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=== Relation to geometry of polytopes ===
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Each row of Pascal's triangle gives the number of elements (such as edges and corners) of each dimension in a corresponding [[simplex]] (such as a triangle or tetrahedron). In particular, for {{math| ''k'' > 0}}, the {{mvar|k}}th entry in the {{mvar|n}}th row is the number of {{math|(''k'' − 1)}}-dimensional elements in a {{math|(''n'' − 1)}}-dimensional simplex. For example, a triangle (the 2-dimensional simplex) one 2-dimensional element (itself), three 1-dimensional elements (lines, or edges), and three 0-dimensional elements ([[Vertex (graph theory)|vertices]], or corners); this corresponds to the third row 1, 3, 3, 1 of Pascal's triangle. This fact can be explained by combining Pascal's rule for generating the triangle with the geometric construction of simplices: each simplex is formed from a simplex of one lower dimension by the addition of a new vertex, outside the space in which the lower-dimensional simplex lies. Then each {{mvar|d}}-dimensional element in the smaller simplex remains a {{mvar|d}}-dimensional element of the higher simplex, and each {{math|(''d'' − 1)}}-dimensional element when joined to the new vertex forms a new {{mvar|d}}-dimensional element of the higher simplex.<ref>{{Cite book |last=Coxeter |first=Harold Scott Macdonald |url=https://books.google.com/books?id=iWvXsVInpgMC |title=Regular Polytopes |date=1973-01-01 |publisher=Courier Corporation |isbn=978-0-486-61480-9 |edition=3rd |pages=118–144 |language=en |chapter=Chapter VII: ordinary polytopes in higher space, 7.2: Pyramids, dipyramids and prisms}}</ref>
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