Put on my engineer hat, and decided to try and design a wheel with hinging segments, that can be used to expand a circle into a square. Pink line is a flexible membrane like an elastic band wrapped around the hinges which are shown in green. The aim was for the outside square to mantain it's shape as the inner circle is rotated. The idea is you could have two of these next to each other yet they could freely rotate without overlapping! This allows for packed circles which fill the space between them. Hinges can be connected to compressive springs, so as to avoid motors.
