John Horton Conway calls it an oblate tetrahedrille or shortened to obtetrahedrille.
Each edge of the tessellation is surrounded by either four or six disphenoids, according to whether it forms the base or one of the sides of its adjacent isosceles triangle faces respectively.
When an edge forms one of the two equal sides of its adjacent isosceles triangle faces, the six disphenoids surrounding the edge form a special type of parallelepiped called a trigonal trapezohedron.
It can be seen as a cubic honeycomb with each cube subdivided by a center point into 6 square pyramid cells.
John Horton Conway calls it an oblate octahedrille or shortened to oboctahedrille.
It can be seen as a cubic honeycomb with each cube subdivided by a center point into 6 square pyramid cells.
There is one type of plane with faces: a flattened triangular tiling with half of the triangles as holes.
There are also square tiling plane that exist as nonface holes passing through the centers of the octahedral cells.
It is one 1/6 of a smaller cube, with 6 phyllic disphenoidal cells sharing a common diagonal axis.