Let $\Delta\subseteq\mathbb{R}^{4}$ be the convex hull of the points

		$\displaystyle(-1,-1,-1,-1),$
		$\displaystyle(4,-1,-1,-1),$
		$\displaystyle(-1,4,-1,-1),$
		$\displaystyle(-1,-1,4,-1),$
		$\displaystyle(-1,-1,-1,4).$

Choose a triangulation $\mathscr{P}$ of $B=\partial\Delta$ into standard simplices; this can be done in a regular way so that the restriction of $\mathscr{P}$ to each two-dimensional face of $\Delta$ is as given by the light lines in Figure 1. This gives a discriminant locus $\Gamma$ depicted by the dark lines in the figure; the line segments coming out of the boundary of the two-face are meant to illustrate the pieces of discriminant locus contained in adjacent two-faces. The discriminant locus there is not contained in the plane of the two-face. In particular, the discriminant locus is not planar at the vertices of $\Gamma$ on the edges of $\Xi$ with respect to the affine structure we define. Note $\Gamma$ is a trivalent graph, with two types of trivalent vertices, the non-planar ones just mentioned and the planar vertices contained in the interior of two-faces.

For an affine manifold, the monodromy of the local system $\Lambda$ is an important feature of the affine structure. In this example, it is very useful to analyze this monodromy around loops about the discriminant locus. If $v$ is a vertex of $\Gamma$ contained in the interior of a two-face of $\Delta$, one can consider loops based near $v$ in $B_{0}$ around the three line segments of $\Gamma$ adjacent to $v$. It is an enjoyable exercise to calculate that these monodromy matrices take the form, in a suitable basis,

$$T_{1}=\begin{pmatrix}1&0&0\\ 1&1&0\\ 0&0&1\end{pmatrix},T_{2}=\begin{pmatrix}1&0&0\\ 0&1&0\\ 1&0&1\end{pmatrix},T_{3}=\begin{pmatrix}1&0&0\\ -1&1&0\\ -1&0&1\end{pmatrix}.$$

They are computed by studying the composition of transition maps between charts that a loop passes through. These matrices can be viewed as specifying the obstruction to extending the affine structure across a neighbourhood of $v$ in $\Gamma$. Of course, the monodromy of $\check{\Lambda}$ is the transpose inverse of these matrices. Similarly, if $v$ is a vertex of $\Gamma$ contained in an edge of $\Delta$, then the monodromy will take the form

$$T_{1}=\begin{pmatrix}1&-1&0\\ 0&1&0\\ 0&0&1\end{pmatrix},T_{2}=\begin{pmatrix}1&0&-1\\ 0&1&0\\ 0&0&1\end{pmatrix},T_{3}=\begin{pmatrix}1&1&1\\ 0&1&0\\ 0&0&1\end{pmatrix}.$$

So we see that the monodromy of the two types of vertices are interchanged between $\Lambda$ and $\check{\Lambda}$.