  This paper aims to show that there exists a triangulation of the Heisenberg
group $\mathbb{H}^n$ into singular simplexes with regularity properties on both
the low-dimensional and high-dimensional layers. For low dimensions, we request
our simplexes to be horizontal while, for high dimensions, we define a notion
of straight simplexes using exponential and logarithmic maps and we require our
simplexes to have high-dimensional straight layers. A triangulation with such
simplexes is first constructed on a general polyhedral structure and then
extended to the whole Heisenberg group. In this paper we also provide some
explicit examples of grid and triangulations.
