Combining rhombic dodecahedrons to form a bigger rhombic dodecahedron

Question

the classic rhombic dodecahedron (RD) can be used to fill space as shown here

I know I cannot recreate a bigger RD by stacking RD together because you never have two touching faces on the sames plane.

I am wondering if joining the center of stacked RD you would end up with a bigger RD ?

I've tried making origami model and it looks like it is not the case.

Answer 1

If I understand your question correctly it is possible. When you take a grid made up of Rhombic Dodecahedron you can create an RD which has edges four times as long as your unit RDs by joining up the centers of certain RDs in your grid.

The image shows how this works. Five (slightly bluish) RDs have been drawn with their centers on a (pretty random) selection of corners of a larger red RD. Between these corner RDs I have drawn a few connecting RDs to show that they are all correctly drawn in the grid.

Large Rhombic Dodecahedron in a grid of RDs

Answer 2

Hint. I think not. The centers of the rhombic docedahedra in that packing are the centers of every other cube in the normal cube packing. I doubt that joining those centers through the faces of the rhombic dodecahedral packing will form another rhombic dodecahedron.

Answer 3

This won't work because the rhombic dodecahedron is not vertex-transitive.

You can, however, pull this off with with the body-centred cubic lattice as shown here: Robert Williams, The Geometrical Foundation of Natural Structure: A Source Book of Design, p. 118.