Tiling the Plane: making patterns like MC Escher

Funny to look back on this blog.  It certainly exemplifies its name, although I can't see that aspect attracting people to read it.  I mean really!  Posts in 2011, and then 2014?  I'll never have regular readers, because my posts aren't regular.

Perhaps I don't need people to read it.  This is not to say that you, the random person who is peeking at this right now who is not me, are not welcome.  Welcome!  But personally I wonder if I am more likely to write or be creative here if I just assume that no one will see it unless over my shoulder in the kitchen.  Which is where my computer is, by the way.

That's all right. 

So it turns out I like to teach things that I know.  When my husband's brother visited to see Hubby's play, one evening I taught him how to design, carve and print "linoleum block prints."  I put that in quotes because nowadays it's not actually linoleum, thank GOD!  If I had to stick with linoleum my hands would be much more scarred than they are now.  Nowadays it's that white eraser material.

ANYWAY.  So last night Hubby and I went to see a play performed in someone's living room.  Who knew that kind of thing happens?  Afterwards there was schmoozing, and I ended up chatting with a pleasant lady about theater and glass and writing and inspiration.  Turned out she had to leave very soon, so I decided to give her a quick art lesson. 

...I had had a glass of wine.  I don't know if she really wanted to know this.

But, I gave her the briefest lesson I could in "tiling the plane."  One minute.  Way shorter than this post.  On my hand I drew an invisible square and gave instructions; she nodded like a very kind person who might or might not have any idea what I was talking about.

So, "tiling the plane."  Ever heard of it?  I learned it in math class in junior year of high school.  Many thanks to my wonderful math teacher from that year; I would mention him by name but I don't know if he would care for that.  Anyway, it's a fun little activity if you want to inspire your creativity, and you don't actually have to know math to do it.  It is just a good illustration of geometric concepts I can't remember now.

Remember MC Escher's patterns of birds and fish (etc)?  For copyright reasons I can't just show you one, but have a look at this link: http://www.mcescher.com/gallery/ink/no-41-two-fish/  He is using the method of "tiling the plane" to create that pattern.  You can too.

So here's what you do. 

Choose a geometric shape.  A square is your best bet to start with, although triangles and hexagons work.  I don't think I would suggest a pentagon because of the odd number, nor octagon because of the complexity.  But they might work.  *shrug*  NO to a circle.  Won't work.

Next, draw a squiggle or shape along one side.  Don't make it too drastic and dramatic at first; you'll see why later. I would say you should draw in black so that you can color your shape in later, but I will use blue and green so that it's easier to refer to.

Now you need to copy that squiggle to the opposite side.  If you're doing this on paper, just trace your blue line and mark where the corners of the square are, then use those corner marks to help you see where to trace it onto the opposite side of your square.  In Microsoft's Paint program I had to select the blue line and copy it, then paste it onto the other side.  Imperfect process but you'll get the idea.

Repeat the process for the remaining side of the square: draw a line on one side, then copy it to the other.  The trick here is paying attention to your blue lines.  Since your green line is going to repeat, you don't want to draw it so that it will cross over the blue line.  Imagine how it will repeat below as you draw it above.

Now, the reason this shape is fun is that you can use it as a repeating pattern.  In the same way that you can have a grid made up of squares, you can use your underlying square from this image to create a grid of your new shape.  Look at my image below: see how my squares are in there too?

This is what I meant about "you'll see later:" if your lines are too dramatic before you understand the way they will repeat, or if you don't know to take the previous (here, blue) lines into account, you might make something that is impossible to repeat without drawing over itself.

 I suggest coloring in the tiles in a checkerboard pattern to make the shapes easier to make out.  Then you can see what the overall impression of your pattern will be.

And that is "tiling the plane."

If you're like my Gramma and enjoy finding pictures in random shapes, this might work for you: stare at your shape and try doodling in it.  Whatever it might look like.  You can repeat that over and over again, or find a new doodle for each shape.

I'm pretending it looks like some sort of cartoon dinosaur.  Don't pop my bubble.  I know it's a stretch.

That doodle method is kind of working backwards: finding an image in your shape.  Working forwards, you would be trying to actually create the image from the start.  This is what Escher did, but I'm afraid I just don't know how. I wonder whether he experimented a LOT, or whether he had a mathematical formula he used.  You can look that up if you want.

Escher got much more complicated about it in order to have more than one shape that repeated (like the birds and fish together, etc).  I don't how to do that right now (I think maybe I could figure it out) so I'm not going to try to teach you.

You know enough for some doodling time now anyway.  Live it up!