Minimizing the Ratio of
Maximum to Minimum Distance

The following pictures show n points in the plane so that the ratio r of maximum distance (shown in red) to minimum distance (shown in blue) is the smallest known.
2.
r2 = 1
Trivial.
3.
r2 = 1
Trivial.
4.
r2 = 2
Proved by Paul Bateman
and Paul Erdös in 1951.

5.
r2 = (3 + √5) / 2 = 2.618+
Proved by Paul Bateman
and Paul Erdös in 1951.
6.
r2 = (5 + √5) / 2 = 3.618+
Proved by Paul Bateman
and Paul Erdös in 1951.
7.
r2 = 4
Proved by Paul Bateman
and Paul Erdös in 1951.

8.
r2 = 1 / (2 - 2cos(π/7)) = 5.048+
Proved by Andras Bezdek
and Ferenc Fodor in 1999.
9.
r2 = 6.600+
Found by David Cantrell
in February 2009.
10.
r2 = 7.713+
Found by David Cantrell
in February 2009.

11.
r2 = 8.222+
Found by David Cantrell
in February 2009.
12.
r2 = 5 + 2√3 = 8.464+
Found by David Cantrell
in February 2009.
13.
r2 = 9.930+
Found by David Cantrell
in February 2009.

14.
r2 = 10.994+
Found by David Cantrell
in February 2009.
15.
r2 = 12.038+
Found by David Cantrell
in February 2009.
16.
r2 = 12.890+
Found by David Cantrell
in February 2009.

17.
r2 = 14.090+
Found by David Cantrell
in February 2009.
18.
r2 = 14.725+
Found by David Cantrell
in February 2009.
19.
r2 = 8 + 4√3 = 14.928+
Found by David Cantrell
in February 2009.

20.
r2 = 16.734+
Found by David Cantrell
in February 2009.
21.
r2 = 17.775+
Found by David Cantrell
in February 2009.
22.
r2 = 19.054+
Found by David Cantrell
in February 2009.

23.
r2 = 20.039+
Found by David Cantrell
in February 2009.
24.
r2 = 14 + 4√3
Found by Charles Audet, Xavier Fournier,
Pierre Hansen, and Frédéric Messine
in January 2010.
25.
r2 = 22.121+
Found by David Cantrell
in February 2009.

26.
r2 = 22.810+
Found by Charles Audet, Xavier Fournier,
Pierre Hansen, and Frédéric Messine
in January 2010.
27.
r2 = 23.083+
Found by Charles Audet, Xavier Fournier,
Pierre Hansen, and Frédéric Messine
in January 2010.
28.
r2 = 24.981+
Found by Charles Audet, Xavier Fournier,
Pierre Hansen, and Frédéric Messine
in January 2010.

29.
r2 = 25.928+
Found by Charles Audet, Xavier Fournier,
Pierre Hansen, and Frédéric Messine
in January 2010.
30.
r2 = 26.879+
Found by Charles Audet, Xavier Fournier,
Pierre Hansen, and Frédéric Messine
in January 2010.

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