Definition

A face is any flat side of a 3-D shape; a cube has $6$ square faces, and a triangular prism has $5$ faces ($2$ triangles and $3$ rectangles). Formally, a face of a polyhedron is a polygonal region forming part of its boundary, bounded by edges and meeting adjacent faces along edges; the five Platonic solids have faces that are all the same regular polygon. In the theory of convex polytopes, a face of a polytope $P$ in $\mathbb{R}^n$ is a subset of the form $P \cap \{x : c^Tx = \max_{y \in P} c^Ty\}$ for some nonzero vector $c$ (the maximizers of a linear functional); 2-dimensional faces, edges, and vertices are the 2-, 1-, and 0-dimensional faces respectively, and the face lattice of a polytope encodes all its combinatorial structure.