For the conceptual characterization of the data structures and
operations of the VISTA visualization, the notion of simplexes
is used.
We call a 
-dimensional simplex
or -simplex ``the most simple'' finite -dimensional geometrical
object defined by
-dimensional sub-simplexes
A zero-dimensional geometrical object is a point.
According to this definition,
a single point is a zero-simplex; it is
obviously the only zero-dimensional geometrical object.
A straight line segment is a
one-simplex; it is the simplest one-dimensional geometrical
object. A planar triangle, bounded by three straight line
segments is a two-simplex; it is again the simplest two-dimensional
finite geometrical object. This recursive hierarchical construction
can be formally continued to arbitrary simplex dimensions.
Table 5.3.1 gives an overview of all practically relevant
simplexes.
Leaving performance and compatibility issues entirely out of considerations, the mere fact that the triangle is a basic primitive of modern graphics hardware for three-dimensional problems confirms that the simplex approach is a sound and feasible choice for the efficient representation of visualizable data.
Table 5.1: Simplex objects and their constituent sub-simplexes
Figure 5.1 shows a three-simplex and its sub-simplexes. The tetrahedron consists of four planar triangles, which consist of six distinct straight lines, which in turn consist of four distinct points.
Figure 5.1: A 3-simplex (a tetrahedron) consists of
four 2-simplexes (triangles),
six 1-simplexes (lines), and four 0-simplexes (points).
A mechanism to attach visual properties to the simplexes and to proliferate and process this information is required. This is done by an integer attribute that can be assigned to every simplex. This integer attribute is used as index for visual properties (like line style, color, or texture) in the output modules as the mapping depends on the capabilities of the output device. The value of the simplex attributes can be set and changed in the visualization modules, depending on geometrical or non-geometrical information. For example, to visualize a quantity together with its tensor-product mesh, the mesh must be decomposed into triangles but the resulting artificial diagonal edges should not appear in the final visible output. So the (PIF) input module sets the attribute of the diagonal lines (1-simplexes) to ``invisible''. This information then propagates unchanged through all intermediate operations to the output module which does not draw the artificial diagonals.