Making the Initial Folds of a 6-sided Hexaflexagon
Cut a strip of paper. Your strip should be at least 12 times as long as it is wide. Construction paper will hold the folds better than standard notebook paper.
Fold equilateral triangles into your paper. Equilateral means that all 3 sides are the exact same length. Make your first triangle by marking in about one knuckle. This will be the top point of your triangle. Make a diagonal fold, starting at this mark, with the long side of your strip. When you do so, you’ll see an equilateral triangle. There will be some excess paper on the top left of your triangle: cut this off. Open your fold to reveal an equilateral triangle.
Repeat this 19 times. You want to finish with a strip of paper that has 19 equilateral triangles. If you started with a rectangular strip, be sure to cut off the triangles from the edges—you’ll recognize them as being different because they won’t be equilateral. Try folding each triangle both ways to make flexing easier later on.
Label your triangles 1-3. Flip your strip so that the first triangle is pointing upward. Label this “1.” Your next triangle is pointing downward: label this “2” and the next one “3.” Your next one is once again pointing downward, label it “1.” Continue labeling 1-3 all the way to the 18th triangle. You will have one extra triangle. This won’t get a number.
Label your triangles 4-6. Flip over your strip. Skip the first triangle. The next triangle is pointing upward: label this “4.” The next triangle is also labeled “4.” The next two are both “5” and then the next two are both “6.” Repeat this pattern to the final triangle on your strip. You can color coordinate your triangles if it helps you identify which is which. The blank triangles are on opposite ends of the strip, on opposite sides.
Constructing the Hexaflexagon
Fold your back numbers to face their partners. Take the first two “4” triangles and fold them into one another. Next, fold the two “5” triangles like this, then “6,” and so on down your entire strip. In effect, you’ve folded a flattened spiral. When you flip it over, your paper should read 1-2-2-3-3-1-1-2-2-3.
Fold between the second set of 1’s and 2’s. With your strip flipped so that the lower numbers are toward you, find the second set of triangles that read 1-1 and 2-2. Make a fold between the two: you should fold the longer side under so that you’re looking at triangles “2,” “2,” and “3.
Fold between your 1’s and 2’s again. When you made the last fold, you’ll notice that you made a cane shape. The corner of your cane is made up of four adjoining “2” triangles. Under your fourth one, make another fold: once again, folding the “1” toward the back. When you do this, slide the end above the end of the cane to cover the lone “3” triangle. Your product should now have 6 “2” triangles forming a hexagon and a “1” triangle as an extra flap on the top right of your hexagon.
Glue your blank triangles together. Flip your hexagon over. The extra “1” triangle that is attached to the top should have a blank face on the back of it. You’ll see the other blank-faced triangle to the left of it. Fold it down and glue the pieces together. This is your end product. It should have the same outer shape as a hexagon, but have layers within.
Flexing Your Hexaflexagon
Push the outer corners in. Start by pinching two triangles together along one of the edges. Do the same on the opposite side and push the corners toward each other.
Open opposite triangles. Take the center of these triangles and open them to reveal the face of your hexaflexagon. If they refuse to open, try pinching an adjacent pair of triangles to loosen the folds.
Try folding different sides. Now that you’ve seen how to open the hexaflexagon, try pinching different sides together to reveal all sides within. Faces 1, 2, and 3 will appear more often than 4, 5, and 6. Color or paste pictures on the six different faces and flex away.
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