Mathematics for the Liberal Arts

Creating Escher-Like Tilings with Software

What we’re going to learn to day is how to take your computer and create some art, and what you will make will astound you. First things first. What we are going to create will look a lot like wallpaper, with repeated shapes. However, the more someone looks at it, the more they’ll see. In fact, what we shall make are called tilings or tessellations. A tiling is a way to divide up the plane into geometric figures, each of which is of an identical shape (you may also have a few different shapes, but they must repeat in a pattern and fill the plane).

In the text you’ve seen many tilings. One example is with squares. Another one can be achieved with hexagons (six sided figures) and triangles. I’ve created one of my own below.

Note that there are two different variations of one tile, in different colours. I'm going ot describe how to use software to create you own tiling. You'll need a painting program. There are many free ones; we'll use thePaint program available in Windows under the “Accessories” menu.

STEP ONE
Start up your paint program. If you’re using Paint, it will automatically open with a blank page to work with. You’ll probably want to increase the size of the page by dragging the bottom right corner of the page out as far as you can.

STEP TWO
Now you want to draw a black rectangle, one that is of medium size (in the end, they should be small enough that you should be able to place at least six of the squares across and down without any overlapping). In Paint this is fairly straightforward: After making sure that both "Color 1" and "Color 2" are black (if one iusn't, select it and then select the black box to the right), all you do is click on the rectangle shape and drag out a rectangle on the page (if you want the rectangle to be a square, hold down the Shift key whie dragging).

STEP THREE
This is the key step. What you want to do here is alter the shape of your tile from a square to a more interesting one. If you think of what you need to be able to put your tiles together, what you’ll see is that whatever pieces are indented into the left side must project from the right side in the same way, and vice versa. The same is true for the top, and as long as this happens, the tiles should fit together like a hand in a glove!

Here is how to proceed. You need to click on the Free-from selection tool under the "Select" menu (by clicking on the arrow beneath it). Now click with the mouse to the left of the rectangle you’ve drawn, and drag out some closed shape inside the left side of the square, making sure to get a bit of the top or bottom edge (this will make the subsequent alignment easier later). By closed I mean that when you let the mouse up, the end point of the mouse is close to where you started dragging.

Now choose Control-X to cut the piece out, and then Control-V to paste it. You'll need to drag the pasted piece to the right side of the tile and carefully release it so that it is exactly fits. If you click on the page away from the rectangle, you should see how the left side of your tile will fit exactly into the right side. Now go about and do the same with the top and bottom (select, cut, paste and drag a piece from the bottom up to the top, or vice versa). An example is shown below. You’ve just drawn your basic tile.

STEP FOUR
Now select your tile with the rectangular selection tool (it is the dotted rectangle on the left), drag a rectangle around your tile, and then click down in the middle of your tile and drag the whole tile down to the bottom of the page. You can take your tile and paint on it however you like with the painting tools. If you don’t know how to use the paint tools, ask someone, or just try them out – it’s pretty easy! Make sure all your painting is within the tile. Here is how I painted mine:

STEP FIVE
Use the rectangular selection tool to select your tile, cut and paste a copy of the tile. Drag it far enough away that you’ve left plenty of space between it and the original tile. Now you can recolour the new tile. One of the best ways is simply to change some of the colours. The easiest way to do this is first to click on the bucket tool. Then select a color for "Color 1". Select the piant bucket tool and click on any colour on your copy of the tile; the colour you selected should replace the old one. Do this several times until you’re happy with how the second tile looks. (You can use Control-Z to undo any choices you regret!)

STEP SIX
Select aone of the two tiles with the rectangular selection tool and move it to fit into the other tile; this is your two tile pattern. Move the copy down to the bottom of your window.Then copy and paste it. Move this to the left side to be you first tile. You can then repated paste and drag this combined tile several times to fill in across the page, fitting it in snuggly as you go, and then continue beneath the first row, and so on. Continue to alternate, dragging copies of the two tiles at the bottom and placing them successively across the top so that they fit neatly into one another. The tiles will always be of opposite colour to the ones directly beside it (on all sides).

STEP SEVEN
Save your picture and print or post it. It should look wonderful! Feel free to start all over again. Here are some ideas to explore once you’ve got the idea:

• You don’t need to start off with a black tile; any solid colour will do.
  Try colouring the second tile quite differently from the original (see my Elephants and Ghosts tiling).
• In most painting programs, you can “crop” your picture (that means, you can cut it down in size to a smaller rectangle). You can then add a frame so it really looks like a picture!
• Finally, you can start with a parallelogram rather than a rectangle provided your program allows you to create one. The hard part will be aligning the selected parts as you move them across the screen (here’s a hint how to do this – find a way to draw lines joining the midpoints of opposite sides with a brand new colour. When you drag the pieces, as long as you’ve selected some of these lines, you’ll be able to line things up properly).