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.

This three dimensional example is taken from [10] §19.3. Let $\Xi$ be the $4$-simplex in $\mathbb{R}^{3}$ spanned by

	$\displaystyle P_{0}=(-1,-1,-1,-1),\ P_{1}=(4,-1,-1,-1),\ P_{2}=(-1,4,-1,-1),$
	$\displaystyle P_{3}=(-1,-1,4,-1),\ \ P_{4}=(-1,-1,-1,4).$

Let $B=\partial\Xi$. Denote by $\Sigma_{j}$ the open $3$-face of $B$ opposite to the point $P_{j}$ and by $F_{ij}$ the closed $2$-face separating $\Sigma_{i}$ and $\Sigma_{j}$. Each $F_{ij}$ contains $21$ integral points (including those on its boundary). These form the vertices of a triangulation of $F_{ij}$ as in Figure 7. By joining the barycenter of each triangle with the barycenters of its sides we form a trivalent graph as in Figure 7. Define the set $\Delta$ to be the union of all such graphs in each $2$-face. Denote by $I$ the set of integral points of $B$. Just as in the previous example, we can form a covering of $B_{0}=B-\Delta$ by taking the open $3$-faces $\Sigma_{j}$ and small open neighborhoods $U_{Q}$ inside $B_{0}$ of $Q\in I$. A coordinate chart $\phi_{i}$ on $\Sigma_{i}$ can be obtained from its affine embedding in $\mathbb{R}^{4}$. If we denote again by $R_{Q}$ the linear space spanned by $Q\in I$, as a chart on $U_{Q}$ we take the projection $\phi_{Q}:U_{Q}\rightarrow\mathbb{R}^{4}/R_{Q}$. A computation shows that this affine structure is simple. In fact the vertices of $\Delta$ which are contained in the interior of each $2$-face are of negative type and those which are contained in the $1$-faces are of positive type.

In the previous example, all edges of $\Delta$ were straight lines, but one can perturb them in the sense of Examples 3.9, 3.11 and 3.13. In fact we can form a new $\Delta$ by keeping the vertices fixed and connecting them through smooth curves, which are small perturbations of the straight edges of the previous example. If these curves stay inside the $2$-faces of $B$, then the affine structure on $B-\Delta$ can be defined just like above.