The ordinary [circle packing problem](https://en.wikipedia.org/wiki/Circle_packing) in the variant with equal radii asks for the largest radius $r\_{max}$ that allows placing $n$ non-overlapping circles with radius $r\_{max}$ e.g. in the unit square, in the unit circle or, on the unit sphere ([Tammes' problem](http://en.wikiyy.com/wiki/Tammes_problem)).


Now, I would like to solve a somehow opposite problem:


  


> 
> **Question:**
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> 
> given a number $n\in\mathbb{N}$, what is the smallest radius $r\_{min}\in\mathbb{R}^+$ that permits a non-overlapping, rigid placement of $n$ circles with radius $r\_{min}$ in the unit square, or in the unit circle or, on the unit sphere?
> 
> 
> 


Under a rigid configuration I understand a configuration, where every open halfplane, resp. hemisphere defined by a hyperplane through a circle's center contains at least one contact point with another circle or, with the boundary of the containing region.


Are there already algorithms and/or theoretical results available for that problem?



Name and Algorithms for a Sparsest Circle Packing
An overview of this "sparse packing problem" is given by Matthew Kahle in [Sparse locally-jammed disk packings](https://pdfs.semanticscholar.org/2827/b52f9626d1f09b918208d793a7ea4975439b.pdf). Quite generally, $r\_{\rm min}=O(1/n)$, so the fraction of the unit square or unit disc covered by the $n$ circles goes to 0 as their number $n$ goes to infinity. In a disc you just line up the circles along the perimeter, and this works for any smooth boundary. For a polygon, such as a square, the construction is more involved to avoid loose circles at the corners.


![](https://ilorentz.org/beenakker/MO/sparsediscs.png)


These configurations are called "rigged" or "jammed", meaning that no single circle can move, each is held in place by its neighbors. (It is possible for several circles to move together, they are not "collectively jammed".) Kahle also discusses Monte Carlo algorithms to find such jammed sparse packings.


