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User:Joseph Myers/Poly-T-tiles
Grünbaum and Shephard, Tilings and Patterns, section 9.4, define poly-T-tiles for an isohedral tiling T as polyforms based on that underlying tiling. This page enumerates related sequences for the 93 types of marked isohedral tilings in the plane. The types of tiling are as enumerated in section 6.2 of Tilings and Patterns. Grünbaum and Shephard originally enumerated types of isohedral tilings (marked and unmarked) in The eighty-one types of isohedral tilings in the plane, Math. Proc. Camb. Phil. Soc. 82 (1977), 177–196; I think the numbering is the same, but have not verified this and have used Tilings and Patterns as my reference in calculating these sequences.
Normal polyforms
The sequences here are for polyforms that must be connected by edges but, unlike the definition in Grünbaum and Shephard, may contain holes (as is the case for the most common definitions of polyforms). For some of the tilings, it matters whether the internal divisions of the tile or only its shape are considered significant; for the sequences here, the internal divisions are significant. (The twelve types with no unmarked representatives cannot meaningfully be considered with shape alone being significant. Of the other tilings, it makes a difference for IH30, IH37, IH38, IH40, IH54, IH56, IH77, IH78, IH81, IH82 when the tilings are considered in their generic forms; it may also make a difference for particular instances of some other tilings.)
If a tiling does not have any reflections or glide reflections as symmetries, the numbers of free and one-sided shapes are the same. If it has glide reflections but not reflections as symmetries, the number of one-sided shapes is twice the number of free shapes. If it does not have any reflections or rotations as symmetries, then no polyforms have any symmetries and the numbers of one-sided and fixed shapes are constant multiples of the number of free shapes.
