A Geometric Higher Dimensional Phenomenon?

Geometry Level 3

In 3D \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 nN n \in \mathbb{N} is it possible to construct a regular nn-dimensional simplex with integer co-ordinates that lie on an n n-dimensional hypercube in Zn \mathbb{Z}^n ?

Details and Assumptions:

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

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

  • Note that the analogous phenomena is not possible in 22 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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