The tetratwigs

All polytwigs of order 4 — figures built from 4 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 4 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 6 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 36 polytwigs

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