Got a ruler, a pencil and a piece of paper? You have everything you need to delve into the fascinating world of tessellation with pentagons.
Cairo Tiling is a fascinating pattern named after some street pavers in the Egyptian city and part of a fascinating set of space-filling Pentagons that has been the subject of research by mathematicians and amateurs over the last century.
That master of patterns, Escher has taken the pattern to another level.
Writing on Medium, Catherine Halloway @femion, has provided some relatively simple programming code to generate Cairo Tiling. Her published pattern is shown above, the code for creating it is linked here and is reproducible by anyone who knows how to program.
This article shows a very simple way to generate the pattern with no mathematics at all.
You can do it at home with a ruler and some graph paper, or you can use computer drawing software to generate it. The great thing about this method is that you can fiddle a little bit to create many other related patterns.
The first step is to rule a bunch of squares. At a minimum you will need four, I found it easier to start with nine.
Now put a dot in one square, preferably half way between the corner of the square and the middle.
… and repeat for every square.
Now join that dot to the three closest corners
… and repeat. There are your first set of hexagons.
Now copy that and flip it over.
If you are using pen and paper you can skip the next step.
If you are using graphics software. Draw the dots and the lines on a separate layer from the squares, cut and paste to a third layer, then flip the third layer.
Now move the flipped layer so its corners are in the middle of the first layer.
You can see that I have moved my example half a square to the right and half a square down.
If you are using paper and pencil, photo copy your hexagons, and put the two copies on a window with one of them face down (upside down or back to front), shift the top one around until it looks like the drawing below and trace the pattern onto the top piece of paper.
Now trim it and remove the squares. Bingo!
I have breaks at the corners of my tiles where I removed the square lines.
Depending what you are planning to do with it, you can avoid that, or use that as a guide to generate square tiles, prints or fabric that repeat to produce the Cairo tiling pattern when they are joined together.
Note that the pentagons are not symmetrical. That is because I have told you to put the dot in the middle of one corner of the square. With a bit of fiddling you can use variations on this approach to produce many of the patterns described in the Wikipedia article on Cairo Tiling. The point of this article is to give you a quick way to produce a fascinating pattern.