The octiamonds

All polyiamonds of order 8 — figures built from 8 cells. Edge-joined equilateral triangles of the triangular lattice, alternating point-up and point-down.

✓ 15/15 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 A000577 (free)
• PASS · matches A006534 (onesided)
• PASS · matches A001420 (fixed)

drawing…

Free 66 polyiamonds

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 120 polyiamonds

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

Fixed 675 polyiamonds

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