A vertex is anywhere that two or more lines meet. A square, for example, has four corners and thus has four vertices (plural of vertex). Any polytope has at least one vertex.

Vertices on graphs

On a cartesian grid there are a plethora of graphs that have vertices. Any graph with the equation 0=Ax^n+...Bx^2+Cx+D will have an amount of vertices equal to or less than the highest power of the equation.

Vertices on polygons, polyhedron, and polychorons

On a polygon, polychoron, or polyhedron there are as many vertices as there are corners.

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