
The package considers several levels of sameness of two cells:
The following table lists the number of cells of sameness 1 (but not sameness 2) where V is the total number of boundary vertices (excluding the interior vertex). In constructing this table the package considers two cells equivalent if they have sameness 3. Using instead sameness 4 would increase the numbers of cells for a given value of V.
V: | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 |
number of cells: | 1 | 1 | 2 | 5 | 13 | 33 | 85 | 199 | 437 | 936 | 1,878 | 3,674 | 6,910 | 12,638 | 22,536 |
4 triangles: 0: 1 2 3 1: 2 3 4 2: 1 3 4 3: 1 2 4 5 vertices: 0: 4 1: 3 - 0 -> 3(2) 3(3) 3(4) 2: 3 - 0 -> 3(1) 3(3) 3(4) 3: 3 - 0 -> 3(1) 3(2) 3(4) 4: 3 - 0 -> 3(2) 3(3) 3(1) Points: 0: 0.0 0.0 0.0 ... 0.0 1: 0.0 0.0 1.0 ... 1.0 2: 0.866 0.0 -0.4999 ... 1.0 3: -0.433 0.75 -0.4999 ... 1.0 4: -0.433 -0.7499 -0.4999 ... 1.0 areas: 0: 1 2 3 3.5218 1: 2 3 4 2.0008 2: 1 3 4 3.5218 3: 1 2 4 3.5218 total area = 12.5663The interior point (always) has index 0, and there are four boundary points labeled 1, 2, 3, 4. There are four boundary triangles whose vertices are defined by the listed indices. There is a total of 5 vertices. The interior vertex, 0, has degree 4. The boundary vertices all have degree 3. They all have degree sequences 3-3-3. The numbers in parentheses indicate the indices of the actual neighbors. The numbers enclosed by - and -> (in this case all zeros) are codes that identify the degree sequence The coordinates of the points and their norms are given. The areas of the individual spherical triangles, and the sum of these areas are also listed.
Cell 1 is obtained by splitting vertex 1:
6 triangles: 0: 2 3 5 1: 2 3 4 2: 1 3 4 3: 1 2 4 4: 1 2 5 5: 1 3 5 6 vertices: 0: 5 4: 3 - 7 -> 4(2) 4(3) 4(1) 5: 3 - 7 -> 4(2) 4(3) 4(1) 1: 4 - 5 -> 4(3) 3(4) 4(2) 3(5) 2: 4 - 5 -> 4(3) 3(5) 4(1) 3(4) 3: 4 - 5 -> 4(2) 3(5) 4(1) 3(4) Points: 0: 0.0 0.0 0.0 ... 0.0 1: -0.25 -0.433 0.866 ... 1.0 2: 0.866 0.0 -0.4999 ... 1.0 3: -0.433 0.75 -0.4999 ... 1.0 4: -0.433 -0.7499 -0.4999 ... 1.0 5: 0.25 0.433 0.866 ... 1.0 areas: 0: 2 3 5 2.2604 1: 2 3 4 2.0008 2: 1 3 4 2.5575 3: 1 2 4 2.5575 4: 1 2 5 1.595 5: 1 3 5 1.595 total area = 12.5663 splitting history: 0 1: 2 3 2 3 4Thus vertex 1 is split into vertex 1 and the new vertex 5, and neighbors 2 and 3 of the old vertex 1 are now attached to both vertex 1 and 5.
The splitting history described how this cell was obtained from the original cell. The given numbers indicate:
8 triangles: 0: 2 3 5 1: 2 3 4 2: 3 4 6 3: 1 2 4 4: 1 2 5 5: 1 3 5 6: 1 3 6 7: 1 4 6 7 vertices: 0: 6 5: 3 - 17 -> 4(2) 5(3) 5(1) 6: 3 - 17 -> 5(3) 4(4) 5(1) 2: 4 - 23 -> 5(3) 3(5) 5(1) 4(4) 4: 4 - 23 -> 4(2) 5(3) 3(6) 5(1) 1: 5 - 38 -> 4(2) 4(4) 3(6) 5(3) 3(5) 3: 5 - 38 -> 4(2) 3(5) 5(1) 3(6) 4(4) Points: 0: 0.0 0.0 0.0 ... 0.0 1: 0.0019 -0.3915 0.9201 ... 1.0 2: 0.866 0.0 -0.4999 ... 1.0 3: -0.433 0.75 -0.4999 ... 1.0 4: -0.433 -0.7499 -0.4999 ... 1.0 5: 0.25 0.433 0.866 ... 1.0 6: -0.4849 -0.4449 0.7528 ... 0.9999 areas: 0: 2 3 5 2.2604 1: 2 3 4 2.0008 2: 3 4 6 2.009 3: 1 2 4 2.5726 4: 1 2 5 1.2507 5: 1 3 5 1.1831 6: 1 3 6 0.8861 7: 1 4 6 0.4033 total area = 12.5663 splitting history: 0 1: 2 3 2 3 4 1 1: 2 4 3 4 2 5Cell 3 is the Octahedron. Vertex 1 is split twice, but the second time three of its neighbors are attached to the new vertex. Two of those, and the remaining neighbor, are attached to the old vertex.
8 triangles: 0: 2 3 5 1: 2 3 4 2: 1 3 4 3: 2 4 6 4: 2 5 6 5: 1 3 5 6: 1 4 6 7: 1 5 6 7 vertices: 0: 6 1: 4 - 15 -> 4(3) 4(4) 4(6) 4(5) 2: 4 - 15 -> 4(3) 4(5) 4(6) 4(4) 3: 4 - 15 -> 4(2) 4(5) 4(1) 4(4) 4: 4 - 15 -> 4(2) 4(3) 4(1) 4(6) 5: 4 - 15 -> 4(2) 4(3) 4(1) 4(6) 6: 4 - 15 -> 4(2) 4(4) 4(1) 4(5) Points: 0: 0.0 0.0 0.0 ... 0.0 1: -0.4656 -0.2888 0.8365 ... 0.9999 2: 0.866 0.0 -0.4999 ... 1.0 3: -0.433 0.75 -0.4999 ... 1.0 4: -0.433 -0.7499 -0.4999 ... 1.0 5: 0.25 0.433 0.866 ... 1.0 6: -0.0173 -0.5476 0.8365 ... 1.0 areas: 0: 2 3 5 2.2604 1: 2 3 4 2.0008 2: 1 3 4 2.2224 3: 2 4 6 2.2224 4: 2 5 6 1.5165 5: 1 3 5 1.5165 6: 1 4 6 0.5235 7: 1 5 6 0.3034 total area = 12.5663 splitting history: 0 1: 2 3 2 3 4 1 1: 3 4 4 2 5 3