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.
I found I can make the funny tiling from a trihexagonal tiling, by removing one edge between a triangle and hexagon, which expands each hexagon into heptagons. And apply dual twice adjusts the geometry like the enterprise tiling.
It can also be made from a truncated square tiling, by pinching pairs of vertices together, reducing squares to triangles and octagons to heptagons.
Even better, I made equilateral forms, first with a heptagonal dodecatile, 2.1.3.1.1.3.1, and triangles 4^3. It can be decomposed into a snub square tiling.
And another with hexatiles: a trapezoid 2.1.0.1.2.0.0, and triangles, 2^3. It can be decomposed into a triangular tiling.
And another with hexatiles: a trapezoid 2.1.0.1.2.0.0, and triangles, 2^3. It can be decomposed into a triangular tiling.
One more example, an octadecatile, -1.-1.7.4.-2.4.7, concave. Maybe asymmetric solutions as well with same topology??
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