The simplexes are
-dimensional manifolds (subsets) of 
. A triangle in
two-dimensional space is conceptually different from a triangle in
three-dimensional space. In order to distinguish these to cases and to
unambiguously denote a -dimensional
simplex in 
-dimensional space we write symbolically
has the following basic properties
The simplex is defined by its bounding 
sub-simplexes.
and, vice versa
Furthermore, we may confine our considerations to
as we have no useful semantics for, e.g., two-dimensional tetrahedrons.
Visualization does of course not work with single simplexes, but with sets of several simplexes which approximate more complex geometrical structures. The visualization operations are always performed on simplex-sets and yield other simplex sets (with different dimensions) as result.
We denote
a set of -simplexes in -dimensional space which can be imagined as
a piecewise linear approximation of a general -dimensional manifold
(curved objects) in 
.
The input/output requirements of the operations define their relationships and hence imply the possible combinations of the visualization modules.