## Abstract

Consider a set of *n* finite set of 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.

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## Notes

- 1.
The value of \(\delta \) is assumed to be the same for all robots. However, since the robots are finitely many, nothing changes if each robot has a different \(\delta \): all the executions in this model are compatible with a “global” \(\delta \) that is the minimum of all the “local” \(\delta \)’s.

- 2.
Roughly the same mechanism has been used in [3], with some technical differences.

- 3.
The results in [19] seem to imply that the Uniform Circle Formation problem can be solved for any odd number of robots in \(\mathcal{ASYNC}\). A proof for the \(\mathcal{SSYNC}\) model is given, but its generalization to \(\mathcal{ASYNC}\) is missing some crucial parts. No extended version of the paper has been published, either. Hence, for completeness, we provide our own solutions for the special cases \(n=3\) and \(n=5\).

- 4.
The proof of Lemma 21 also provides a way of constructing such intervals with a compass and a straightedge, and hence by algebraic functions.

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## Acknowledgements

The authors would like to thank the anonymous reviewers for helping improve the readability of the paper, Marc-André Paris-Cloutier for many helpful discussions and insights, and Peter Widmayer and Vincenzo Gervasi for sharing some of the fun and frustrations emerging from investigating this problem. This work has been supported in part by the Natural Sciences and Engineering Research Council of Canada under the Discovery Grants program, by Professor Flocchini’s University Research Chair, and by project PRA_2016_64 *Through the fog* funded by the University of Pisa.

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## Additional information

Some of these results have been presented at the 18th International Conference on Principles of Distributed Systems [15].

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Flocchini, P., Prencipe, G., Santoro, N. *et al.* Distributed computing by mobile robots: uniform circle formation.
*Distrib. Comput.* **30, **413–457 (2017). https://doi.org/10.1007/s00446-016-0291-x

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### Keywords

- Autonomous mobile robots
- Uniform circle formation