According to this paper by Diane Maclagan (https://arxiv.org/abs/1207.1925, p.10), a polyhedral complex is pure if the dimension of every maximal polyhedron is the same, which is shown in the figure 11. 
However, I don't understand why the complex on the left is pure and the one on the right isn't pure. I assumed that we are only talking about the grey-ish areas; so, on the left, I thought the maximal polyhedra would be the ones more in the center. But according to this theory, I would only have one polyhedron on the right and this would be maximal as the only one?!
So, it would be great if anyone could solve my (possible) misunderstanding of "maximal polyhedra" and additionally explain the figure. Thank you!
Points inside the 2 lower edges in the right pic do not belong to any gray face. Thus the local dimension is just 1 (at maximum). Whereas all other edges (in fact of both pics) belong at least to one gray face at minimum. Thus their local maximal dimension is 2. Therefore in the left pic all points belonging to the structure do have maximal dimension 2, whereas in the right pic some points have maximal dimension 1, others 2.
--- rk