The normal English definition by tijlan says that the shape is a "threesided shape". gleki'so Simple English definition says that it is a triangle/tetrahedron/simplex.
These are not the same thing. The former definition does not require x1 to be closed or convex. The latter definition does not require x1 to be threesided.
Which is it? I propose that gleki's version be adopted, since it seems rather more useful. If desired, we can mention that there is no contraint that x1 be convex, nonselfintersecting, and closed. I am not sure how useful this loosening of the term would be in normal life (a lot more qualifiers would need to be employed in normal situations), but I suppose that it is mathematically more general. (And the speaker can always intend it to her restricted appropriately.)
Also the normal English definition, x3 is a specification for the aides which define x1, or the dimension(ality) in which it lives. The latter case makes sense to me, especially since x2 gives a set of the vertices (which should define the convex hull of the shape at the least) and, in the light of gleki's definition, the dimensionality really determines which simplex it is. On the other hand, the vertices will carry the dimensionality information with them, if they are specified, so the "sides" meaning is more useful, especially if we allow the shape to be more ugly than a traditional triangle (convex, etc.); in fact, if it must have three angles (in accordance with the first definition), then the number of dimensions must be less than or equal to four, so a dimensionality Terri is not as useful (although probably still good to have).
The point is that these are NY the same meaning for x3.
I propose that, in any case, x3 be reserved for specifying the sides of the shape (which is useful, especially if we want some way to discuss them directly as sumti), and I propose that it must be supplied with a partial (but not overspecifying) set of these sides; then I also propose that a new terbri, x4, be introduced for the dimensionality, which must be greater than or equal to 1 (if we assume the simplex interpretation) and is usually an integer.
