pictorial illustration of simplicial complexes

Question

Consider the following two complexes (Bruns&Herzog p.215):

By just looking at the complex on the left, i am not sure how to read its faces. Surely its vertices are $v_1,v_2,v_3,v_4,v_5$. The fact that e.g. $v_1$ and $v_5$ are connected via an edge, leads me to believe that $\left\{v_1,v_5\right\}$ is a face. The fact that the triangle $\left\{v_1,v_2,v_5\right\}$ is dotted leads me to believe that $\left\{v_1,v_2,v_5\right\}$ is another face. But how about e.g. $\left\{v_1,v_2,v_4,v_5\right\}$? Is this a face? Or how about $\left\{v_1,v_2,v_3,v_4,v_5\right\}$? Is this a face? And how can we distinguish between the two complexes where e.g. $\left\{v_1,v_2,v_4,v_5\right\}$ is face in the first but it is not a face in the second?

On the other hand, by looking at the complex on the right, i can tell that e.g. $\left\{u_2,u_4,u_5\right\}$ is not a face.

Answer 1

they are not faces: if $\left\{v_1,v_2,v_4,v_5\right\}$ be a face there was a pyramid ($\left\{v_1,v_4\right\}$ is missing). $\left\{v_1,v_2,v_3,v_4,v_5\right\}$ being a face can not be shown in a picture, because it is of dimension 4.
the complex on the right: $\left\{u_2,u_4,u_5\right\}$ is not a face

Answer 2

USe the definiton of simplicial complex to read the faces. A simplex S of $k+1$ point say $p_i$ is the affine combination with non negative coefficient $\alpha_i$. A face is a subsimplex that is $T\subset S$. Convex hull of T is a simplex that is $\alpha_i=0$ if $p_i$ does not belong to T. Definiton: A Simplicial complex is a finite collection K of simplex such that 1) $\sigma\in K$ and $\tau\leq\sigma$ ($\tau$ is a subsimplex of $\sigma$)implies $\tau\in K$, and 2) $\sigma$,$\upsilon\in K$implies $\sigma\cap\upsilon\leq\sigma,\upsilon$.