Vertex (3-D)
A 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 of a 3-D shape where at least three edges (and at least three faces) meet; a cube has $8$ vertices, one at each corner. By Euler's formula $F + V - E = 2$, knowing any two of faces, vertices, and edges determines the third. Formally, a vertex of a convex polytope $P$ is a point $x \in P$ such that $P \setminus \{x\}$ is still convex, equivalently a point that is not a convex combination of other points in $P$; vertices are the extreme points of the polytope and generate it as a convex hull, and in linear programming, the optimal solution is always at a vertex of the feasible polytope.
Example
The tip of a pyramid is a vertex where all triangular sides meet, and the corners of a cube are vertices where $3$ edges and $3$ faces meet at one point. A square pyramid has $5$ vertices, $4$ base corners and $1$ apex, with $F=5$, $E=8$, $V=5$: check $5 + 5 - 8 = 2$. The simplex algorithm for linear programming moves from vertex to adjacent vertex along edges of the feasible polytope, always improving the objective function, with the worst-case number of vertices of an $n$-dimensional polytope given by the upper bound theorem (McMullen, 1970).
Key Insight
The more edges that meet at a vertex, the "sharper" or more complex the corner. 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 ($3 \times 60 = 180 < 360$), and this angular deficit determines the polyhedron's curvature. 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, fundamental in functional analysis, optimization, and quantum mechanics.