What's the secret to the concentric squares in the Weeping Waterfall stage?

These are decorative Easter eggs, not a puzzle. Your map node is white most likely because you have another room to find. (Perhaps this one.)

The concentric squares have no bearing on the game or its completion. There are many of them scattered around with different numbers of rings: in Gomez's village (1), atop the broken arch stage (2), atop the bell stage (3), on the 4-cube door's floor (5), atop the waterfall stage (6), on the 16-cube door's temple-under-construction ledge (7), and on the same ledge in the temple warp zone (8), for example.

As far as I know this has no purpose. I've tried to stand on them while inputting the burned treasure map code to no avail. The stage is gold for me, since I found all the doors out, so I don't think it has a bearing on your completion.