Octahedron

The octahedron is one of the five Platonic solids. It consists of 8 equilateral triangles, joined 4 to a vertex. It is the 3 dimensional orthoplex.

Vertex coordinates
An octahedron of side length 1 has vertex coordinates given by all permutations of (±$\sqrt{2}$/2, 0, 0).

Tetratetrahedron
The tetratetrahedron is a variant of the octahedron with A3 symmetry. It consists of two types of equilateral triangles. It can be constructed as a rectification of the tetrahedron.