vertices

Vertices (singular vertex) are fundamental points in geometry and graph theory. In geometric figures they are the corner points where edges or faces meet; in graph theory, a vertex is a node connected by edges. The plural form is vertices.

In polygons, every vertex marks a corner of the shape. A polygon with n sides has n vertices, and the interior angles sum to (n − 2) × 180 degrees. In polyhedra, a vertex is a point where at least three edges converge; the degree of a vertex equals the number of incident edges, and regular polyhedra have vertices that all share the same degree.

In graph theory, a vertex is a discrete object that can be connected to others by edges. The degree of a vertex is the number of incident edges, with loops or multiple edges treated according to the context. The total of all vertex degrees equals twice the number of edges. For planar graphs, relationships among vertices, edges, and faces are captured by Euler’s formula: V − E + F = 2 for connected graphs. Vertex connectivity describes how many vertices must be removed to disconnect the graph.

Vertices can be annotated with coordinates in space, such as 2D (x, y) or 3D (x, y, z), especially in computer graphics and modeling. Beyond pure geometry, the concept appears in networks, where vertices represent entities and edges represent relations or interactions.

Examples include a triangle with three vertices, a square with four, a cube with eight vertices, a tetrahedron with four, and a dodecahedron with twenty.