Circles Covering Circles

The following pictures show n unit circles covering the largest known circle (of radius r ).


n=1

r = 1
Trivial.
n=3

r = 2 / √3 = 1.154+
Trivial.
n=4

r = √2 = 1.414+
Trivial.
n=5

r = 1.641+
Proved by Károly Bezdek
in 1983.


n=6

r = 1.798+
Proved by Károly Bezdek
in 1979.
n=7

r = 2
Trivial.
n=8

r = 2.246+
Proved by Gábor Fejes Tóth
in 1996.
n=9

r = √2 + 1 = 2.414+
Proved by Gábor Fejes Tóth
in 1996.


n=10

r = 2.532+
Proved by Gábor Fejes Tóth
in 2005.
n=11

r = 2.631+
Found by Hans Melissen
in 1997.
n=12

r = 2.769+
Found by Hans Melissen
in 1997.
n=13

r = 2.884+
Found by Jeremy Tan
in 2018.


n=14

r = 3.014+
Found by Jeremy Tan
in 2018.
n=15

r = 3.143+
Found by Jeremy Tan
in 2018.
n=16

r = 3.244+
Found by Jeremy Tan
in 2018.
n=17

r = 3.349+
Found by Jeremy Tan
in 2018.


n=18

r = 3.446+
Found by Jeremy Tan
in 2018.
n=19

r = √13 = 3.605+
Found by Jeremy Tan
in 2018.
n=20

r = 3.692+
Found by Jeremy Tan
in 2018.
n=21

r = 3.804+
Found by Jeremy Tan
in 2018.


n=22

r = 3.948+
Found by Jeremy Tan
in 2018.
n=23

r = 4.000+
Found by Jeremy Tan
in 2018.
n=24

r = 4.076+
Found by Jeremy Tan
in 2018.
n=25

r = √6 + √3 = 4.181+
Found by Jeremy Tan
in 2018.


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