If two convex polygons tile the plane, how many sides can one of them have?

The set of convex polygons which tile the plane is, as of $$2017$$, known: it consists of all triangles, all quadrilaterals, $$15$$ families of pentagons, and three families of hexagons. Euler's formula rules out strictly convex $$n$$-gons with $$n\ge 7$$. (The pentagonal case is by far the most difficult one.)

I am interested in pairs of convex polygons that can collectively tile the plane. Specifically, I am curious how many sides can be in a polygon which is part of such a tiling.

Here are some conditions to impose on such a tiling, from weakest to strongest:

• There is at least one copy of each tile. (Without this condition, one can trivially take a pair consisting of a tiling polygon and any other convex polygon, and just never use the latter shape.)

• There are at least $$k$$ copies of each tile.

• There are infinitely many of each tile.

• Every tile borders a tile of the other type.

• The tiling is $$2$$-isohedral, i.e., every tile can be carried to any other tile of the same shape by a symmetry of the tiling.

Each of these conditions implies those above it.

In the weakest case, the number of sides is unbounded, as exhibited by the following example:

(The tiling is constructed by decomposing "wedges" of central angle $$2\pi/N$$ into congruent isosceles triangles, and then combining the central triangles to yield an $$N$$-gon in the center.)

Requiring at least $$k$$ of each tile still yields arbitrarily high numbers of sides, by taking the above construction for $$N=M\cdot k$$ and subdividing the $$N$$-gon into $$k$$ "wedges" which are $$(M+2)$$-gonal.

On the other end of the spectrum, I have found a $$2$$-isohedral tiling using regular $$18$$-gons, shown below:

After consulting this paper, it seems that the tiling pictured above is of type $$4_2 18_{12}-1\text{a}\ \text{MN}\ \text{p}6\text{m}$$ in their classification scheme (shown at the bottom of page 109); there are no $$2$$-isohedral tilings which allow for any higher number of contacts between different shapes, although type $$3_1 18_{12}-1\text{a}\ \text{MN}\ \text{p}6\text{m}$$ also works (and can be obtained from the above construction by cutting each kite-shaped tile in two). Thus, it is maximal among $$2$$-isohedral tilings.

What are the maximal tilings under weaker conditions? The maximal number of sides under each successively stronger restriction is a weakly decreasing sequence which goes $$\infty, \infty, ?, ?, 18$$. So far, I have no bounds on the missing two terms except that they are each at least $$18$$.

Some notes on this problem:

• It is not necessarily the case that one of the tiles may tile the plane on its own; see this math.SE question for a counterexample.

• If convexity is relaxed for either piece, the number of sides is unbounded even in the $$2$$-isohedral case (in fact, both pieces can simultaneously have arbitrarily many sides).

Edit: Crossposted to Math Overflow here.