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Take a flat surface for example... make it a triangular shape. This shape has one face called a 'polygon'. For a polygon to have a shape it must contain at least 3 vertices. A vertex is the 3 points of which make up the polygon. Remove one vertex and you no longer have a polygon, but an empty space between 2 vertices.

Imagine a spider web with very straight strands. Wherever the strands meet is like a vertex that can have many strands leading from it. If that vertex is removed, every strand that links to it will collapse, or in the sense of the polygons, any polygon directly related to that vertex will disappear.

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Q: Definition of vertex in Autodesk 3ds Max?
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