Triangles in Squares

The following pictures show n equilateral triangles with side 1 packed inside the smallest square (of side length s).

1. 2. 3.
s = (√2 + √6) / 4 = .965+
Trivial. s = √6/2 = 1.224+
Trivial. s = √3/2 + √6/4 = 1.478+
Found by Erich Friedman in 1996.

4. 5. 6.
s = 1 + 1 / √3 = 1.577+
Found by Erich Friedman in 1996. s = 1.803+
Found by Erich Friedman in 1996. s = 9 / 2 - 3 √3 / 2 = 1.901+
Found by Erich Friedman in 1996.

7. 8. 9.
s = 2
Found by Erich Friedman in 1996. s = 3 √3 / 2 - 1 / 2 = 2.098+
Found by Erich Friedman in 1996. s = 2.291+
Found by David Cantrell in July 2002.

10. 11. 12.
s = 2.377+
Found by David Cantrell in July 2002. s = 2.497+
Found by David Cantrell in July 2002. s = 7√3/2 - 7/2 = 2.562+
Found by David Cantrell in July 2002.

13. 14. 15.
s = 2.595+
Found by David Cantrell in July 2002. s = 2.726+
Found by David Cantrell in July 2002. s = 27/4 - 9√3/4 = 2.852+
Found by David Cantrell in July 2002.

Back to Erich's Packing Center.

1.		2.		3.
s = (√2 + √6) / 4 = .965+ Trivial.		s = √6/2 = 1.224+ Trivial.		s = √3/2 + √6/4 = 1.478+ Found by Erich Friedman in 1996.

4.		5.		6.
s = 1 + 1 / √3 = 1.577+ Found by Erich Friedman in 1996.		s = 1.803+ Found by Erich Friedman in 1996.		s = 9 / 2 - 3 √3 / 2 = 1.901+ Found by Erich Friedman in 1996.

7.		8.		9.
s = 2 Found by Erich Friedman in 1996.		s = 3 √3 / 2 - 1 / 2 = 2.098+ Found by Erich Friedman in 1996.		s = 2.291+ Found by David Cantrell in July 2002.

10.		11.		12.
s = 2.377+ Found by David Cantrell in July 2002.		s = 2.497+ Found by David Cantrell in July 2002.		s = 7√3/2 - 7/2 = 2.562+ Found by David Cantrell in July 2002.

13.		14.		15.
s = 2.595+ Found by David Cantrell in July 2002.		s = 2.726+ Found by David Cantrell in July 2002.		s = 27/4 - 9√3/4 = 2.852+ Found by David Cantrell in July 2002.