# coarser than triangulations “almost partitions” into simplices

The total space $T$ of an embedded into $\mathbb{R}^n$ pure $n$-dimensional simplicial complex (in other words, the union of finitely many $n$-dimensional compact convex polytopes) sometimes admits an "almost partition" into $n$-simplices (i.e. $T$ equals the union of these simplices, and the interiors of these simplices do not, pairwise, intersect) which is coarser than a triangulation.

For instance, consider the union $T$ of two tetrahedra $T_1$, $T_2$ in $\mathbb{R}^3$ which intersect at the mid-point $M=E_1\cap E_2$ of edges $E_i\in T_i$, $1\leq i\leq 2$. Then any triangulation of $T$ will have to include $M$, and have at least 4 simplices. On the other hand, there exist an obvious "almost partition" of $T$ into just 2 simplices, $T_1$ and $T_2$.

We wonder whether there exists any research on these kinds of "almost partitions", and whether there is any well-established terminology here.

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The term "triangulation" tends to be ambiguous, as they appear both in a geometric and topological context. In this case, what you want is called a dissection, at least in the discrete geometry literature (see e.g. here and there).

P.S. Oh, yes, plenty of research. See e.g. here for whether dissections of convex polytopes can be smaller than triangulations.

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Thanks a lot. It's very useful. Especially the fact that two dissections are connected by elementary moves (unlike triangulations) is simplifying our problems a lot. Are there any plans for your book to be published? –  Dima Pasechnik Jul 27 '12 at 4:14