How to Divide a Piece of Paper in Thirds (by folding alone)

Picture the following scene:

You've decided to start that new origami model.  Great!  It's something you've been putting off for a while, but you finally have some time to devote to your craft.  This is the perfect opportunity to hone your skills.  You'll start small, with a fairly simple model, but you'll gradually move along and do something more advanced.  You take your 15x15 cm. piece of kami, open the origami book to the right page, and read the first step.

And that's when you see it:  "Divide the paper in thirds."

You scratch your head in confusion.  What are you supposed to do?  You can divide a piece of paper in half--that's unimaginably easy.  You can divide a piece of paper in fourths, too--just divide in half, and then divide the two halves in half.  But thirds remain frustratingly out of reach, like Tantalus trying to grab fruit from an origami tree branch.

Well, fear not!  Here are five different methods for dividing a square in thirds.  Each one uses folding alone (we are purists, after all--no cheating with a ruler and a pencil), taking advantage of the unique geometry of the square.  There is no perfect one (each method has weaknesses, which I will be going over), so look through each and see which one works for you.

Number One: The Diagonal

I first saw this one in the book "Origami Polyhedra Design," by John Montroll.

Procedure: Fold a small halfway mark on one side.  Fold a line connecting this mark and one of the opposing corners.  Lastly, fold a diagonal that intersects this second line.  The resulting intersection is at the coordinates (1/3, 1/3).

Weaknesses: Creates a mark in the center of the paper, rather than the edge.  If this is what you want, such as making a 3x3 grid, this is good, because you don't have to repeat the same procedure on two different sides.  It can be annoying for other purposes.

Number Two: The Japanese Way

Unlike most of these, I discovered this in the book "Real Origami: Flying Creatures" by Fukui Hisao.  It's entirely in Japanese.  Hence the name.

Procedure: Fold the paper in half lengthwise and widthwise.  Fold a line dividing the top section in half lengthwise.  Then, fold a line from the bottom corner through the intersection point you just made.

Weaknesses: Creates a total of four extra lines to make the final 1/3 mark, more than any other method.  These lines can easily clutter up the paper and make the model harder to fold.  On the other hand, the folding sequence is the simplest to understand, given that you mostly just fold the paper in half.

Number Three: The Triangle

Another one from the John Montroll book, this method revolves around the properties of a 30-60-90 right triangle.  See if you can figure out the geometry behind this one!

Procedure: Fold a mark on the left side of the square.  Make a crease going through the top right corner such that the bottom right corner lands on top of the mark on the left (make sure it's long enough).  Fold the top right corner down on top of the mark at the bottom.  The right side should be divided in thirds.

Weaknesses: The mark on the left tends to stick out a bit, which gets a little annoying.

Number Four: The Edge Method

Yet another from John Montroll--in Origami Polyhedra Design, he detailed numerous methods for dividing a square into various fractions, not just thirds.

Fold a halfway mark on the edge.  Make another fold such that 1) the bottom edge touches this mark and 2) the bottom right corner touches the top edge.  The top edge is divided into thirds.

 

 

 

Weaknesses: This method undeniably has the most complex fold, requiring the folder to align two sets of lines at once.  It was certainly difficult for me when I tried it the first time around.

Number Five: The God Method

Self-explanatory.

Procedure: Just approximate!  Make small adjustments before forming sharper creases.  It's easy to understand, it's fairly precise, and you can scale it up to arbitrarily small divisions.  It is truly a folding technique worthy of the gods.  What's not to like?

Weaknesses: Weaknesses?  What weaknesses?

Anyway, I hope you enjoyed this little experiment.  I'm hoping to do more talky posts every once in a while.  Ideally, these will be more frequent than the once-a-month model posts.  Maybe twice a month, or once a week, or something within that range.

Thanks for reading, and keep coming back for more awesome stuff!