HexaFlexagons are weird. No sense denying it; they have a weird property that makes these shapes so interesting to us in the first place. Their amazing flexablilities are what draw us in to create one. Above are the videos that Vi Hart created on YouTube that briefly goes over the history of the HexaFlexagon. Please watch them and then come back to this intro so that you get an idea of what HexaFlexagons are.
________________________________________________

You'll notice that this didn't start out as a current trend, but actually starts way back in the 1900s with one person that was just fiddling around with extra paper that he had to trim out of his older English binder. when he started fiddling with it, however, he created a hexagon shape that wasn't all that interesting....at least, at first. Once he started folding it, however, he noticed a little hole in the center which opened it right back up, but on a different side. That became interesting enough to experiment with, and that led to what we now call the HexaFlexagon.

In this Instructable, I'll be covering how to make a normal HexaFlexagon, a Hexa-HexaFlexagon (which is a HexaFlexagon with more sides to open up), some safety guidelines on usage of the HexaFlexagon, and a neat little holiday craft you can (attempt) to create just in time for the holidays using this interesting geometric shape.

Fair warning, though: this takes a lot of effort to really understand. This is why I have to make this as extensive as it is; there's so much on HexaFlexagons that we could cover.

So, with that said, let's get started! What do we need to start?

Step 1: Supplies, Materials, and Tips

Everything you are going to need for this project is shown in the above picture, sans glue sticks and just regular paper if you're trying this without a template. For the first time HexaFlexagon maker, I don't recommend making these without a template, but you could if you were feeling bold.

You are going to need the following tools:

Paper (preferably a template, like these ones online: http://vihart.com/hexaflexagons/ or

https://upload.wikimedia.org/wikipedia/commons/e/e... for a Hexa-HexaFlexagon, but regular paper works too!)

Scissors (for cleaner lines and construction)
tape or a glue-stick (to attach the HexaFlexagon together)
folding abilities (trust me, you'll need them! :-) )

Once you have all these tools and materials, a few tips should be mentioned before we begin:

*DON'T USE STAPLERS! This is a BIG no-go! Stapling your fully folded HexaFlexagon will not allow HexaFlexagation to occur, as you are stapling all the folds together and not just one to keep it together. Use tape if you are really desperate and there is no glue-sticks in sight. This has occurred in the past, so to prevent it from happening again, this tip will be mandatory to remember.

*Fold ALL the triangles, and try to get them all equilateral. The trick to getting a perfect HexaFlexagon is by making 60 degree angles with your folds. The easiest way to do this is to fold your best guess, then crease VERY LIGHTLY so that you don't make mistakes later. If it doesn't look like an equilateral triangle, keep adjusting your angle until it is a perfect triangle. then you can self-replicate them, but make sure to fold all the triangles for smooth flowing HexaFlexagation. Otherwise, your geometric shape is going to be stiff and harder to flip back and forth.

*It's okay to flex backwards! If you accidentally flex your HexaFlexagon backwards, that's totally okay. It can go both ways.

*Decorate them! Other than to show which side you are on, this is a way to let your creative side show. color them one solid color. Make patterns. Go crazy here--there is no wrong way to color a HexaFlexagon.

*Have fun with it!

Step 2: What NOT to Do When Creating a HexaFlexagon: the Mistake Phase

Before we can truly get into making a perfect HexaFlexagon, you should know that you WILL make mistakes. You WILL have to go back and start over. But if you didn't make mistakes, you wouldn't learn how to do it properly, and no progress would be made. the pictures above were taken as a bit of a "How Did I Get To the Perfect HexaFlexagon" montage.

As you can see, it started with lots of paper cutting from index cards, because I was too bold in trying to create the HexaFlexagon first go. Part of the problem, I believe, was not folding the triangles into 60 degrees correctly. So while they did fold into a hexagon, it just wasn't folded correctly and didn't work into a Flexagon. After two tries, I realized that index cards are probably way to rigid for actual Flexigation, so it was scrapped and I went to print out templates.

The second try didn't really fair much better.

Yes, I know what you must be thinking: "But you used a template! You had to have gotten it right that time!" Except that's not always how it works. Once again, you will make mistakes, and that also applies to using templates, because even then you could fold it incorrectly (Which I didn't know was possible, but there you go. It is). The second time I tried, I thought folded it correctly, but for some reason, it doesn't Flexigate the way I had seen it do. I did this twice with tape, with the robust and non robust version until I just figured it wasn't going to Flexigate the way I thought it should. Maybe you can see the mistakes and try to avoid them.

One more mistake that might pop up is the trapezoid formation. If your hexagon shape isn't all rhombus shaped and has a trapezoid somewhere in it, the Flexigation won't work and you'll have to start over.

So, with that covered, I think it's time we went over safety protocols.

Step 3: HexaFlexagon Safety Guide

This video speaks for itself. Please watch this video before moving on to the next step, where we will finally get to the creation of the HexaFlexagon.

Step 4: HexaFlexagon Creation

Ah, here we are! I knew I had these instructions somewhere. :)

The first thing you are going to want to do is cut out the template you have printed out beforehand (or cut a rectangular strip out of a regular piece of paper, just be sure to make it a skinny rectangle).

If you're using a template, once you cut it out, it's vital to note the equilateral triangles that appear on the template. These are vital for the Flexagon; they can only be created using 60 degree angles. For the template, You should start folding on one edge and zig-zag fold your way to the other end.

(Zig-zag folding is just folding on one side but then repeating that action on the other side of the paper. if you fold a triangle towards you, or on top of the paper, then you should fold it so it's underneath the paper, or away from you. You should also be sure to fold it both ways for proper flexibility).

If your using regular paper, it becomes much more difficult, as you will find very quickly that making 60 degree angles without a guide of some sort takes a lot more effort than you would expect. This is where many mistakes will occur, so it will take several tries to actually get 60 degrees. (Or you can sorta cheat and find a way to draw 60 degrees onto the paper so you don't have to do all that extra folding! :P)

Once you have your nine triangles folded up all nicely (the amount needed for a Flexagon to work), you're going to need to fold it around into a hexagon-shaped twisted loop. Since this is both easy and difficult, there are a few things you need to be sure of before you seal the hexagon twisted loop for good:

If it has a trapezoid along with diamonds, it's the wrong shape to Flexigate properly and you'll have to re-wrap it. In order to have a three diamond shape, you are going to have to tuck one of the flaps underneath another.
Whichever way the Flexagon is wrapped (with three diamonds instead of two and a trapezoid) determines it chirality, or whichever way the water would flow on a Flexagon which is determined by which way the flaps of the diamonds are shown on the side of the Flexagon shape. This doesn't affect Flexagation, but it's good to know when folding the hexagon twisted loop shape.
You are going to have an extra triangle. This is important for the last step of construction.

Now that you have a proper twisted loop, you need to glue or tape the final extra triangle to the triangle it looks like it's going to land on.

(BIASED OPINION ALERT!) I prefer gluing to taping my Flexagons, as after I glue them I'm not worried about them falling apart after the tape has finished its usage. Both methods are fine, but the glue-stick method is the one I prefer. (!!!)

With that done, your HexaFlexagon is ready for the Flexigation to occur! :)

Step 5: How to Flip a HexaFlexagon

Okay. If you've gotten this far, then you have managed to create the perfect Flexagon. But how do you flex it, anyway? It's cool, but I have no idea how to properly flex!

The best method for flexing is to pinch three folds together, which is what I am doing in the first picture. The second picture is the same Flexagon, but I have turned it down so you can see how they close up on the opposite side.The best part is seeing it open up from the middle, which happens when you take one of the closed sides and open up the flappy bit. The rest opens up by itself.

There is no end to opening up a Flexagon; even if it seems that way when you color it. It just keeps flipping inside out over and over.

Speaking of coloring...

Step 6: Decorating Your HexaFlexagon

There is no law in decorating your Flexagon. Maybe you want all swirls. Maybe you need more character references in your life. Or maybe you just don't care and went all solid colors. the only limit is your imagination.

Here, I've shown you different examples of what I did for the Flexagon. For the first one, I did the color wheel, in case I needed a reminder on it. But the more robust version I went ham on, because I wanted patterns. I drew flowers on the first side, then moons and stars on the next one, and hearts on the last one because I had absolutely no idea what I was doing. But when you flip them, you'll always come to find them at some point, which can be loads of fun.

Step 7: Creating a Hexa-HexaFlexagon

Okay, so you've gotten the HexaFlexagon down to an art. You've folded them, you've colored them, you know what you're doing. Maybe you want MORE sides to flip to and color and mess around with. Ladies and Gentlemen, I give you the Hexa-HexaFlexagon. In the pictures above, I've shown you three sides of a Hexa-HexaFlexagon that I worked on for a friend of mine who wanted it Undertale-themed. There are six sides, though, which means that three sides isn't going to be all for you. Here's how to do it:

Take a piece of paper and cut it out so you are going to have 18 triangles instead of 9. Or find a 18-triangle template and work from there. Fold all the triangles again--it'll work best for Flexagation. Instead of just making a twisted loop, however, you'll need to wrap the triangles around so you have diamonds on either side (exceptions: the ends of the diamond loop, where the triangles are). Now you wrap it in a twisted loop, remembering to tuck on side on itself so there is no trapezoid shape and glue or tape the sides.

Because Hexa-HexaFlexagons have more flaps, you'll have to color the first primary sides (like I did, kinda shown with orange and yellow) before taking the quest to find the rest. It's not as easy as it looks, it took me about a day or so to search around and flex the object properly to get them all. But once you do, you'll find all different ways to flex in different ways before! So have fun with it!

Step 8: HexaFlexaflakes: a Holiday Craft

Well, since this is going up during the holiday season, I also felt it would be appropriate to add the holiday craft that is the HexaFlexaflake.

HexaFlexaflakes are slightly more complicated than you would expect, because of all those extra folds being there, which means that if you cut along all the folds, you'll have a very HexaFlexa-gone object instead of a HexaFlexaflake. (BTW, all credit to that pun goes to Vi Hart, I wish I was that creative with puns).

There is no limit besides cutting the whole fold to how you create your HexaFlexaflake with the power of scissors, but make sure all the triangles line up so that you have symmetry on all the sides. Be aware that Flexigating your HexaFlexaflake might be more difficult, as you now have holes in your shape and may want to be more careful. Also, some centers may be left unobserved because it seems to be unopenable for observing, so If you wanted a specific center in your snowflake-y thing, you might not get it.

Either way...

Now you know all the tips and tricks of HexaFlexagons! I hope you enjoyed learning about these fun little shapes!