Welcome to another tutorial from the How To Origami Studio! This Instructable will teach you how to make the Origami Flexagon, a geometrically spectacular object that I still say is better than a fidget spinner. It moves through itself infinitely and is mind-blowing to witness.This tutorial is moderately difficult, so it may take a few tries to get right. I highly recommend watching the embedded video tutorial, however, if you prefer there is also this instructable with written instruction and images. Please feel free to comment any suggestions, concerns, or questions and I will do my best to get back to you. Good luck!
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Step 1: Create a Paper With a 1:2 Size Ratio
This can be easily done by taking a square origami paper and ripping/cutting it perfectly in half. If you do this option, you are done with this step and may move on. If you want to do this with a piece of printer paper, start by holding the paper in landscape orientation and folding in half from left to right (image 1 - 2). Unfold the paper, then fold the bottom left and right edges up to the center crease (image 3 - 4). The edges when folded should create a horizontal line. Fold as accurately as you can on that edge line (image 5 - 6). Fold back and forth on this most recent crease and cut/rip along this line(image 7). Discard the smaller strip, and keep the larger piece. This piece should now have a perfect 1:2 ratio (last image).
Step 2: Begin the Flexagon
Take your 2:1 paper and begin by folding it in half lengthwise, then unfold. Once you've done that, fold both the left and right edges into the center crease (last image). Do not unfold.
Step 3: Create a 2 by 8 Grid
Now we need to make a 2 by 8 grid of perfect squares. Do this by holding the paper so it lays horizontal, then folding both the left and right edges into the center crease (image 1 - 2). Then unfold. Now take the right edge and fold it into the nearest crease (image 3), then unfold and fold it again to the farthest crease (image 4). Unfold. Lastly, you will do the same thing with the left edge, the only difference being that you won't fold to the farthest crease. It will be the second farthest crease. The result should be a series of perfect squares.
Step 4: Create the Diamond Grid
In my opinion, the best way to conceptualize this step is to position the paper vertically. Now imagine that the square grid we created is a graph. Now take a pencil and draw in lines that are angled down with a slope of -2. In other words, start at one corner of a square and mark it. Then from that point, move down two squares and over one. Then connect the dots and do this for every square in both directions. The end result should look like the second image. Once this is done, fold along those lines (image 3). This is the step that makes this a moderate to difficult tutorial. I highly recommend watching the video if this step doesn't make sense to you.
Step 5: Form the Ring
In the first image, I am opening up a sort of pocket formed by the two flaps in the bottom. The idea in this step is that one end must go inside the other one to create a ring. The best thing to do first is to see if one end is slightly more narrow than the other. If that is the case, the narrower side will go inside the wider end. The narrow end must go two square lengths into the wider end until the edge is perfectly aligned with the second crease. The end result should form a hexagonal ring (image 3). Again, the video probably does a better job explaining this.
Step 6: Begin Shaping the Flexagon
Hold the ring in the orientation shown in the first image (vertical). There should be a total of six horizontal lines around the ring when held in this position. Press inwards on either side of the horizontal line as shown in the first image. If the model does not easily collapse inwards (second image), then move on to the next horizontal line and try again. This will only happen on every other line (three horizontal lines total). When all three lines have been pressed inwards the model should look like the last image.
Step 7: Finish Shaping the Flexagon
Move the ring to the orientation shown in the first image (horizontal). There should be three points on the top and bottom of the ring (last image of the last step). Pinch two opposing points (one point on top and the bottom one directly below) together while pushing them back to the middle of the ring. They should easily fold along the premade folds as shown in the first three images. Repeat on the other two sets of points until the flexagon takes shape (last image).
Step 8: Begin the Movement
Congratulations, you have finished! To work the model into moving, Start by slowly rolling the object through itself. This will allow the creases to align and decreases your chance of tearing the paper. Once the movement becomes easier, you can go as fast as you want. I really hope this was a comprehensible tutorial, as this is a very difficult model to teach. Let me know what you think, and comment any suggestions or questions. If you liked this tutorial or want to see more like this, check out my channel How to Origami on Youtube. Likes, comments, and subscriptions are welcome. Have a great day!