Symmetry/Tetrahedron/Example

We consider a tetrahedron, that is, a pyramid with four equilateral triangles as faces. The simplest model for this arises when we take every second vertex of a cube; for example, the vertices

${\displaystyle (1,1,1),\,(-1,-1,1),\,(1,-1,-1),\,(-1,1,-1).}$

The distance of the vertices to the origin is ${\displaystyle {}{\sqrt {3}}}$, and the edges have length ${\displaystyle {}{\sqrt {2}}}$. A proper symmetries of the tetrahedron is also a proper symmetry of the corresponding cube.