Circles in Hexagons

The following pictures show n unit circles packed inside the smallest hexagon (of side length s.)

1. 2. 3.
s = 2/3 = 1.154+
Trivial. s = 1 + 2/3 = 2.154+
Trivial. s = 4/3 = 2.309+
Trivial.

4. 5. 6.
s = 2/3 + 4/7 = 2.666+
Found by Shahriar Manzoor, April 2002. s = 2.999+
Found by Erich Friedman, 1999. s = 2 + 2/3 = 3.154+
Found by Erich Friedman, 1999.

7. 8. 9.
s = 2 + 2/3 = 3.154+
Found by Erich Friedman, 1999. s = 3.709+
Found by David Cantrell, July 2002. s = 20/7 + 2/3 = 4.011+
Found by David Cantrell, July 2002.

10. 11. 12.
s = 4.119+
Found by David Cantrell, July 2002. s = 2 + 4/3 = 4.309+
Found by Erich Friedman, 1999. s = 2 + 4/3 = 4.309+
Found by Erich Friedman, 1999.

13. 14. 15.
s = 8/3 = 4.618+
Found by Erich Friedman, 1999. s = 2 + 2/3 + 4/7 = 4.666+
Found by David Cantrell, July 2002. s = 4.961+
Found by David Cantrell, January 2006.

16. 17. 18.
s = 4.999+
Found by David Cantrell, July 2002. s = 5.133+
Found by David Cantrell, January 2006. s = 4 + 2/3 = 5.154+
Found by Erich Friedman, 1999.

Back to Erich's Packing Center.

1.		2.		3.
s = 2/3 = 1.154+ Trivial.		s = 1 + 2/3 = 2.154+ Trivial.		s = 4/3 = 2.309+ Trivial.

4.		5.		6.
s = 2/3 + 4/7 = 2.666+ Found by Shahriar Manzoor, April 2002.		s = 2.999+ Found by Erich Friedman, 1999.		s = 2 + 2/3 = 3.154+ Found by Erich Friedman, 1999.

7.		8.		9.
s = 2 + 2/3 = 3.154+ Found by Erich Friedman, 1999.		s = 3.709+ Found by David Cantrell, July 2002.		s = 20/7 + 2/3 = 4.011+ Found by David Cantrell, July 2002.

10.		11.		12.
s = 4.119+ Found by David Cantrell, July 2002.		s = 2 + 4/3 = 4.309+ Found by Erich Friedman, 1999.		s = 2 + 4/3 = 4.309+ Found by Erich Friedman, 1999.

13.		14.		15.
s = 8/3 = 4.618+ Found by Erich Friedman, 1999.		s = 2 + 2/3 + 4/7 = 4.666+ Found by David Cantrell, July 2002.		s = 4.961+ Found by David Cantrell, January 2006.

16.		17.		18.
s = 4.999+ Found by David Cantrell, July 2002.		s = 5.133+ Found by David Cantrell, January 2006.		s = 4 + 2/3 = 5.154+ Found by Erich Friedman, 1999.