Disk covering problem

Unsolved problem in mathematics:
What is the smallest real number 
  
    
      
        r
        (
        n
        )
      
    
    {\displaystyle r(n)}
  
 such that 
  
    
      
        n
      
    
    {\displaystyle n}
  
 disks of radius 
  
    
      
        r
        (
        n
        )
      
    
    {\displaystyle r(n)}
  
 can be arranged in such a way as to cover the unit disk?
<a href="/facts/List_of_unsolved_problems_in_mathematics/RroxGPpP">(more unsolved problems in mathematics)</a>

<p>The disk <a href="/facts/Covering_problem/4BIHqE9K">covering problem</a> asks for the smallest <a href="/facts/Real_number/R02gw5Pb">real number</a>  
  
    
      
        r
        (
        n
        )
      
    
    {\displaystyle r(n)}
  
 such that 
  
    
      
        n
      
    
    {\displaystyle n}
  
 <a href="/facts/Disk_(mathematics)/DgMVr97H">disks</a> of radius 
  
    
      
        r
        (
        n
        )
      
    
    {\displaystyle r(n)}
  
 can be arranged in such a way as to cover the <a href="/facts/Unit_disk/phrUu7yC">unit disk</a>. Dually, for a given radius <i>ε</i>, one wishes to find the smallest integer <i>n</i> such that <i>n</i> disks of radius <i>ε</i> can cover the unit disk.
</p><p>The best solutions known to date are as follows.
</p>
<table><tbody><tr><th>n</th><th>r(n)</th><th>Symmetry</th></tr><tr><td>1</td><td>1</td><td>All</td></tr><tr><td>2</td><td>1</td><td>All (2 stacked disks)</td></tr><tr><td>3</td><td>                                          3                                    /                2              {\displaystyle {\sqrt {3}}/2}   = 0.866025...</td><td>120°, 3 reflections</td></tr><tr><td>4</td><td>                                          2                                    /                2              {\displaystyle {\sqrt {2}}/2}   = 0.707107...</td><td>90°, 4 reflections</td></tr><tr><td>5</td><td>0.609382... <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A133077</td><td>1 reflection</td></tr><tr><td>6</td><td>0.555905... <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A299695</td><td>1 reflection</td></tr><tr><td>7</td><td>                    1                  /                2              {\displaystyle 1/2}   = 0.5</td><td>60°, 6 reflections</td></tr><tr><td>8</td><td>0.445041...</td><td>~51.4°, 7 reflections</td></tr><tr><td>9</td><td>0.414213...</td><td>45°, 8 reflections</td></tr><tr><td>10</td><td>0.394930...</td><td>36°, 9 reflections</td></tr><tr><td>11</td><td>0.380083...</td><td>1 reflection</td></tr><tr><td>12</td><td>0.361141...</td><td>120°, 3 reflections</td></tr></tbody></table>

Disk covering problem open-in-new

Disk covering problem