MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
up vote 4 down vote accepted

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.

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.