The hexatwigs

All polytwigs of order 6 — figures built from 6 cells. Unit line segments of the hexagonal (honeycomb) lattice joined end to end; only three segments meet at each vertex.

✓ 14/14 verification checks passed

• PASS · all forms have n cells
• PASS · all forms connected
• PASS · fixed forms distinct
• PASS · orbit count: free -> fixed
• PASS · orbit count: free -> one-sided
• PASS · orbit count: one-sided -> fixed
• PASS · free <= one-sided <= fixed
• PASS · canonical form idempotent
• PASS · canonical form group-invariant
• PASS · encoding round-trips
• PASS · free forms expand to the fixed set
• PASS · free forms expand to the one-sided set
• PASS · matches Polyform Puzzler (free)
• PASS · matches Polyform Puzzler (onesided)

drawing…

Free 27 polytwigs

Two figures are the same free form if one can be turned into the other by translation, rotation or reflection. This is the usual count — the one that gives 12 pentominoes.

One-Sided 49 polytwigs

One-sided forms may be translated and rotated but not flipped over, so a chiral figure and its mirror image are counted separately.

Fixed 281 polytwigs

Fixed forms may only be translated. Every distinct rotation and reflection is counted as its own figure.