Squares in Squares

Summary

5 $s = 2 + {1\over 2}\sqrt 2 = \Nn{2.70710678118654}$Rigid.Proved by Frits Göbelin early 1979. 10 $s = 3 + {1\over 2}\sqrt 2 = \Nn{3.70710678118654}$Found by Frits Göbel in early 1979.Proved by Walter Stromquist in 2003.Explore group 11 $s = {}^{8}🔒 = \Nn{3.87708359002281}$ $s^8 - 20s^7 + 178s^6 - 842s^5 + 1923s^4 - 496s^3 - 6754s^2 + 12420s - 6865 = 0$ Rigid.Found by Walter Trumpin 1979. 17 $s = {}^{18}🔒 = \Nn{4.67553009360455}$ $4775s^{18}-190430s^{17}+3501307s^{16}-39318012s^{15}+300416928s^{14}-1640654808s^{13}+6502333062s^{12}-18310153596s^{11}+32970034584s^{10}-18522084588s^9-93528282146s^8+350268230564s^7-662986732745s^6+808819596154s^5-660388959899s^4+358189195800s^3-126167814419s^2+26662976550s-2631254953=0$ Found by John Bidwellin 1998.Based on packing found by Pertti Hämäläinen in 1980. 18 $s = {7\over 2} + {1\over 2}\sqrt 7 = \Nn{4.82287565553229}$Found by Pertti Hämäläinenin 1980.Pictured alternative with minimal rotated squares found by Mats Gustafsson in 1981. 19 $s = 3 + {4\over 3}\sqrt 2 = \Nn{4.88561808316412}$Found first by Robert Wainwrightin late 1979.Based on packing found by Charles F. Cottingham in early 1979. 26 $s = {7\over 2} + {3\over 2}\sqrt 2 = \Nn{5.62132034355964}$Found by Erich Friedmanin 1997.Unextends the $s(37)$ found by Evert Stenlund in early 1980. 27 $s = 5 + {1\over 2}\sqrt 2 = \Nn{5.70710678118654}$Found by Frits Göbelin early 1979.Explore group 28 $s = {}^{6}🔒 = \Nn{5.82444461667405}$ $s^6-24s^5+212s^4-812s^3+1025s^2+882s-1615=0$ Rigid.Found by David Ellsworthin December 2025, using his modified version of Thomas Schadt's simulated annealing program, starting from randomness. 29 $s = \Nn{5.93383346267692}$Found by Thomas Schadt in December 2025, using a simulated annealing program he wrote, starting from randomness.Similar to the packing found by Thierry Gensane and Philippe Ryckelynck in April 2004, using a computer program they wrote.Optimized by David Ellsworthin December 2025. 37 $s = {}^{8}🔒 = \Nn{6.59861960924436}$ $6s...