Vertex (geometry)
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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.
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On A Graph
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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.
On A Polygon
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On a polygon there are as many vertices as there are corners.
On a Polyhedron
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On a polyhedron there are as many vertices as there are corners.
On a Polychoron
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On a polychoron there are as many vertices as there are corners.