To start building this space-filling honeycomb of three Archimedean solids, I begin with a great rhombcuboctahedron. This polyhedron is also called the great rhombicuboctahedron, as well as the truncated cuboctahedron.

Next, I augment the hexagonal faces with truncated tetrahedra.

The next polyhedra to be added are truncated cubes.

Now it’s time for another layer of great rhombcuboctahedra.

Now more truncated tetrahedra are added.

Now it’s time for a few more great rhombcuboctahedra.

Next come more truncated cubes.

More great rhombcuboctahedra come next.

More augmentations using these three Archimedean solids can be continued, in this manner, indefinitely. The images above were created with Stella 4d: Polyhedron Navigator, a program you may try for yourself at http://www.software3d.com/Stella.php.