Vertex (3-D)
Geometry & MeasurementA vertex of a 3-D solid is a corner point where three or more edges meet.
Definition
A vertex (plural: vertices) is a corner point on a 3-D shape where edges meet. A cube has $8$ vertices, one at each corner of the box.
Example
The tip of a pyramid is a vertex where all the triangular sides meet. The corners of a cube are vertices where $3$ edges and $3$ faces all meet at one point.
Key Insight
In 2-D, a vertex is a corner of a polygon. In 3-D, it is a corner of a solid where at least $3$ edges come together. The more edges that meet at a vertex, the "sharper" or more complex the corner.
Definition
A vertex of a polyhedron is a point where three or more edges meet. At each vertex, at least $3$ faces also meet. By Euler's formula $F + V - E = 2$, knowing any two of $\{\text{faces, vertices, edges}\}$ determines the third.
Example
A square pyramid has $5$ vertices: $4$ base corners (each where $2$ base edges and $1$ lateral edge meet) and $1$ apex (where $4$ lateral edges meet). $F=5$, $E=8$, $V=5$: check $5 + 5 - 8 = 2$.
Key Insight
At each vertex of a convex polyhedron, the sum of the face angles is less than $360$ degrees. For a regular tetrahedron, three $60$-degree triangles meet at each vertex: $3 \times 60 = 180 < 360$. This "angular deficit" at vertices determines the curvature of the polyhedron.
Definition
A vertex of a convex polytope $P$ is a point $x \in P$ such that $P \setminus \{x\}$ is still convex; equivalently, $x$ is not a convex combination of other points in $P$. The vertices are the extreme points of the polytope and are the generators of $P$ as a convex hull. In linear programming, the optimal solution is always at a vertex of the feasible polytope.
Example
The simplex algorithm for linear programming moves from vertex to adjacent vertex along edges of the feasible polytope, always improving the objective function. The worst-case number of vertices of an $n$-dimensional polytope with $m$ faces is given by the upper bound theorem (McMullen, 1970).
Key Insight
The Krein-Milman theorem generalizes the vertex concept to infinite-dimensional convex sets: any compact convex set in a locally convex space is the closed convex hull of its extreme points. This is fundamental in functional analysis, optimization, and quantum mechanics.