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) = 1.43488+
Trivial.
3.

s = 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 = 2.92141+
Found by Jonathan Viquerat
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 = 3.28390+
Found by Jonathan Viquerat
in May 2026.

16.

s = 3.37010+
Found by Jonathan Viquerat
in May 2026.

17.

s = 3.47696+
Found by Jonathan Viquerat
in May 2026.

18.

s = 3.56009+
Found by Jonathan Viquerat
in May 2026.

19.

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

20.

s = 3.72606+
Found by Jonathan Viquerat
in May 2026.