Making 4-in-1 Rounds
To begin with, you'll want to have plenty of cut rings. Follow the first few directions from the basic 4-in-1 instructions for details.
To begin with, make an 8-in-1 by feeding eight closed rings onto one open ring and closing it. Depending on your wire size, ring diameter, and the size of your pliers, this may be a little hard to do. The end result can be flattened out to resemble the second picture above.
Now we add a layer to the outer edge of this center piece, with each individual ring feeding through two other rings. The left picture shows how to add the first ring, and the center one shows the piece with a second outer ring added. Continue around the outer edge with six more rings, to get a piece that looks like the right picture. Notice that we still have eight rings around the outer edge.
Now we need to add some rings to allow for expansion of the pattern. For this layer, we want to add four rings to the existing eight, for a total of twelve in the outside row. These expansion rings should be added evenly around the perimeter, and should only connect to a single ring each. The images illustrate where the rings should be added, how they should be connected, and what your piece should look like once they go in.
From here, everything is just repetition. Add new layers, adding expansion rings with each layer as you build outward. The only difference is that as you add further expansions, you should add six rings per layer, rather than four, which is used only on the first layer. Thus, your ring-count around the outside edge should go: 8, 12, 18, 24, 30, 36, 42, 48, etc.