Robot Search in 3D Swarm Construction

Robots in a swarm construction system need to be able to find their way to all potential places to add material to a structure. In two dimensions, a (trivial) procedure exists for a stateless robot moving along the outside of a structure, that will let it start at any point, take it to all potential attachment sites exactly once, and return it to its starting point, regardless of what part of the structure has already been built. Here we show that for three-dimensional structures made from cubic building blocks, where each exposed face of a block is a potential attachment site, no such procedure exists. Any robot search scheme must then necessarily involve excess movement and/or increased component complexity (e.g., memory or communication).