  Consider a set of $n$ simple autonomous mobile robots (asynchronous, no
common coordinate system, no identities, no central coordination, no direct
communication, no memory of the past, non-rigid, deterministic) initially in
distinct locations, moving freely in the plane and able to sense the positions
of the other robots. We study the primitive task of the robots arranging
themselves on the vertices of a regular $n$-gon not fixed in advance (Uniform
Circle Formation). In the literature, the existing algorithmic contributions
are limited to conveniently restricted sets of initial configurations of the
robots and to more powerful robots. The question of whether such simple robots
could deterministically form a uniform circle has remained open. In this paper,
we constructively prove that indeed the Uniform Circle Formation problem is
solvable for any initial configuration in which the robots are in distinct
locations, without any additional assumption (if two robots are in the same
location, the problem is easily seen to be unsolvable). In addition to closing
a long-standing problem, the result of this paper also implies that, for
pattern formation, asynchrony is not a computational handicap, and that
additional powers such as chirality and rigidity are computationally
irrelevant.
