Minimizing the Ratio of Maximum to Minimum Distance

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 Ingo Rechenberg
in 2006.

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 Timo Berthold et al
in January 2026.

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 = 17.774+
Found by Timo Berthold et al
in January 2026.

22.

r² = 19.053+
Found by Timo Berthold et al
in January 2026.

23.

r² = 20.039+
Found by David Cantrell
in February 2009.

24.

r² = 14 + 4√3
Found by Charles Audet et al
in January 2010.

25.

r² = 22.121+
Found by David Cantrell
in February 2009.

26.

r² = 22.810+
Found by Charles Audet et al
in January 2010.

27.

r² = 23.083+
Found by Ingo Rechenberg
in 2006.

28.

r² = 24.981+
Found by Charles Audet et al
in January 2010.

29.

r² = 25.924+
Found by Timo Berthold et al
in January 2026.

30.

r² = 26.879+
Found by Charles Audet et al
in January 2010.

31.

r² = 28
Found by Gianluca Costantino
in August 2026.