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To make a basic hexaflexagon, fold a strip divided into ten equilateral triangles, matching the labels in order, then secure the matching ends. For a printed version, align the front and back patterns carefully before gluing. The exact pattern depends on the construction method: Vi Hart’s tutorial points to printable patterns, while a separate simple-strip method provides a useful hands-on route.
What you’re making
A hexaflexagon is a paper puzzle whose folded structure lets it flex to reveal faces hidden from view. Vi Hart’s “How To Make a Hexaflexagon: The Definitive Guide” explores the mathematics behind the folds; the instructions below focus on making a working paper model. TED-Ed’s lesson says the puzzle was invented in 1939 by Arthur H. Stone.
There are two practical routes: build and label a plain strip, or print a two-sided pattern. The strip route requires careful labeling and folding; the template route depends on aligning the printed sides correctly. Neither route is established as faster or more reliable than the other.
| Route | Preparation | Assembly | What it depends on |
|---|---|---|---|
| Ten-triangle strip | Prepare a strip and label both sides | Crease and fold the triangles in sequence; secure the matching ends | Accurate labels and folds |
| Printed template | Print the matching front and back | Align the patterns, glue the sides together, then fold as directed by the template | Access to the correct pattern and precise front-to-back alignment |
What you need
- Paper, or gummed mailing or adding-machine tape about 1.5 inches (4 cm) wide for the strip method described in the Cambridge University Press construction excerpt.
- A ruler and pencil if you are drawing and labeling your own strip.
- White glue or a glue stick if your strip is not gummed; a paper clip is another way to secure the matching ends in that method.
- A printer and glue stick for the two-sided printed-template route. Coloring tools are optional.
Vi Hart’s November 24, 2017 post mentions paper, markers, glue sticks, and a printer among the materials used while making the video. You do not need all of them: the construction route determines what is necessary.
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Make a basic hexaflexagon from a strip
The Cambridge excerpt describes a simple construction using a straight strip of ten equilateral triangles. Follow a specific labeled pattern rather than improvising the numbering: the labels tell you which triangles must meet, and an incorrect sequence can prevent the strip from folding as intended.
- Prepare the strip. Use a strip divided into ten equal equilateral triangles. The cited method suggests gummed tape about 1.5 inches (4 cm) wide; if using ordinary paper, have glue or a paper clip ready for joining the ends.
- Label both sides precisely. Copy the labels for the ten-triangle pattern onto both faces of the strip. Keep the orientation consistent; the pattern, not just the triangle count, matters.
- Crease every fold line both ways. Fold each boundary forward and backward so the paper can move readily in either direction.
- Fold matching labels together in order. Bring the numbered triangles together as specified by the pattern, working along the strip. Avoid forcing a crease where the labeled triangles do not meet.
- Join the matching ends. Glue or paper-clip the ends that the pattern identifies. With gummed tape, use its adhesive as appropriate.
- Test the flex gently. Pinch and open the folded model gently to find the motion that turns the visible face. If it resists, check the label sequence, fold directions, and end connection rather than pulling harder.
The ten-triangle sequence is one construction reference, not a universal recipe for every hexaflexagon in Vi Hart’s video. The video listing separately points viewers to printable patterns at Vi Hart’s Vimeo listing.
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Make one from a two-sided printed template
A printed pattern can save you from drawing and labeling each triangle, but its two faces must be positioned correctly relative to each other. FutureLearn’s hexa-hexa-flexagon activity asks learners to print the front and back templates in color, then glue them together with the patterns aligned.
- Get the matching front and back templates. Use the template intended for the model you want to make. Vi Hart’s Vimeo listing points to printable patterns at vihart.com/hexaflexagons; availability of a particular file can change.
- Print both sides in color. Check that both pages are the corresponding faces of the same pattern.
- Check the alignment before gluing. Compare the color sequences on the front and back with the instructions accompanying that template. The triangles must line up in the specified order; do not assume that matching page edges alone guarantees correct alignment.
- Glue the sides together and fold as the template directs. Let the glue set enough to keep the layers from shifting, then make the marked creases and assemble the model according to its own pattern.
- Flex gently to reveal the faces. Use the template’s fold sequence if the model does not open smoothly; different patterns can fold differently.
FutureLearn provides a detailed color-sequence alignment check in its hexa-hexa-flexagon instructions. Use that page and the chosen template for the full pattern rather than substituting the ten-triangle strip sequence.
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Why the folds reveal other faces
The visible faces are determined by how the paper is folded inside the model. A flexagon is not simply a flat drawing folded into a triangle: its inner arrangement lets parts of the paper turn through the structure when you flex it.
The Flexagon Portal’s generalized construction method describes using n−2 construction mapping units for an n-sided hexaflexagon. In that method, the larger triangles must match and the inner triangles must form one connected path. Different valid arrangements can produce different flexing characteristics. Those rules explain why patterns are not interchangeable, but you do not need to prove them to make a model from a correctly labeled strip or template.
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If your hexaflexagon won’t flex
- It will not close into the intended form: recheck that the labels on both faces were copied accurately and folded in the specified order.
- The paper fights the folds: crease each line in both directions before assembling, as the strip method recommends.
- The printed layers do not match: compare the front and back color sequences against the template instructions before gluing. If already glued incorrectly, separating the layers may damage the paper; starting with a fresh print can be the cleaner fix.
- The model seems stuck: flex it gently and inspect the folds and end connection. Do not yank it; a forceful pull can tear a narrow paper hinge.
Vi Hart’s place in the story
Vi Hart’s video presents the puzzle as a way into flexagon mathematics. In a 2017 post about making it, Hart noted, “As I worked on this video I realized there’s a lot of assumed knowledge in simple things like folding paper.” That observation is a useful reminder to take the alignment and creasing steps slowly, especially when following a diagram.
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