Note: This is an extended version of a piece I wrote for the Pacific Coast Origami Conference (PCOC) 2023 Diagram Book, available for purchase here if you're interested.

The World’s Strangest Paper Size

by Marcus Ho

The great thing about being a paperfolder is that paper is everywhere. Truly everywhere. The bad thing about being a paperfolder is that most of that paper isn’t square. Luckily, there’s a simple way to convert rectangles into squares. Fold the top corner of in half, fold the excess paper at the bottom behind, and cut or tear across the line. Voila! But with this comes a secondary problem: what do you do with the extra bit that’s left over?

I’ve never seen anything in the origami literature on how to fold leftover strips–a strange thing, given that every folder has probably encountered this problem at some point. In this article, I’m going to try to remedy that. I will take you through my journey, trying to come up with as many designs as I can and harnessing the unique properties of these neglected pieces of paper. Welcome to my story of exploring the world’s strangest paper size.

To begin with, I started thinking about the geometry of the strip. I live in the United States, where standard letter paper has dimensions 8.5” by 11”. Removing the largest square leaves a rectangle with dimensions of 2.5” by 8.5”, or 5 by 17. My first thought was a grid-based model, which led me to make a Christmas Tree:

Feeling a bit inspired by Jeremy Shafer, I turned it into a Dead Fish:

 

Unfortunately, for whatever reason, the paper strip doesn't quite have integer proportions; it's very slightly shorter than 5 by 17. Take a look at the head of the Dead Fish: even though the horizontal lines intersect the corners in the body, they fail to reach the ones at the head. For something simple, this isn't really a problem (and I'd say it's actually a positive in this particular model), but it would seriously hinder the making of any more complex grid-based designs.

So I got stuck on my first approach. Thankfully, it turns out the proportional error leads to a weird coincidence. While the paper strip isn't quite a 5 by 17 rectangle, it very closely approximates the more mathematically significant 1 by 2 + √2. That's interesting because it can be further subdivided into a square and a very convenient rectangle (note the 22.5° angle):

This reminded me of a model by Peter Engel: the One-Dollar Boat, folded from a dollar bill, whose crease pattern roughly matched that of the convenient rectangle part. The extra square would naturally lend itself to becoming a head, and the following model seemed obvious:

While I’m not much of a money folder, this method could likely be applied to a great many dollar bill models, adding more details on something like a head. But I didn’t stick around to find out, because I soon came across the idea of reversing the strip to make a model with a large head and a small body. I thought this would lend itself well to an animal with complex horns, and thus the Leftover Strip Deer was born.

 The base for the deer is surprisingly versatile. I've made a warthog, a mammoth, and a Triceratops (no pictures, sorry) and I can see a hippo or a bighorn sheep coming out of this one. It's a bit front-heavy, but careful planning and balancing can alleviate most of those issues.

So after that mammal spree, I decided to switch things up and turn the strip into two conjoined squares. After a few false starts, I managed to come up with a doubled crane, shown here.

 But now I was on a bird spree, and given the shape of the strip, I thought I'd do something with long wings. A rough equilateral triangle-based design (based on another weird mathematical coincidence) gave me the Leftover Strip Albatross.

I wanted to explore the geometry of the albatross design further and ended up making a Leftover Strip Kite. But kites need strings, so I ended up attaching a second strip as a string.

This is where I had to stop. If I was willing to make modular designs, there were simply too many possibilities open, and I had no idea how or where to continue. I could very easily just keep folding forever, and I didn’t want to take too long to write the article! This model provided a natural endpoint for my project.

So, big question: which of the models might I actually keep in my repertory? Apart from the deer (my favorite one by a decent margin), probably none of them. That isn’t a bad thing, though. These were improvisations, meant to be spontaneous and off-the-cuff rather than carefully constructed. They’re no less artistic for that. I’ve been trying to make improvisation a larger part of my folding experience, and I’ve had a great success in that regard.

This project has been, more than anything else, a lot of fun. I think that’s the message of the whole thing: go out and explore stuff! Fold from weird paper dimensions. Start wet-folding if you haven’t, or stop wet-folding if you do it constantly. Fold insects out of three-inch tissue (if you dare). Origami is art, and art should capture all of human experience. That includes the simple, pure joy of discovering things. Even if this article is the first in-depth exploration of leftover paper strips, I hope it encourages someone else to write a second.

But I’ll give you a warning before you try anything: I’ve ended the project facing precisely the opposite problem from the one I started with. I keep cutting thin strips off pieces of letter paper and now I have no idea what to do with the leftover squares.