Twin Simplex

Geometry Level 5

Let \(e_1, \ldots , e_{n+1}\) be the standard basis in \(\mathbb R^{n+1}\). We define an \(n\)-dimensional simplex as the convex hull of the endpoints of the vectors \(e_1, \ldots , e_{n+1}\). That is, the \(n\)-dimensional simplex can be defined as the convex hull of \(n + 1\) affinely independent points.

We can thus clearly see that the standard 1-simplex is a line segment, the standard 2-simplex is an equilateral triangle, the standard 3-simplex is a regular tetrahedron, and so on.

How many 4-dimensional faces does the 197-dimensional simplex have?

Image: orthogonal projection of the 4-simplex (5-cell) performing a simple rotation. Credit for the image goes to Jason Hise.

Bonus: Generalize the problem. How many \(k\)-dimensional faces does an \(n\)-dimensional simplex have?


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