The pentahexes

All polyhexes of order 5 — figures built from 5 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 22 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 33 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 186 polyhexes

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