# Minimizing the Ratio ofMaximum 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 = 1Trivial. 3. r2 = 1Trivial. 4. r2 = 2Proved by Paul Batemanand Paul Erdös in 1951.

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

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

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

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

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

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

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

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

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

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