  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, we investigate the plane
formation problem that requires a swarm of robots moving in the three
dimensional Euclidean space to land on a common plane. The robots are fully
synchronous and endowed with visual perception. But they do not have
identifiers, nor access to the global coordinate system, nor any means of
explicit communication with each other. 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 for fully-synchronous robots to solve the
plane formation problem that does not depend on obliviousness i.e., the
availability of local memory at robots. 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 form a plane from every regular
polyhedron (except a regular icosahedron), whose symmetry is usually considered
to be higher than any semi-regular polyhedrdon.
