Are all convex hexagonal space tilings either double lattices or triple lattices?

Question

Let $H$ stand for a convex hexagon with the following property:

It is possible to tile the space with $H$ using only translations and rotations.

There trivially are $H$ that tile the space with a double lattice. They can be constructed with two following methods:

1. By cutting in half any centrally symmetric convex hexagon through its center (red hexagon 1).
2. By creating a convex hexagon whose opposing edges* are parallel and have equal lengths (red hexagon 2).

By opposing edges I mean edges that don't share adjacent edges.

There also are $H$ for which double lattice tiling does not exist, but exists a tiling with the following allowed operations (which I call a triple lattice):

• Translation.
• $120^\circ$ rotation .

In order to construct them, divide a regular hexagon into three equal pieces through its center, in such a way that it's not a parallelogram (red hexagon 3).

However, are there any $H$ whose tilings are not double or triple lattices?

Answer 1

Your first example appears to be a pentagon, rather than a strictly convex hexagon.

The hexagons that can tile the plane monohedrally come in three categories, as described on Wikipedia here:

I'm not entirely sure what you mean by your definitions in the original post, but I think the second type of tiling here does not meet either of your criteria since four orientations are required to tile the plane.