Triangles Covering Squares

The following pictures show n unit equilateral triangles covering the largest known square of side s.


1.            2.            3.
s = 2√3 - 3= .464+
Trivial.
s = √3 - 1 = .732+
Found by David Cantrell
in August 2002.
s = 4√3 - 6 = .928+
Found by David Cantrell
in August 2002.


4.            5.            6.
s = (3√3 - 3) / 2 = 1.098+
Found by Erich Friedman
in August 2002.
s = (15√3 - 21) / 4 = 1.245+
Found by David Cantrell
in July 2005.
s = (107√3 - 85) / 71 = 1.413+
Found by Maurizio Morandi
in April 2009.


7.            8.            9.
s = (7√3 - 6) / 4 = 1.531+
Found by Maurizio Morandi
in April 2009.
s = (12 - 5√3) / 2 = 1.669+
Found by Maurizio Morandi
in April 2009.
s = (139 - 69√3) / 11 = 1.771+
Found by Maurizio Morandi
in April 2009.


10.            11.            12.
s = 5 / √3 - 1 = 1.886+
Found by Maurizio Morandi
in April 2009.
s = √3 + 1/4 = 1.982+
Found by Maurizio Morandi
in April 2009.
s = (7√3 + 15) / 13 = 2.086+
Found by Maurizio Morandi
in April 2009.


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