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How to Make a Paper Icosahedron (AKA: 20-sided die)

Say you're a Geometry teacher who is trapped at home because COVID-19 has shut down your school district. Say that same school district has decided that in order to make virtual education easier and more standardized, they've purchased a year-long subscription to an online learning platform that has all instructional videos and activities already pre-loaded so you don't have to.

Your job has now been changed from educator into data-entry, and even now, there is so little data to enter that mostly you're sitting around in the optional Zoom classroom waiting for a student, ANY student to show up just so you can interact with a non-family member.

Welcome to my life.

And if you're anything like me, you probably don't handle boredom well, so you'll need some sort of an outlet to get you through the day.

Enter - the paper icosahedron.

Before I begin, it should be noted that I first learned how to make the modules for this piece of origami several years back in one of my Master's classes for my education degree, and I honestly do not remember the person who taught me. Several years later, I found out that one could glue these modules together to create an icosahedron, but again, the name of the person who told me that has slipped my memory.

So while I'd love to get credit to those math-artists before me who taught me these skills, alas, I cannot. Just know that I didn't come up with this all on my own, and I'm a terrible example at citing one's sources.

Now, to begin:

First, you will need 20 congruent (same size!) circles. You can make 20 of the same color, 20 different colors, or some grouping in between. I chose to use 4 different colors because of the Four Color Theorem, which states that any map can be colored using 4 colors so that no two abutting pieces share the same color. (https://mathworld.wolfram.com/Four-ColorTheorem.html)

Once you've cut out your 20 circles (I made 40 so I could make two in case I didn't like the first one) you'll need to fold each one as follows:

Fold the edge of the circle to the center.

Do this three times so you get an equilateral triangle.

From there, you'll fold each corner into the center to make a hexagon:

Unfold these flaps and fold each corner to the opposite side, crease, then unfold.

You're now ready to assemble your truncated triangular pyramid. Do this by lifting each flap up. You'll see that each corner has a little pocket. Slide one corner into another corner's pocket.

This makes kind of a pita looking thing. The third corner can just be tucked into the middle (inside the pita).

And you have your truncated pyramid. You'll need 20 of these pieces.

I've found that sometimes one of the corners folds better than the others, so play around with it, to see which arrangement fits tightly but not strained.

Remember, you need 20 of these, so get to folding!

Now you're ready to assemble. You'll need some glue. I debated using Modge Podge but in the end settled for the classic Elmer's School Glue. Nothing fancy. We're just glueing paper, right?

You start with 5 of your truncated pyramids arranged in a circle.

Each piece has a small triangle which will be inside of the icosahedron, and a larger triangle which will form an outside face. The sides are made of identical trapezoids which you will glue together like so:

For my first icosahedron, I was trying to make stripes of colors, all my greens on the bottom, then yellow, black, blue. You'll find as you start glueing that the pieces just naturally fit together. So it just becomes a matter of gluing the next piece on, waiting for the glue to dry, then moving to the next piece

I occasionally added a reinforcing stripe of glue along the cracks. Not sure how helpful it was.

Keep adding pieces...

And eventually it all comes together! You can see the completed first icosahedron here:

After I finished the first one, I wasn't happy how the colors turned out. You can see that the blue and green are all piled on each other, but the black and yellow alternate. It didn't look how I wanted it, so I decided to make a second using the Four Color Theorem mentioned earlier.

Started with a base, then tried to add colors in alternating patterns. I got to a point where I realized not every triangle will have three different colors around it (you can see the black triangle has 2 yellow triangles touching it) but as long as now two same-color triangles are touching, I'm happy:

Almost done!

And we have a completed icosahedron!

This icosahedron for whatever reason didn't come together as easily. Part of the problem could be my dog.

Your dog? You ask. Well, funny story. At some point in between the making of the first icosahedron and the second, my dog got into my bag of truncated pyramids and pulled a handful of them out. Luckily, he didn't rip them to shreds, but he did squish a few in his excitement, and well... I think their structural integrity may have been compromised. I, of course, decided to still use them, because I am, after all, a math teacher, not an engineer.

My final two icosahedrons:

All I need to do is number the faces and I'll be ready for a giant game of D'n'D!

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