In $\mathbb{R}^{3}$ consider the $3$-dimensional simplex $\Xi$ spanned by the points

$$P_{0}=(-1,-1,-1),\ \ P_{1}=(3,-1,-1),\ \ P_{2}=(-1,3,-1),\ \ P_{3}=(-1,-1,3).$$

Let $B=\partial\Xi$. We explain how to construct a simple affine structure with singularities on $B$. Each edge $\ell_{j}$ of $\Xi$ has $5$ integral points (i.e. belonging to $\mathbb{Z}^{n}$), which divide $\ell_{j}$ into $4$ segments. For each $j=1,\ldots,6$ denote by $\Delta^{j}_{k}$, $k=1,\ldots,4$ the four barycenters of these four segments. We let

$$\Delta=\{\Delta^{j}_{k};j=1\ldots 6\ \text{and}\ k=1,\ldots,4\}.$$

A covering of $B_{0}=B-\Delta$ can be defined as follows. The first four open sets consist of the four open faces $\Sigma_{i}$, $i=1\ldots,4$ with the affine coordinate maps $\phi_{i}$ induced by their affine embeddings in $\mathbb{R}^{3}$. Denote by $I$ the set of integral points of $B$ which lie on an edge. For every $Q\in I$ we can choose a small open set $U_{Q}$ in $B_{0}$ such that $\{\Sigma_{i}\}_{i=1,\ldots,4}\cup\{U_{Q}\}_{Q\in I}$ is a covering of $B_{0}$. Let $R_{Q}$ denote the $1$-dimensional subspace of $\mathbb{R}^{3}$ generated by $Q\in I$. One can verify that if $U_{Q}$ is small enough, the projection $\phi_{Q}:U_{Q}\rightarrow\mathbb{R}^{3}/R_{Q}$ is an homeomorphism. A computation shows that the atlas $\mathscr{A}=\{\Sigma_{i},\phi_{i}\}_{i=1,\ldots,4}\cup\{U_{Q},\phi_{Q}\}_{Q\in I}$ defines an affine structure on $B_{0}$ making $(B,\Delta,\mathscr{A})$ simple.