The 2-simplices are pairs ([phi]|[psi]) consisting of a 2-simplex [phi] of [M.sub.1] and a 2-simplex [psi] of [M.sub.2] with common boundary {[A.sub.12], [A.sub.02], [A.sub.01]}, such that the diagram

The 3-simplices are pairs ([[phi].sub.123], [[phi].sub.023], [[phi].sub.013], [[phi].sub.012]| [[psi].sub.123], [[psi].sub.023], [[psi].sub.013], [[psi].sub.012]) consisting of a 3-simplex ([[phi].sub.123], [[phi].sub.023], [[phi].sub.013], [[phi].sub.012]) in [M.sub.1] and a 3-simplex ([[psi].sub.123], [[psi].sub.023], [[psi].sub.013],[[psi].sub.012]) in [M.sub.2] such that ([[phi].sub.ijk]|[[psi].sub.ijk]) is a 2-simplex in [M.sub.12] for each ijk, and the diagram

and recovers (16) for any point u within the 2-simplex [[u.sub.1], [u.sub.2], [u.sub.3]].

4, we depict a straight line segment inside a 2-simplex with a point u marked on it.

In this case, each pixel is on the standard 2-simplex in [R.sup.3] due to [p.sub.s] + [p.sub.v] + [p.sub.d] = 1.

The standard 2-simplex in [R.sup.3] is showed in Fig.

In Figure 4, the 1-simplex spanned by the vertexes {P0, Q1} is a proper face of the 2-simplex {P0, Q1, R2}.

Figure 33 shows a chromatic subdivision of a complex S defined on a 2-simplex S.

For each pair of elements [alpha], [beta] [member of] [[pi].sub.1] (X) we fix a singular 2-simplex r([alpha], [beta]):[[DELTA].sub.2] [right arrow] X such that [0,1], [1,2] and [0,2] map according to r([alpha]), r([beta]) and r([alpha][beta]).

For each triple ([alpha],[beta];a) [member of] [[pi].sub.1](X) x [[pi].sub.1](X) x [[pi].sub.2](X) we consider a singular 2-simplex r([alpha], [beta]; a):[[DELTA].sub.2] [right arrow] X such that [0,1], [1,2] and [0,2] map according to r([alpha]), r([beta]) and r([alpha][beta]) and when we glue r([alpha], [beta]; a) with r([alpha], [beta]) along the boundary we get a [member of] [[pi].sub.2](X).