example, a 7-simplex is (1,1)⁸ = (1,2,1)⁴ = (1,4,6,4,1)² = (1,8,28,56,70,56,28,8,1).

The number of 1-faces (edges) of the -simplex is the -th triangle number, the number of 2-faces of the -simplex is the th tetrahedron number, the number of 3-faces of the -simplex is the th 5-cell number, and so on.

An -simplex is the polytope

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