This question arose out of my attempts to understand [another question](http://mathoverflow.net/questions/24453/). The most popular construction for the chain complex for defining singular homology uses the $n$-simplex.

But it is also possible to use other spaces. For example, one can use the $n$-cubes instead, as done in certain books. Also it occurs to me that one can use the discs $D^n$, with orientation specified at the boundary. I haven't checked all the details; but I am hopeful that it can be made to work and made to prove that this homology is the same as the homology constructed with cubes or with simplexes.

So, why is the simplex the most used choice? Granted, it has a certain symmetry in all directions and so it is aesthetically somehow more satisfying. But are there other reasons?