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.
|number of cells:||1||1||2||5||13||33||85||199||437||936||1,878||3,674||6,910||12,638||22,536|