15 of 15 added 6 characters in body

Another dissection puzzle

enter image description here

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 distinct lines do not intersect except at mutual end points.

    The region contained by the lines.

  • The hexagon, triangle and tiles 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 the copy of any other tile, but may share boundary points with one or more.

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