Circles in Triangles

The following pictures show n unit circles packed inside the smallest equilateral triangle (of side length s ). Most of these have been proved optimal. When m = (n²+n)/2, s = 2(n-1) + 2√3. It is conjectured that one less circle does not change this.

1. 2. 3.
s = 2√3 = 3.464+
Trivial. s = 2 + 2√3 = 5.464+
Trivial. s = 2 + 2√3 = 5.464+
Trivial.

4. 5. 6.
s = 4√3 = 6.928+
Proved by Milano in 1987. s = 4 + 2√3 = 7.464+
Proved by Milano in 1987. s = 4 + 2√3 = 7.464+
Proved by Oler/Groemer in 1961.

7. 8. 9.
s = 2 + 4√3 = 8.928+
Proved by Melissen in 1993. s = 2 + 2√3 + 2√33/3 = 9.293+
Proved by Melissen in 1993. s = 6 + 2√3 = 9.464+
Proved by Melissen in 1993.

10. 11. 12.
s = 6 + 2√3 = 9.464+
Proved by Oler/Groemer in 1961. s = 4 + 2√3 + 4√6/3 = 10.730+
Proved by Melissen in 1993. s = 4 + 4√3 = 10.928+
Proved by Melissen in 1994.

13. 14. 15.
s = 4 + 2√6/3 + 10√3/3 = 11.406+
Found by Melissen in 1993. s = 8 + 2√3 = 11.464+
Found by Erdos/Oler in 1961. s = 8 + 2√3 = 11.464+
Proved by Erdos/Groemer in 1961.

Back to Erich's Packing Center.

1.		2.		3.
s = 2√3 = 3.464+ Trivial.		s = 2 + 2√3 = 5.464+ Trivial.		s = 2 + 2√3 = 5.464+ Trivial.

4.		5.		6.
s = 4√3 = 6.928+ Proved by Milano in 1987.		s = 4 + 2√3 = 7.464+ Proved by Milano in 1987.		s = 4 + 2√3 = 7.464+ Proved by Oler/Groemer in 1961.

7.		8.		9.
s = 2 + 4√3 = 8.928+ Proved by Melissen in 1993.		s = 2 + 2√3 + 2√33/3 = 9.293+ Proved by Melissen in 1993.		s = 6 + 2√3 = 9.464+ Proved by Melissen in 1993.

10.		11.		12.
s = 6 + 2√3 = 9.464+ Proved by Oler/Groemer in 1961.		s = 4 + 2√3 + 4√6/3 = 10.730+ Proved by Melissen in 1993.		s = 4 + 4√3 = 10.928+ Proved by Melissen in 1994.

13.		14.		15.
s = 4 + 2√6/3 + 10√3/3 = 11.406+ Found by Melissen in 1993.		s = 8 + 2√3 = 11.464+ Found by Erdos/Oler in 1961.		s = 8 + 2√3 = 11.464+ Proved by Erdos/Groemer in 1961.