# Permutations for tilings / tessellations

I'm an artist, many years past my maths a-level - so apologies for any idiocy up front.

At the moment, I'm working with tilings/tessellations of right-angled isosceles triangles. I have two designs, and am currently working out the permutations possible for an eight tile square. Importantly, I'm aiming to find the number of unique possibilities.

With 2 choices and 8 spaces to fill, the simple starting answer would be 2^8 - 256 permutations. But of course it isn't that simple...for a start, there are six different ways to arrange 8 of the tiles to achieve a square

So, the answer grows - potentially to 256 x 6 (1,536).

But then some of these layouts have rotational symmetry - so some of the tiles would not be unique. At this stage I got excited, and thought, well just divide the 256 permutations for each layout by the order of rotational symmetry. So 256/4 + 256/4 + 256/2 + 256 + 256 + 256 = 1,024 unique tilings.

But then I realised that some permutations aren't repeated the same number of times as the order of symmetry (the most obvious example being where all the tiles used are the same type, but there are others involving alternating even & equal numbers of tile type). So 1,024 is little low (if not by much).

I don't really want to have to work through all 256 permutations (in the three layouts which have rotational symmetry...) to figure out which ones are the rotated repeats. So I was wondering if there was a specific area of maths that I should be looking at?

While this specific problem is certainly something I'd like an answer to, I'm keen to gain a better understanding of the area of maths involved - as I'd like to be able to work the same thing out for larger numbers of tiles (and other types of tessellation).

In addition, with the eight tile square, I managed to work out that the number of potential layouts was 6 relatively easily (by drawing them). But as the number of tiles grows, the number layouts will also grow. So I'd like to be able to figure this out too.

Thanks for any pointers/input.

## 1 Answer

I think the branch of mathematics you are looking is Combinatorics: https://en.wikipedia.org/wiki/Combinatorics Also Group Theory https://en.wikipedia.org/wiki/Frieze_group

I am working on something simpler so I don't really know much other than places to look

• Cheers, Edward! Yes, Combinatorics seems spot on - thank you for the much-needed vocabulary. Also found Grunbaum and Shepherd, 'Tilings and Patterns' by looking at other problems on Stack Exchange - and that's certainly going to be useful and inspiring. But it will require work to fully understand!! – Georgia Jan 29 at 8:05
• Have a look at this video about superpermutations I think it may be illuminating to your search for an ideal/minimum number of permtations: youtube.com/watch?v=OZzIvl1tbPo – Edward Vogel Jan 30 at 3:50
• Ooh, great! So much to explore... – Georgia Jan 31 at 8:20