A Geometric Higher Dimensional Phenomenon?

Geometry Level 3

In \( \text{3D} \) it's possible to make a regular tetrahedron with integer coordinates that all lie on the vertices of a cube.

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.


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