Edge
An edge is a line segment where two faces of a 3-D solid meet.
Definition
An edge is where two flat faces of a 3-D shape meet, a straight line along the corner between them; a cube has $12$ edges. Formally, an edge is the line segment formed by the intersection of two adjacent faces of a polyhedron, and each edge is shared by exactly two faces; for a convex polyhedron, Euler's formula $F + V - E = 2$ constrains the relationship between faces, vertices, and edges, so $E = F + V - 2$ can always be computed once faces and vertices are known. In the language of CW-complexes, an edge is a 1-cell, a 1-dimensional face of the polytope bounded by exactly two vertices, and the edge graph (1-skeleton) of a convex polytope is always 3-connected (Steinitz's theorem), meaning no two vertex removals can disconnect it.
Example
Think of a cardboard box: each fold line where two sides meet is an edge, and a cube has $12$ of them ($4$ on top, $4$ on bottom, $4$ vertical). A triangular prism has $9$ edges: $3$ on the top triangle, $3$ on the bottom, and $3$ vertical, checking against Euler's formula: $F=5$, $V=6$, $E=9$; $5 + 6 - 9 = 2$. The Petersen graph cannot be the edge graph of any convex polyhedron; Steinitz's theorem gives necessary and sufficient conditions, a graph is realizable as a convex polytope skeleton if and only if it is 3-connected and planar.
Key Insight
Edges connect vertices (corners) and separate faces; for any simple polyhedron (no holes), the Euler characteristic $F + V - E$ always equals $2$, while for a torus (donut shape) it equals $0$. Steinitz's theorem is fundamental in combinatorial geometry, completely characterizing convex polyhedra in purely graph-theoretic terms and showing that topology and combinatorics together determine what 3-D shapes are possible.