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A flexagon
is a polygon folded from paper with a very remarkable quality; you can
turn it inside out to make it reveal hidden surfaces.
History
It all started in 1939 when Arthur H. Stone,
a twenty-three-year-old student, was cutting American papers to make them
fit into an English file. Left over were some strips of paper that he began
to fold in different ways. One of the figures that he made caught his attention.
He had folded a strip to a hexagon and joined the ends and when he put
three corners together the hexagon turned around and showed a hidden side.
He had discovered the tri-hexaflexagon.
Arthur suspected that it should be possible
to make more complex hexagons with more hidden sides. The next day he managed
to make the hexa-hexa-flexagon which has six sides (four hidden). He showed
some friends what he had discovered and soon the whole school was making
flexagons and many new were discovered.
How to Make a Hexa-hexa-flexagon
You will need a strip of paper, a pair of
scissors and some glue to make a hexa-hexa-flexagon. To make it easier
to follow the instructions I have painted the strip with six different
colors, one for each side of the flexagon. Just follow the instructions
below:
| Front
side: |
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| Back side: |
 |
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Cut and paint a strip of paper.
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Take a strip of paper and paint 19 equilateral
triangles on each side. Then cut the paper according to the figure above.
If you want to you can use numbers or other markings instead of colors
separate the different sides of the flexagon.
Fold the paper in both directions along
the edges of the triangles to make the folds straight and sharp. This will
help a lot later when the flexagon is complete and we start to flex it.
Fold the strip to a spiral.
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Then fold
the strip to a spiral. One side of the strip will be on the outside and
one will be on the inside. There should be no gaps between each turn. The
gaps in the figure are for visibility only. |
| Now we should
fold the spiral to the shape of a hexagon: Fold the A- and B-fold in a
way that hide the red triangles and make one side of the hexagon orange
and one side yellow. |
Fold the spiral to a hexagon.
|
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Notice that
the end of the strip goes on top of the first end.
Make a final fold to make the glue marks
touch each other and then glue them together. Ta-da! You have made yourself
a hexa-hexa-flexagon. |
Flexing the Flexagon
| Now to the
funny part. Bring three alternating corners together (A) by folding them
in to the picture. Then the flexagon will open like a flower at B and reveal
hidden surfaces. The B corners will now be at the edge and the A corners
will be at the center on the back of the flexagon. If the flexagon doesn't
open like a flower, you have to flatten it again and try with the other
three alternating corners (unmarked). The hexa-hexa-flexagon has six different
faces. Can you find them all? |
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Books
Martin Gardner (1959), The Scientific American
Book of Mathematical puzzles and Diversions (Rolig matematik in Sweden),
Simon and Shuster, Inc., New York.
Madeline Jones (1966), The Mysterious Flexagons,
Crown Publishers, Inc., New York.
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