# A Geometric Higher Dimensional Phenomenon?

**Geometry**Level 3

Does this phenomenon occur in any other dimensional space?

Or specifically, for how many \( n \in \mathbb{N} \) is it possible to construct a regular \(n\)-dimensional simplex with integer co-ordinates that lie on an \( n\)-dimensional hypercube in \( \mathbb{Z}^n \)?

**Details and Assumptions**:

A regular \(n\)-dimensional simplex in \( \mathbb{Z}^n \) has \(n+1\) vertices that are all an equal distance apart. (It's like an \(n\)-dimensional version of an equilateral triangle!)

Here is the Wikipedia article on hypercubes. (It's like an \(n\)-dimensional version of a square!)

Note that the analogous phenomena is not possible in \(2\) dimensions. That is, we cannot create an equilateral triangle with integer co-ordinates that all lie on a square. Check out why, here.