Can one always tile the plane with a regular polygon + a single other shape

euclidean-geometrygeometrytiling

We know that regular triangles, squares and hexagons can tile the plane without leaving any "hole".

However, I've noticed that many regular polygons can tile the plane if we allow for a single type of "hole" (i.e., another shape) to be present.

The following image contains an example with pentagons and rhombi:

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What we obtain in this case is not a periodic tiling, but rather an aperiodic one: Still, we are able to tile the plane with these two shapes.

Another example, this time with decagons + "concave hexagons":

enter image description here

My question is: Can we always tile the plane by combining a regular polygon and a single other shape?

I'm also interested in the extension to star polygons.

Edit

As pointed out in the comments, the way I asked the question was imprecise. I guess that we should add the additional constraint that no "hole" can be in contact with another "hole", otherwise there will be trivial solutions to the problem.

Best Answer

Yes, it is always possible.

Suppose $n>3$ is odd. Then we arrange the polygons as shown (examples below for $n=5,7,9$):

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Note also that $n=3,4$ are trivially possible.

Now suppose $n>4$ is even. Then we take the construction from $n/2$ (which we may assume exists by induction) and simply truncate the corners of the $n/2$-gons to produce $n$-gons which share the same adjacency graph, while causing equal perturbations to the "holes" so that they remain congruent to one another.

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