Triangles in Triangles

The following pictures show n equilateral triangles with side 1 packed inside the smallest known equilateral triangles (of side length s ). Most of these have not been proven.


1.
           2.
           3.
s = 1
Trivial.
s = 2
Trivial.
s = 2
Trivial.


4.
           5.
           6.
s = 2
Trivial.
s = 1 + √3 = 2.732+
Proved by Erich Friedman
in 1997.
s = 13/8 + 3√13/8 = 2.977+
Found by Maurizio Morandi
in August 2008.


7.
           8.
           9.
s = 3
Found by Erich Friedman
in 1997.
s = 3
Found by Erich Friedman
in 1997.
s = 3
Trivial.


10.
           11.
           12.
s = 3.5
Found by Erich Friedman
in 1997.
s = 9/4 + 9√21/28 = 3.722+
Found by Erich Friedman
in 1997.
s = 2 + 2cos(π/9) = 3.879+
Found by David W. Cantrell
in July 2007.


13.
           14.
           15.
s = 3.992+
Found by Maurizio Morandi
in May 2008.
s = 4
Trivial
s = 4
Trivial


16.
           17.
           18.
s = 4
Trivial
s = 4.465+
Found by Maurizio Morandi
in May 2008.
s = 4.5
Found by Erich Friedman
in 1997.


19.
           20.
           21.
s = 14/3 = 4.666+
Found by Maurizio Morandi
in May 2008.
s = 33/8 + 9√21/56 = 4.861+
Found by Maurizio Morandi
in May 2008.
s = 4.923+
Found by Maurizio Morandi
in May 2008.


22.
           23.
           24.
s = 4.996+
Found by Maurizio Morandi
in May 2008.
s = 5
Trivial
s = 5
Trivial


25.
           26.
           27.
s = 5
Trivial
s = 5.408+
Found by Maurizio Morandi
in May 2008.
s = 5.5
Found by Erich Friedman
in 1997.


28.
           29.
           30.
s = 5.5
Found by Erich Friedman
in 1997.
s = 17/3 = 5.666+
Found by David W. Cantrell
in July 2007.
s = 23/4 = 5.75
Found by Maurizio Morandi
in May 2008.


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