Fold-and-cut Pi

Have you ever played a fold-and-cut game, where you fold a sheet of paper, make a single straight cut, and after unfolding it, a shape appears—perhaps a star or a butterfly? In this tutorial, I'll teach you how to make a Pi symbol using the fold-and-cut method.

This tutorial was inspired by Erik Demaine. He created a font featuring the alphabet that can be made entirely by fold-and-cut. This got me thinking: could I push this idea further to create the most elegant symbol in math, π? Fortunately, he also created a web editor where you can draw a shape to generate the correct crease pattern. (My creases pattern not only use it but also use Affinity to design the Pi shape and adjust some creases)

1. A sheet of paper printed with the crease pattern.
2. A pair of scissors. (You will need to cut through 12 layers of paper at once in the final step. However, the section with 12 layers only spans about 0.5 cm, while the rest averages only 6 to 8 layers thick.)

Notes:

The color PI crease pattern is for those who are "rich," as it requires color printing, while the grayscale version is more economical.

Additionally, since the printed crease lines will remain on the paper after you cut the shape out, you might want to flip the paper over for a clean look—only to find the PI shape reversed! To solve this, I have also provided reversed (flipped) crease pattern files. This allows you to get a perfectly clean, correctly-oriented PI shape after flipping the final piece over.

Make all the creases based on the crease pattern.

Red lines denote mountain folds, while blue lines denote valley folds.

If you are using the grayscale pattern, the Dashed line lines are valley folds, and the Dash-dotted line lines are mountain folds.

When pre-creasing, try to fold strictly along the specific lines drawn on the pattern. If your creases extend beyond these lines, the subsequent steps will be much harder to do.

If you really need step-by-step instructions for pre-creasing, I recommend completing all the valley folds first before moving on to the mountain folds.

Tips:

If you printed a grayscale version but still want to clearly distinguish the mountain and valley folds, you can try tracing the lines with colored pens.

While folding, if you notice that the edges of the PI shape (the purple lines in the image) all land on the same straight line after flattening the paper, you are highly likely doing it right. This is because all the edge lines must align perfectly so that you can cut out the PI shape with a single straight cut at the end.

You can drop "PI crease.svg" into origami simulator (in top menu: Files>Imports>select your file>Import SVG) to help you understand how to fold by adjusting the slide bar at the bottom though it's not very accurate.

Notes:

The paper in this image contains some unused creases because it was a trial version. The file in the attachment is the final version, which does not contain these unnecessary creases. Additionally, there is a misdrawn crease in the top-right corner, but it doesn't matter as I already fixed the error before that crease is actually used (in Step 6).

The following steps all come with a crease pattern image, highlighting the creases used in that specific step in color, along with numbered folding orders. Note that the folding order is just a reference if you are unsure how to proceed; you can adjust it based on your own habits. Ultimately, only the correct outcome matters. If you are extraordinarily talented, you might even be able to skip straight to Step 10, as you would know how to fold it just by looking at the complete crease pattern.

This pattern is "flat-foldable," meaning the paper can be collapsed completely flat once all the creases are made.

As you start folding, you might feel like it's impossible and begin to doubt whether my pattern is actually correct. Rest assured, it is!

Before proceeding, make sure the "feet" of the π (PI) symbol are pointing toward you. The terms "top," "bottom," "left," and "right" in the following steps will only make sense if your paper is placed in this correct orientation.

For the top-left section, fold up the corner first (it should be easy to flatten), and then finish the valley fold below it. At this point, this corner should be able to collapse flat against the paper. The top-right section will feel a bit awkward and won't lie flat, but that's fine. It will be addressed later, so there is no need to flatten that part right now.

(Optional) You can unfold this section temporarily if it gets in the way during Step 3.

1. First, make the three main vertical creases across the middle and bottom of the paper. Fold them at a wider angle (around 90 degrees). The bottom section doesn't need to be flattened yet, as we will deal with it later.
2. Next, fold crease No. 3 down completely (180 degrees). Once done, you can pull the paper down along the middle to complete creases No. 4, 5, and 6.
3. Finally, fold down both creases labeled No. 7. Pressing down the left one should allow you to completely flatten the structure from Step 2 that felt awkward earlier. The right side still won't flatten perfectly because it will be blocked by the right edge, but don't worry, just leave it as is for now.

This step is very straightforward—just fold it up.

Note that at this stage, the entire left half of the paper should be able to collapse completely flat.

At this point, you should see that all the edges of the PI shape (the purple lines) on the left side of the paper are perfectly aligned on a single straight line.

After completing the left side, flip to the right side.

The first obvious issue is that while the bottom and middle sections can be flattened, the top side cannot be flattened without creating an unwanted crease (shown as a green line in the image). Therefore, we'll fold the top-right corner first, which should make the unnecessary crease disappear naturally.

As you pull the blue valley folds (No. 1, 2, and 3) toward the bottom left, the blue and red lines should naturally fold upwards.

Creases No. 1, 4, 5, and 6 should all have a smaller paper angle, while creases No. 2 and 3 should have a larger angle. However, none of them can be fully flattened at this stage.

You may notice that crease No. 7 hasn't been folded correctly. Instead, a crease that shouldn't exist appears (orange line in the crease pattern image). Don't crease it firmly yet – it will be resolved in the next step. If you crease it firmly, you'll notice that the edges of the PI shape no longer align, indicating an error.

Before working on the bottom section, fold creases No. 1 and 2 first, making their paper angle around 90 degrees. Then, collapse crease No. 3 in the up-left direction. Fix the intersection of No. 3 and 6 before folding No. 4 down completely. At this moment, No. 5 should naturally fold inward towards the up-right direction. As you pull it further back, meanwhile, pull No. 3 toward the right as well. The edge of the PI shape crossing No. 5 should slowly approach the already existing PI edge in the middle of the paper. Finally, the two PI edges should overlap, and No. 6 will be able to be flattened.

After completing this step, the central area of the right side of the paper will likely still be uneven. We'll address that next.

During the collapsing process in the previous step, the paper angle of creases No. 1 and 2 gradually went from a right angle to 0 degrees.

When No. 1 (crease No. 2 in the last step) is flattened, the two No. 2 and two No. 3 creases in the image will overlap, forming two groups. Since each group contains both a valley fold and a mountain fold, the two layers of paper should be stacked and folded together. Pull the two No. 2 creases down together, then pull the two No. 3 creases (which are sticking up) upwards and flatten them. While flattening them, pull No. 5 down synchronously, and make sure No. 4 remains a valley fold instead of a mountain fold.

Now, you'll find that the unwanted orange crease from Step 6 has disappeared.

The PI edges on both sides should align perfectly. Lift parts of the paper to double-check.

Cut along the PI edge with a straight line, then unfold it to reveal your finished PI!