Circles in Triangles

The following pictures show n circles with the largest possible sum of radii packed inside a unit equilateral triangle.

 1. 2. 3. Σ r = √3/6 = .288+Trivial. Σ r = 2√3/9 = .384+Trivial. Σ r = (3√3 – 3) / 4 = .549+Trivial.

 4. 5. 6. Σ r = .617+Found by David W. Cantrellin June 2011. Σ r = .693+Found by David W. Cantrellin June 2011. Σ r = 6 – 3√3 = .803+Trivial.

 7. 8. 9. Σ r = .855+Found by David W. Cantrellin June 2011. Σ r = .912+Found by David W. Cantrellin June 2011. Σ r = .976+Found by David W. Cantrellin June 2011.

 10. 11. 12. Σ r = (15 – 5√3) / 6 = 1.056+Trivial. Σ r = 1.098+Found by David W. Cantrellin June 2011. Σ r = 1.143+Found by David W. Cantrellin June 2011.

 13. 14. 15. Σ r = 1.191+Found by David W. Cantrellin June 2011. Σ r = 1.243+Found by David W. Cantrellin June 2011. Σ r = (60 – 15√3) / 26 = 1.308+Trivial.

 16. 17. 18. Σ r = 1.343+Found by David W. Cantrellin June 2011. Σ r = 1.381+Found by David W. Cantrellin June 2011. Σ r = 1.420+Found by David W. Cantrellin June 2011.

 19. 20. 21. Σ r = 1.462+Found by David W. Cantrellin June 2011. Σ r = 1.505+Found by David W. Cantrellin July 2011. Σ r = (105 – 21√3) / 44 = 1.559+Trivial.

 22. 23. 24. Σ r = 1.589+Found by David W. Cantrellin July 2011. Σ r = 1.622+Found by David W. Cantrellin July 2011. Σ r = 1.655+Found by David W. Cantrellin July 2011.

 25. 26. 27. Σ r = 1.690+Found by David W. Cantrellin July 2011. Σ r = 1.727+Found by David W. Cantrellin July 2011. Σ r = 1.763+Found by David W. Cantrellin July 2011.

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