The figure shows a hexagon and triangle tiled by six tiles, which are pairwise congruent.

My question is:

What is the smallest number of polygonal tiles that will tile both a regular hexagon and an equilateral triangle in such a way that all the edges of the tiles are parallel to an edge of the tiled figure?

Small print:

• I define a polygonal tile as a figure comprised of:

  A finite set $P$ of at least three points.

  The (straight) lines $X\pi(X)$ for each point $X\in P$ and some cyclic permutation $\pi$ of $P$ such that no point outside $P$ is in distinct lines.

  The region contained by the lines.

• The hexagon and triangle are assumed to include the vertices, edges and interior.

• The figures tiled must be the union of exactly one congruent copy of each tile (though the set of tiles itself is allowed to contain congruent figures). No copy of a tile can share an interior point with any other copy, but may share boundary points with one or more.

• A congruent copy of a tile may be a "flipped" version (e.g. the pair of cyan tiles in the above digram are actually mirror images - this is allowed).