Below are three different views of the tetrahedroid, a 4-Noid with the symmetries of the regular tetrahedron, which also is a special pyramidal 4-Noid

These surfaces can be viewed as a special case of k-Noids with the symmetries of an antiprism with regular base. This then includes the case of an octahedroid as well, which can also be understood as a bipyramidal k-Noid.

Resources

Mathematica Notebook and CDF

PoVRay Sources