Edge
Geometry & MeasurementAn 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. It is a straight line along the corner between two faces. A cube has $12$ edges.
Example
Think of a cardboard box. Each fold line where two sides of the box meet is an edge. A cube has $12$ of those fold lines ($4$ on top, $4$ on bottom, $4$ going up the sides).
Key Insight
Edges connect vertices (corners) and separate faces. Knowing the number of faces, edges, and vertices of any convex shape always gives $F + V - E = 2$. This is called Euler's formula.
Definition
An edge of a polyhedron is the line segment formed by the intersection of two adjacent faces. 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.
Example
A triangular prism has $9$ edges: $3$ on the top triangle, $3$ on the bottom triangle, and $3$ vertical edges connecting them. Check: $F=5$, $V=6$, $E=9$; $5 + 6 - 9 = 2$.
Key Insight
The number of edges can always be computed if faces and vertices are known: $E = F + V - 2$. For any simple polyhedron (no holes), this Euler characteristic equals $2$. For a torus (donut shape), it equals $0$.
Definition
An edge is a 1-cell in the CW-complex structure of a polyhedron, a 1-dimensional face of the polytope bounded by exactly two vertices. The edge graph (or 1-skeleton) of a convex polytope is always 3-connected (Steinitz's theorem), meaning no two vertex removals can disconnect it. This characterizes which graphs can be edge graphs of convex polyhedra.
Example
The Petersen graph cannot be the edge graph of any convex polyhedron (it is not 3-connected in the right way). Steinitz's theorem gives necessary and sufficient conditions: a graph is realizable as a convex polytope skeleton iff it is 3-connected and planar.
Key Insight
Steinitz's theorem is fundamental in combinatorial geometry: it completely characterizes convex polyhedra in purely graph-theoretic terms, showing that topology and combinatorics together determine what 3-D shapes are possible.