Hexagons in Octagons

Hexagons in Octagons
The following pictures show n regular hexagons with side 1 packed inside the smallest known regular octagon (of side length s).

1.

s = 0.82134+
Trivial.
2.

s = 2√3/(1+√2) = 1.43487+
Trivial.
3.

s = 3+2√6–2√2–2√3 = 1.60645+
Found by Jonathan Viquerat
in May 2026.

4.

s = 1.75630+
Found by Jonathan Viquerat
in May 2026.

5.

s = 1.98707+
Found by Jonathan Viquerat
in May 2026.

6.

s = 2.16435+
Found by Nicolas Campailla
in May 2026.

7.

s = 2.18284+
Found by Jonathan Viquerat
in May 2026.

8.

s = 2.40792+
Found by Jonathan Viquerat
in May 2026.

9.

s = 2.62665+
Found by Jonathan Viquerat
in May 2026.

10.

s = 2.74238+
Found by Jonathan Viquerat
in May 2026.

11.

s = 4√3/(1+√2) = 2.86975+
Found by Jonathan Viquerat
in May 2026.

12

s = 2.92139+
Found by Timo Berthold et al
in May 2026.

13.

s = 3.07201+
Found by Jonathan Viquerat
in May 2026.

14.

s = 3.13475+
Found by Jonathan Viquerat
in May 2026.

15.

s = 3.28379+
Found by Timo Berthold et al
in May 2026.

16.

s = 3.37007+
Found by Timo Berthold et al
in May 2026.

17.

s = 3.47655+
Found by Jake Loyd
in June 2026.

18.

s = 3.56008+
Found by Timo Berthold et al
in May 2026.

19.

s = 5√3/(1+√2) = 3.58719+
Found by Jonathan Viquerat
in May 2026.

20.

s = 3.72603+
Found by Joshua Wolk
in September 2026.

21.

s = 3.80996+
Found by Joshua Wolk
in September 2026.

22.

s = 3.91910+
Found by Joshua Wolk
in September 2026.

23.

s = 3.98431+
Found by Joshua Wolk in September 2026.

24.

s = 4.05051+
Found by Joshua Wolk in September 2026.

25.

s = 4.10656+
Found by Joshua Wolk in September 2026.

26.

s = 4.18241+
Found by Joshua Wolk in September 2026.

27.

s = 4.27772+
Found by Joshua Wolk in September 2026.

28.

s = 4.33169+
Found by Joshua Wolk in September 2026.

29.

s = 4.42547+
Found by Joshua Wolk in September 2026.

30.

s = 4.46421+
Found by Joshua Wolk in September 2026.

31.

s = 4.57396+
Found by Joshua Wolk in September 2026.

32.

s = 4.63613+
Found by Joshua Wolk in September 2026.

33.

s = 4.68419+
Found by Joshua Wolk in September 2026.

34.

s = 4.75153+
Found by Joshua Wolk in September 2026.

35.

s = 4.81554+
Found by Joshua Wolk in September 2026.

36.

s = 4.88534+
Found by Joshua Wolk in September 2026.

37.

s = 4.96714+
Found by Joshua Wolk in September 2026.

38.

s = 5.00301+
Found by Joshua Wolk in September 2026.

39.

s = 5.06731+
Found by Joshua Wolk in September 2026.

40.

s = 5.12027+
Found by Joshua Wolk in September 2026.

41.

s = 5.19962+
Found by Joshua Wolk in September 2026.

42.

s = 5.25700+
Found by Joshua Wolk in September 2026.

43.

s = 5.31519+
Found by Joshua Wolk in September 2026.

44.

s = 5.34048+
Found by Joshua Wolk in September 2026.

45.

s = 5.39089+
Found by Joshua Wolk in September 2026.

46.

s = 5.46113+
Found by Joshua Wolk in September 2026.

47.

s = 5.53171+
Found by Joshua Wolk in September 2026.

48.

s = 5.56469+
Found by Joshua Wolk in September 2026.

49.

s = 5.64565+
Found by Joshua Wolk in September 2026.

50.

s = 5.70090+
Found by Joshua Wolk in September 2026.