The only thing you need to know to fold any odd division grid

Posted by Madonna Yoder on

What if I told you that you don’t need to look up how to fold even thirds, or even fifths, or even 17ths ever again? Like you, I used to spend a lot of time looking up how to fold thirds every time I wanted to fold a 48-fold grid. No more.


I learned this method from Elseh (@zamanulli on IG) back in May (it's now October) and I've been using it for my odd grids ever since.

 

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So, let’s get down to it: what’s this awesome piece of knowledge that will change your folding life?


Basic unit of knowledge: The sum of the grid spacings between any two pairs of points on parallel lines gives you the number of divisions you can make between the parallel lines with references from only those four points.


divide paper in thirds, fold accurate thirds, origami grid equal thirds


More specifically: Two horizontal parallel lines have two points each and are separated by height h. The bottom points are separated by distance b, and the top points are separated by distance a. When diagonals are drawn between opposing points on these lines, the diagonals intersect at a height h*b/(a+b). This means that there are b grid divisions below the intersection, and a divisions above the intersection. Scroll down to the last section for a proof.


Ok, but how do I use this?

  1. Pick your points so that the grid spacings on the bottom and the grid spacings on the top add up to your target number. For these examples, I’ll take 2 spacings on the bottom and 1 spacing on the top to get a division by 2+1 = 3.
    1. Caveat: these spacings can’t have common divisors. For example, 6 and 3 will just give you thirds, not ninths, because both 6 and 3 are divisible by 3. For ninths, you’d need to pick 1 and 8, 2 and 7, 4 and 5, or do thirds twice.
  2. Fold the diagonals between your four points. You don’t need to fold all the way - just pinch near the intersection if you want a cleaner fold.
  3. Since you’re folding an odd division in the grid, one of the sides of the intersection will have an even number of divisions.
    1. How do I know this? Only an odd number and an even number can sum to an odd number. Both odd-odd and even-even sums give even numbers.
  4. Fold the line on the even side to meet the point marked by the intersection of the diagonals. In this case, I fold the bottom line up to the intersection.
  5. If the side you just folded still has gaps divisible by 2, keep dividing it until it has odd divisions.
  6. Then take the other side and fold it to the first crease after the marked intersection. This will be an even division since you started with an odd number and you’re adding an odd number.
  7. Continue until you’ve filled in all the creases.

fold even thirds, square grid, triangle grid, divide a square in thirds

 

Let’s run through this for a particularly nasty example: 7-fold divisions with spacings of 4 and 3.

 

divide a square in sevenths, fold even sevenths, accurate sevenths without a ruler

 

I start with a 4x4 grid, and fold my two diagonals. One of my diagonals is the main diagonal of the square, which lets me reuse this intersection point as a reference for the other direction too.

The area below the intersection has 4 divisions, so I fold it in half.

 

origami grid sevenths, equal sevenths, general method to divide a grid

 

The area below the intersection has spacings of 2, so I fold those in half again.

 

grid folding tutorial, even sevenths tutorial, paper folding odd divisions

 

 

Now the distance between the top and the closest crease is even, so I fold that in half.

 

fold equal odd divisons, how to fold a grid, easy origami grid

 

There’s still even gaps in the top, so I fold those in half too. Now the 7ths are done in that direction, and I can repeat the process with the same reference intersection in the other direction!

You might notice that our creases from the 4x4 grid are still hanging out there, not being used. Once you divide your grid of 7ths in four to get a 28x28 grid, all the original lines will be used in the grid.

It’s easier to fold the odd divisions of a grid early in the gridding process, before there’s a ton of lines to keep track of. So, if you’re folding a 48-fold grid, don’t fold four divisions of 2 and then divide by 3 - fold in half once, then divide by 3, then do another three divisions of two.

 

Ok, this sounds pretty great - where’s the proof?

origami proof, origami math, geometry grid math

algebra origami method proof, how to divide a square

 

If you use this in your own folding, go ahead and share it with me (@gatheringfolds) on Instagram or Facebook! I love to see people folding grids - and the tessellations made with them!

 

Also, if you have any questions about this post or any other gridding topics that you’d like me to explore next, send your questions and topics to madonna[at]gatheringfolds.com and I’ll see what I can do to help!


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