It is a well known problem, for example for convex pentagons they are 15 known tiles, which solve the problem, for hexagons they are 3 known tiles, which solve the problem etc. However, it is supposed that a tile can be used from both sides (also a mirrored tile can be used as well). The real tile however (which one uses in a bathroom :-) is only one-sided. The question is therefore, what is the solution to this problem if a tile can ONLY be used one-sided, i.e. only rotated and shifted? In general I would like know what is the solution for triangles, quads, convex/non-convex pentagons, convex/non-convex hexagons and higher polygons. Any answer or link is appreciated!
Why I want to answer this question: MathStackExchange is full of agile people who love to close new questions :-), therefore I would like to explain my motivation for this question. I do numerical mathematics, however not discrete mathematics. I have explained this problem to kids, so I would like to understand the problem myself more (i.e. to know what is known). I do not need it for any of my work.
All triangles can tile without mirroring:
as can quads
and type 1 pentagons.
In fact, the triangle and quad tilings are special cases of the type 1 pentagon tiling where one or two of the sides have length zero.
The pentagon type 2 is complicated. The tiling pattern uses reflected tiles, but there are enough degrees of freedom to make the tile symmetric. This can be done in several ways, but I will have to look into that when I have more time. For reference, here is the standard type 2 tiling:
The pentagon types 3 to 6 can all tile without mirroring.
Type 3:
Type 4:
Type 5:
Type 6:
The type 7 pentagon tiles only in one way which needs reflection, and its one degree of freedom does not make it symmetric at any point.
The type 8 pentagon tiles only in one way which needs reflection, but its degree of freedom can be adjusted until the tile becomes symmetric:
The type 9 pentagon tiles only in one way which needs reflection, and its one degree of freedom does not make it symmetric at any point.
The type 10 pentagon tiles only in one way which needs reflection, but its degree of freedom can be adjusted until the tile becomes symmetric:
The pentagons of types 11 to 15 all tile only in one way which needs reflection. They are never symmetric.
The convex hexagons come in 3 types.
Type 1 tiles without reflections:
As with pentagons, type 2 is complicated. The tiling pattern needs reflection, but the tile can be made symmetric, possibly in several ways. For reference, here is the general tiling pattern:
The type 3 hexagon tiles without reflection.
There are no convex higher polygons that tile. The non-convex polygons would probably need at least as long an answer as this already is.