A simple trapezoid can self-tile the plane in many interesting periodic ways. This trapezoid is half of a regular hexagon. A couple more periodic tilings

Here is an octatile, 24-sided, 1.1.1.-1.-1.-3.3.1.1.3.-1.-1.-1.-1.3.1.1.3.-3.-1.-1.1.1.1, shaped like a starship enterprise, and it can self-tesselate the plane. Surprisingly a tiling of forward facing enterprise, with funny purple gaps does something weird. A topological simplification (by applying dual operator twice) makes a heptagonal tiling with triangles. I don't actually have a name for this arrangement. Third image is single dual.

Four octatiling: with stellaocti -1.2^8, -2.3^8, stellatetrus -1.3^4 with squares and spinners -2.3.1^4 and -3.3.2^4  in the gaps. All are topological truncated square tilings. More periodic octatilings