The tetrahexes

All polyhexes of order 4 — figures built from 4 cells. Edge-joined regular hexagons of the honeycomb tiling; the cell centres form a triangular lattice.

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• 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 A000228 (free)
• PASS · matches A006535 (onesided)
• PASS · matches A001207 (fixed)

drawing…

Free 7 polyhexes

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 10 polyhexes

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

Fixed 44 polyhexes

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