The normal English definition by tijlan says that the shape is a
"three-sided shape". gleki'so Simple English definition says that it is a
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 three-sided.
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, non-self-intersecting, 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.