Squares Covering Circles

The following pictures show n unit squares covering the largest known circle (of radius r).

1.

          

2.

r = 1/2 = .500
Trivial.
r = 2 - 2 = .585+
Proved by Trevor Green in August 1999.

3.

          

4.

r = .794+
Proved by Trevor Green in August 1999.
r = 1
Proved by Trevor Green in August 1999.

5.

          

6.

r = 1.018+
Found by Trevor Green in September 1999.
r = 1.126+
Found by Erich Friedman in August 1999.

7.

          

8.

r = 1.239+
Found by Found by David Cantrell in July 2002.
r = 1.357+
Found by David Cantrell in July 2002.

9.

          

10.

r = 3/2 =1.500
Found by Trevor Green in August 1999.
r = 1.546+
Found by Trevor Green in August 1999.

11.

          

12.

r = 1.604+
Found by Erich Friedman in August 1999.
r = 1.701+
Found by David Cantrell in August 2002.

13.

          

14.

r = 1.758+
Found by Trevor Green in August 1999.
r = 1.883+
Found by David Cantrell in July 2002.

15.

          

16.

r = 1.986+
Found by Trevor Green in August 1999.
r = 2.0004+
Found by David Cantrell in August 2002.

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