Irregular Path Tiles
In my posts a couple of months ago about the number of permutations in various path tile games (link1, link2, link3), I only talked about regular polygons. There are other shapes which tile the plane, and you could use those as tiles in a game. The other day Odinsonnah posted some comments asking about this case. In particular, he was asking about the case I mentioned where you combine octagons and squares. You can do this as two separate tiles, like I showed at the end of this post, but what if you made tiles which were each composed of an octagon and a square glued together. They would tile the plane like so:
But what about the paths? How many different version of the tile would there be? Well, what I think happens in this case is that gluing the two shapes together breaks the rotational symmetry, and you need to take the product of the number of octagonal tiles and the number of square tiles. Since there are 105 octagonal tiles (as opposed to 18 with rotational symmetry) and 3 square tiles (as opposed to 2 with rotational symmetry), we get a total of 315 different tiles.
I created a simple processing sketch which lets you step through all of them. I you click in this window, you can use the left & right arrow keys to cycle through the different octagonal tiles, and the up & down arrow keys to cycle through the different square tiles.
What do you think? Does that look correct? And what other irregular tiles would be interesting? Obviously something like the Penrose tiles would make a pretty cool game.