Plane Formation by Synchronous Mobile Robots in the Three Dimensional Euclidean Space

Creating a swarm of mobile computing entities frequently called robots, agents or sensor nodes, with self-organization ability is a contemporary challenge in distributed computing. Motivated by this, this paper investigates the plane formation problem that requires a swarm of robots moving in the three dimensional Euclidean space to reside in a common plane. The robots are fully synchronous and endowed with visual perception. But they have neither identifiers, access to the global coordinate system, any means of explicit communication with each other, nor memory of past. Though there are plenty of results on the agreement problem for robots in the two dimensional plane, for example, the point formation problem, the pattern formation problem, and so on, this is the first result for robots in the three dimensional space. This paper presents a necessary and sufficient condition to solve the plane formation problem. An implication of the result is somewhat counter-intuitive: The robots cannot form a plane from most of the semi-regular polyhedra, while they can from every regular polyhedron (except a regular icosahedron), which consists of the same regular polygon faces and the robots on its vertices are "more" symmetric than semi-regular polyhedra.