Water can flow, or it can crash. To push the polygon to the right you want to flow past it to the left, then crash against it to the right. If there were a 'move without pushing' it would make the flowing stage a whole lot simpler.
The 'move without pushing' might require a state change (temperature dependent viscosity?). Perhaps we could use 3D robots that have shorter length scale in some direction than the others, and flip them sideways for flowing. Any suggestions?
An obvious physical analogue would be iron-filings suspended in a fluid, with magnetic fields "commanding" all of them to move left, right, etc. In that case, we obviously can't just have the filings move through each other or the workpiece being manipulated.
Two extensions to our control schemes are having alternating implode/explode movement (move to center of flock, move away), and also exert a torque (which is simple to do; if every robot in the swarm has an orientation and knows whether the swarm center is to its left or right, we can have them move forward or backward, and then we get torsion).
It's a really exciting field--email us!