A hexaflexagon may look like a normal hexagon when viewed from the front, but it hides many other surfaces inside. Bending a hexaflexagon is a quick and fun activity that can trigger a higher appreciation of geometry. You can fold some different types of hexaflexagons, of which tri-hexaflexagon and hexaflexagon are the easiest to learn. Read on to learn more.
Steps
Method 1 of 2: Method 1 of 2: Tri-hexaflexagon
Step 1. Draw a strip of 10 adjacent equilateral triangles
The first triangle of the strip should be upside down, with the tip down, while the last one should point up.
- The triangles in the middle alternate between straight and upside down.
- The last triangles have only one edge in common, but all the other triangles in the middle share 2 edges, with only the base being separated from the others.
- Make sure that each edge of each triangle matches all other edges and triangles in length. Accuracy is the key.
- Do not detach the triangles.
Step 2. Label the strip on the front and back
Starting from the left, the front triangles should be labeled with the numbers from 1 to 10. Turn the triangle on the back and label the triangles from 11 to 20.
- Label the triangles lightly so that you can erase the marks later if you want.
- Note that each triangle on the back will be greater than 10 than its respective front.
Step 3. Fold each triangle along the adjacent edge
Fold each shared edge back and forth a couple of times to fold these edges nicely. This will make it easier to fold the hexaplexagon.
Step 4. Fold the first three triangles down and back
Triangles 1 to 3 should be folded back along the edge shared by triangles 3 and 4.
Once the fold is complete, triangles 12 and 11 should be visible under the strip. Triangle 12 will be directly below triangle 4
Step 5. Fold the last four front triangles down and to the front
Triangles 7 to 10 should be folded frontally, along the edge shared by triangles 7 and 6.
- Triangle 6 will be covered by this new fold.
- The only original front triangles remaining will be the 4 and 5. The other triangles currently visible from the front are the triangles that were originally on the back.
- Note that the main figure will now be a hexagon with a small triangle sticking out of the bottom.
Step 6. Slide triangle 11 over triangle 19
Now, the triangle 11 is covered by the triangle 19 in the lower edge of the hexagon. Swap the two triangles so that 11 overlaps 19.
The rest of the figure should remain unchanged
Step 7. Fold triangle 20 in front of triangle 11
Triangle 20 should be folded along its base. As a result, it will cover triangle 11.
- Triangles 11 and 20 are no longer visible, but in their place, triangle 10 should be visible again.
- Note that you will now have a full hex.
Step 8. Apply tape along the edge of triangles 10 and 11
Fold a small piece of tape around the right edges of triangles 10 and 11. The tape should fold around the front and back of these triangles.
This step will help hold the hexaplexagon together
Step 9. Flex the hexaplexagon
Now that the basic tri-hexaflexagon has been completed, you can "flex" it as the name suggests.
- Hold the hexaplexagon in front of you with two hands.
- Squeeze 2 adjacent triangles together. Note that you should choose two triangles that are next to each other but that do not share a folded edge.
- Use your non-dominant hand to squeeze the triangles together.
- Use your dominant hand to push down the extension of the flattened edge. This extension should be a shared edge, and pushing it down should lead you to push it into the middle of the hexaplexagon.
- As the hexaplexagon opens in the center, use your dominant hand to open it on the edge and fold the triangles outwards.
Method 2 of 2: Method 2 of 2: Hexa-hexaflexagon
Step 1. Draw a strip of 19 adjacent equilateral triangles
Both ends of the strip should be in the right direction, with the tip up and the base down.
- The triangles in the middle alternate between straight and upside down.
- The last triangles have only one edge in common, but all the other triangles in the middle share 2 edges, with only the base being separated from the others.
- Each edge of each triangle must match in length with all other edges and triangles. Do not detach the triangles.
Step 2. Label the front and back of the strip
Label the front triangles using a sequence of 1, 2, 3, repeating it six times until the first 18 triangles are labeled with a 1, 2, or 3. On the back, skip the first triangle, and label each triangle in a double sequence of 4, 5, 6 (4, 4, 5, 5, 6, 6, etc.). This sequence should repeat three times before reaching the end of the strip.
- The last front triangle and the first rear triangle will not be numbered.
- Label the triangles lightly so that you can erase the marks later if you want.
Step 3. Fold each triangle along the adjacent edge
Fold each shared edge back and forth a couple of times to fold these edges nicely. This will make it easier to fold the hexaplexagon.
Step 4. Fold the strip so that the triangles labeled 4, 5 and 6 face each other
Fold the first triangle 4 along the edge shared with the second triangle 4. Do the same with the first and second triangles 5 and the first and second triangles 6. Repeat until all three groups of triangles 4, 5 and 6 have been folded.
Essentially, you are folding the strip into a single long spiral. This spiral should create a stripe apparently similar to the initial one of the tri-hexaflexagon. For this reason, the instructions from here on are almost identical to those of the tri-hexaflexagon
Step 5. Fold the “first” three triangles down and back
The first three triangles from the left should be folded back along the edge shared by the third and fourth visible front triangle.
The back of the "first" two triangles you just folded down should now be visible from the front
Step 6. Fold the last four front triangles down and forward
The last four triangles from the right should be folded in front of the strip, along the edge shared by the fourth last and fifth last visible front triangle.
- The fifth to last triangle will be covered.
- At this point, the general shape will be a hexagon with a single triangle protruding from the bottom. If this triangle protrudes from another part of the figure, turn it until it is facing downwards.
Step 7. Swap the overlapping lower triangles
The triangles on the lower edge of the hexagon will overlap. Swap the overlapping triangles so that the back section now passes in front.
Step 8. Fold the excess triangle in front of the lower front triangle
The triangle that extends below the edge of the hexagon should be bent upwards along its shared base.
After making this fold, you should have a full hex
Step 9. Stick some tape along the edge of the lower triangles
Curl a piece of tape around the right edges of the triangles in the lower section of the hexagon. This piece of scotch tape should go from front to back.
This step holds the hexaplexagon together
Step 10. Flex the hexaplexagon
Now that the basic tri-hexaflexagon has been completed, you can "flex" it as the name suggests.
- Hold the hexaplexagon in front of you with two hands.
- Squeeze 2 adjacent triangles together. Note that you should choose two triangles that are next to each other but that do not share a folded edge.
- Use your non-dominant hand to squeeze the triangles together.
- Use your dominant hand to push down the extension of the flattened edge. This extension should be a shared edge, and pushing it down should lead you to push it into the middle of the hexaplexagon.
- As the hexaplexagon opens in the center, use your dominant hand to open it on the edge and fold the triangles outwards.
