Squares in Squares: Rigid packings

This is a list of rigid packings. Most are the best known, but in cases where they are inoptimal, they are shown alongside the best known. For the main list, see Squares in Squares.

A packing is rigid when it cannot be continuously transformed into any other valid packing without changing the size of its enclosing square.

Where the word "alternative" is used, this designates an alternative packing, which is enclosed within the same-sized bounding square as another packing, but cannot be reached by continuously translating and/or rotating the squares in that packing. When the packing can be reached by such continuous transformations, the word "rearrangement" is used.

The property of rigidity is rare among nontrivial best known packings. Even the known recurring patterns only extend finitely before becoming inoptimal.

$s = 2 + { 1\over 2}\sqrt 2 = \Nn{2.70710678118654}$
Proved optimal by Frits Göbel
in early 1979.

11.

$s = {}^{8}🔒 = \Nn{3.87708359002281}$ $s^8 - 20s^7 + 178s^6 - 842s^5 + 1923s^4 - 496s^3 - 6754s^2 + 12420s -6865 = 0$
Found by Walter Trump
in 1979.

18.
$s = 2 + 2 \sqrt 2 = \Nn{4.82842712474619}$ Side length found by Frits Göbel in early 1979. Not optimal overall. Rigid alternative found by David Ellsworth in December 2024.		$s = {7\over 2} + {1\over 2}\sqrt 7 = \Nn{4.82287565553229}$ Found by Pertti Hämäläinen in 1980. Beats the rigid $s(18)$, but is not rigid.

28.

$s = 3 + 2 \sqrt 2 = \Nn{5.82842712474619}$
Side length found by Frits Göbel
in early 1979.
Explore group
Rigid alternative with minimal rotated squares found by David Ellsworth in June 2023.

40.

$s = 4 + 2 \sqrt 2 = \Nn{6.82842712474619}$
Found by Frits Göbel
in early 1979.
Explore group

50.
$s = {11\over 2}\sqrt 2 = \Nn{7.77817459305202}$ Found by David Ellsworth in December 2024. Not optimal overall.		$s = {}^{8}🔒 = \Nn{7.59861960924436}$ $6s^4-(232+64\sqrt{2})s^3+(2718+1042\sqrt{2})s^2-(12700+5550\sqrt{2})s+21371+10074\sqrt{2}=0$ $36s^8-2784s^7+78248s^6-1146800s^5+9944448s^4-53242000s^3+173869324s^2-319180600s+253748689=0$ Found by David W. Cantrell in September 2002, by adding an "L" to the $s(37)$ he found. Improves upon the $s(50)$ found by Evert Stenlund in early 1980. Beats the rigid $s(50)$, but is not rigid.

52.

$s = 7 + {1\over 2}\sqrt 2 = \Nn{7.70710678118654}$
Side length found by Frits Göbel
in early 1979.
Rigid alternative with minimal rotated squares found by David W. Cantrell
in 2005.

149.

$s = 12 + {1\over 2}\sqrt 2 = \Nn{12.70710678118654}$
Extends the $s(52)$ found by
Frits Göbel in early 1979.
Rigid alternative with minimal rotated squares found by David Ellsworth in November 2024, based on the $s(52)$ found by David W. Cantrell in 2005.

200.
$s = {21\over 2}\sqrt 2 = \Nn{14.84924240491749}$ Found by David Ellsworth in December 2024. Not optimal overall.		$s = 9 + 4 \sqrt 2 = \Nn{14.65685424949238}$ Adds two "L"s to the $s(148)$ that continues a pattern found by Frits Göbel in early 1979. Beats the rigid $s(200)$, but is not rigid.

203.
$s = 12 + 2 \sqrt 2 = \Nn{14.82842712474619}$ Combines two copies of the rigid $s(18)$ and one of the rigid $s(28)$ found by David Ellsworth in December 2024 and June 2023, respectively. Not optimal overall.		$s = 7 + {11\over 2}\sqrt 2 = \Nn{14.77817459305202}$ Found by David Ellsworth in December 2024. Based on the $s(233)$ that continues a pattern found by Frits Göbel in early 1979. Beats the rigid $s(203)$, but is not rigid.

233.
$s = 13 + 2 \sqrt 2 = \Nn{15.82842712474619}$ Combines two copies of the rigid $s(18)$ found by David Ellsworth in December 2024 and one of the $s(40)$ found by Frits Göbel in early 1979. Not optimal overall.		$s = 8 + {11\over 2}\sqrt 2 = \Nn{15.77817459305202}$ Continues a pattern found by Frits Göbel in early 1979. Explore group Beats the rigid $s(233)$, but is not rigid.

265.
$s = 14 + 2 \sqrt 2 = \Nn{16.82842712474619}$ Combines three copies of the rigid $s(28)$ found by David Ellsworth in June 2023. Not optimal overall.		$s = 9 + {11\over 2}\sqrt 2 = \Nn{16.77817459305202}$ Continues a pattern found by Frits Göbel in early 1979. Explore group Beats the rigid $s(265)$, but is not rigid.

296.

$s = 17 + {1\over 2}\sqrt 2 = \Nn{17.70710678118654}$
Extends the $s(52)$ found by
Frits Göbel in early 1979.
Rigid alternative with minimal rotated squares based on the $s(52)$ found by David W. Cantrell in 2005.

299.
$s = 15 + 2 \sqrt 2 = \Nn{17.82842712474619}$ Combines two copies of the rigid $s(28)$ found by David Ellsworth in June 2023 with one of the $s(40)$ found by Frits Göbel in early 1979. Not optimal overall.		$s = {33\over 2} + {1\over 2}\sqrt 7 = \Nn{17.82287565553229}$ Extends the $s(86)$ found by Erich Friedman in 1997. Beats the rigid $s(299)$, but is not rigid.

335.

$s = 16 + 2 \sqrt 2 = \Nn{18.82842712474619}$
Combines the rigid $s(28)$ found by
David Ellsworth in June 2023 with
two copies of the $s(40)$ found by
Frits Göbel in early 1979.

373.

$s = 17 + 2 \sqrt 2 = \Nn{19.82842712474619}$
Combines three copies of the $s(40)$ found by Frits Göbel in early 1979.

493.

$s = 22 + {1\over 2}\sqrt 2 = \Nn{22.70710678118654}$
Extends the $s(52)$ found by
Frits Göbel in early 1979.
Rigid alternative with minimal rotated squares based on the $s(52)$ found by David W. Cantrell in 2005.

740.

$s = 27 + {1\over 2}\sqrt 2 = \Nn{27.70710678118654}$
Extends the $s(52)$ found by
Frits Göbel in early 1979.
Rigid alternative with minimal rotated squares based on the $s(52)$ found by David W. Cantrell in 2005.

1037.

$s = 32 + {1\over 2}\sqrt 2 = \Nn{32.70710678118654}$
Extends the $s(52)$ found by
Frits Göbel in early 1979.
Rigid alternative with minimal rotated squares based on the $s(52)$ found by David W. Cantrell in 2005.

1044.

$s = 30 + 2 \sqrt 2 = \Nn{32.82842712474619}$
Combines five copies of the $s(40)$ found by Frits Göbel in early 1979.

1384.

$s = 37 + {1\over 2}\sqrt 2 = \Nn{37.70710678118654}$
Extends the $s(52)$ found by
Frits Göbel in early 1979.
Rigid alternative with minimal rotated squares based on the $s(52)$ found by David W. Cantrell in 2005.

1781.

$s = 42 + {1\over 2}\sqrt 2 = \Nn{42.70710678118654}$
Extends the $s(52)$ found by
Frits Göbel in early 1979.
Rigid alternative with minimal rotated squares based on the $s(52)$ found by David W. Cantrell in 2005.