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.