Possible tiling patterns for equiangular hexagon with alternating side lengths

Question

I want to make a pattern of an equiangular hexagons with side lengths of 1-5-1-5-1-5. Here are 3 examples I made:

Which different patterns exist? Are there more than the ones I show? And which one is the most space-efficient, so that it minimizes the wasted space between the hexagons?

I checked the wikipedia page on hexagonal tilings but I cannot find examples for this specific hexagon.

Answer 1

Here are the patterns I could think of, represented by their unit cells and with their densities listed beside:

The first tiling shown in the question has no code in my diagram and has density $\frac{46}{49}=0.9388$. This is not very interesting because it is a rearrangement of the A0 tiling (the second in the question), which has the same density.

From the A0 tiling we can shift the hexagons on their long edges by one to five units to create the A1 to A5 tilings. As the holes between the tiles grow, so does the density decrease:

• A1 has density $\frac{23}{26}=0.8846$
• A2 has density $\frac{46}{61}=0.7541$
• A3 has density $\frac{23}{38}=0.6053$
• A4 has density $\frac{46}{97}=0.4742$
• A5 has density $\frac{23}{62}=0.3710$

The B tiling is the last one shown in the question and has density $\frac{23}{27}=0.8519$.

Now to answer the question of densest possible tiling. When I started typing this I thought it was tiling M, which has density $\frac{23}{24}=0.9583$ and features rows of tiles that can slide over each other.

Then I realised the rows could be pushed into each other a bit more, resulting in the real winner: tiling N with density $\frac{46}{47}=0.9787$.

So if you want to tile a real wall or floor with this shape, your best bets are tilings M and N. While M is less dense, it isn't chiral like N, and its rows may make it better suited to the (usually) rectangular areas tiles are supposed to cover.