Consecutive Rectangles in Similar Rectangles

The following pictures show similar rectangles with short sides 1, 2, 3, ... n with aspect ratio r packed inside another similar rectangle with short side s with the largest known proportion p of coverage.

 1. r = anys = 1p = 1Trivial. 2. r = 1 + √2s = 1 + √2p = 15 – 10 √2 = .857+Found by Maurizio Morandi in June 2015. 3. r = √15 / 3s = √15p = 14/15 = .933+Found by Maurizio Morandi in June 2015.

 4. r = (3 + √73) / 8s = (3 + √73) / 2p = 15 (3√73 – 41) / 256 = .900+Found by Maurizio Morandi in June 2015. 5. r = 2 √15 / 5s = 2 √15p = 11/12 = .916+Found by Maurizio Morandi in June 2015. 6. r = (√273 – 3) / 12s = (3 + √273) / 2p = 91 (47 – √273) / 2904 = .955+Found by Maurizio Morandi in June 2015.

 7. r = 2s = 12p = 35/36 = .972+Found by Maurizio Morandi in June 2015. 8. r = √210 / 14s = √210p = 34/35 = .971+Found by Maurizio Morandi in June 2015. 9. r = (15 + √365) / 14s = (15 + √365) / 2p = 57 (59 – 3 √365) / 98 = .980+Found by Maurizio Morandi in June 2015.

 10. r = √391 / 17s = √391p = 385/391 = .984+Found by Maurizio Morandi in June 2015. 11. r = 3 √57 / 19s = 3 √57p = 506/513 = .986+Found by Maurizio Morandi in June 2015. 12. r = √165 / 11s = 2 √165p = 65/66 = .984+Found by Maurizio Morandi in June 2015.

 13. r = 5 / √19s = 7 + 5 √19p = 91 (262 – 35 √19) / 10082 = .987+Found by Maurizio Morandi in June 2015. 14. r = 3/2s = 32 p = 1015/1024 = .991+Found by Maurizio Morandi in June 2015. 15. r = 3 √66 / 11 – 1s = 11 + 3 √66p = 1240 (65 – 6 √66) / 20339 = .991+Found by Maurizio Morandi in June 2015.

 16. r = 2 √42 / 9s = 6 √42p = 187/189 = .989+Found by Maurizio Morandi in June 2015. 17. r = √1794 / 39s = √1794 p = 595/598 = .994+Found by Maurizio Morandi in June 2015. 18. r = √2135 / 35s = √2135p = 2109/2135 = .987+Found by Maurizio Morandi in June 2015.

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