# Number of Regular tetrahedron from Unit cube

Can we count number of Regular tetrahedrons formed out of Unit cube?

If vertices of Unit cube are taken as $(0,0,0,)$, $(0,0,1)$, $(0,1,1)$, $(0,1,0)$, $(1,0,0)$, $(1,0,1)$, $(1,1,1)$ and $(1,1,0)$

I have seen a post where any four vertices satisfying $x \le y \le z$ forms a Regular Tetrahedron.

But if i choose vertices as $(0,0,0)$, $(1,1,1)$,$(0,0,1)$,$(0, 1,1)$

will it form Regular tet?

Can i know any logical way to choose vertices of cube to form Tet?

## 2 Answers

No, because, for instance, the distance from $(0,0,0)$ to $(1,1,1)$ is different from the distance from $(0,0,0)$ to $(0,0,1)$.

You'll get a tetrahedron if you take those vertices with an odd number of $0$'s and another one if you take those vertices with an even number of $0$'s.

• Logically will total possible Tets are they $58$ Hidden in cube? Since total ways to choose $4$ vertices are $\binom{8}{4}$ and there are $6$ faces and $6$ planes formed by two Opp Edges , hence Total = $70-6-6=58$ – Ekaveera Kumar Sharma Sep 16 '18 at 16:17
• @EkaveeraKumarSharma That's another question. I suggest that you post it as such. – José Carlos Santos Sep 16 '18 at 16:21

Whenever a polyhedron has faces with even counts of sides only, such as the cube, then you can apply a vertex alternation. Say you mark one initial vertex in red, all edge connected neighbours of red vertices in green, and conversely all edge connected neighbours of green vertices in red. This then describes two enantiomorph alternated polyhedra, defined as the convex hull of the red vertices only, and by the green ones only respectively. - For the cube this results in two mutually inverted copies of regular tetrahedra.

Any other choice of 4 vertices from the vertex set of a cube clearly combinatorically could be structured to become a simplex as well. But those simplices would not be regular tetrahedra (as those would use at least 2 different edge length out of $\{1, \sqrt{2}, \sqrt{3}\}$, i.e. edge of cube, diagonal of square face, body diagonal of cube. In some cases such simplices even would become degenerate (when choosing all 4 vertices from a single square face).

--- rk