Hexagons in Dominoes

The following pictures show n unit hexagons packed inside the smallest known domino (with short side length s).

1.

s = √3 = 1.73205+
Trivial. 		           2.

s = 1.8946+
Found by Jonathan Viquerat in April 2026. 		           3.

s = 6–2√3 = 2.53589+
Found by Jonathan Viquerat in April 2026.

4.

s = 22–11√3 = 2.94744+
Found by Jonathan Viquerat in April 2026. 		           5.

s = 3.24870+
Found by Yanis Gomes
in October 2026. 		           6.

s = 21(15–2√3)/71 = 3.41202+
Found by Jonathan Viquerat in April 2026.
7.

s = 3.48722+
Found by Bhavithran Ananthan in July 2026. 		           8.

s = 3.6906+
Found by Jonathan Viquerat in April 2026. 		           9.

s = (36+5√3)/11 = 4.06002+
Found by Jonathan Viquerat in April 2026.
10.

s = 9(2√3–3) = 4.17691+
Found by Jonathan Viquerat in April 2026. 		           11.

s = 4.5036+
Found by Jonathan Viquerat in April 2026. 		           12.

s = 41(18–5√3)/83 = 4.6136+
Found by Jonathan Viquerat in April 2026.
13.

s = 4.8137+
Found by Jonathan Viquerat in April 2026. 		           14.

s = 25(6–√3)/22 = 4.84994+
Found by Jonathan Viquerat in April 2026. 		           15.

s = 56(27–4√3)/227 = 4.95163+
Found by Jonathan Viquerat in April 2026.
16.-17.

s = 3√3 = 5.19615+
Trivial. 		           18.

s = 31(4–√3)/13 = 5.40818+
Found by Jonathan Viquerat in April 2026. 		           19.

s = 5.65220+
Found by Joshua Wolk in September 2026.
20.

s = 5.8012+
Found by Jonathan Viquerat in April 2026. 		           21.

s = 5.85640+
Found by Thomas Greenleaf in June 2026. 		           22.

s = 6.06942+
Found by Aapo Lipponen
in October 2026.
23.

s = 6.18125+
Found by Joshua Wolk in September 2026. 		           24.

s = 6.19594+
Found by Joshua Wolk in September 2026. 		           25.

s = 477/46–53√3/23 = 6.37831+
Found by Yanis Gomes
in October 2026.
26.

s = 6.40192+
Found by Joshua Wolk in September 2026. 		           27.

s = 6.49866+
Found by Joshua Wolk in September 2026. 		           28.

s = 1391/157–214√3/157 = 6.49898+
Found by Yanis Gomes
in October 2026.
29.

s = 6.83667+
Found by Joshua Wolk in September 2026. 		           30.

s = 6.92820+
Found by Joshua Wolk in September 2026. 		           31.

s = 6.95439+
Found by Yanis Gomes
in October 2026.
32.

s = 6.98004+
Found by Joshua Wolk in September 2026. 		           33.

s = 7.24872+
Found by Joshua Wolk in September 2026. 		           34.

s = 7.42562+
Found by Joshua Wolk in September 2026.
35.

s = 172/13–43√3/13 = 7.50167+
Found by Yanis Gomes
in October 2026. 		           36.

s = 7.55423+
Found by Yanis Gomes
in October 2026. 		           37.

s = 7.70206+
Found by Joshua Wolk in September 2026.
38.

s = 1512/131–288√3/131 = 7.73411+
Found by Yanis Gomes
in October 2026. 		           39.

s = 329√3/37–282/37 = 7.77958+
Found by Yanis Gomes
in October 2026. 		           40.

s = 7.78193+
Found by Joshua Wolk in September 2026.
41.

s = 7.94217+
Found by Yanis Gomes
in October 2026. 		           42.

s = 7.94895+
Found by Joshua Wolk in September 2026. 		           43.

s = 7.95390+
Found by Joshua Wolk in September 2026.
44.

s = 8.15857+
Found by Joshua Wolk in September 2026. 		           45.

s = 368√3/61–138/61 = 8.18679+
Found by Yanis Gomes
in October 2026. 		           46.

s = 8.47613+
Found by Yanis Gomes
in October 2026.
47.

s = 8.60495+
Found by Joshua Wolk in September 2026. 		           48.-50.

s = 8.66025+
Found by Joshua Wolk in September 2026.