What is a Flexagon?
A flexagon is a hexagon, which you can make from
a strip of triangles.
The point is: If you open the flexagon in the
middle, then a new face, which was hidden before appears.
How to make a Trihexaflexagon
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The simplest flexagon is a trihexaflexagon with three faces.
(The colours in the drawings show you the front and the reverse side.)
(1) Draw a strip of ten equilateral triangles with compasses and ruler.
The length of a triangle is e.g. 4 cm. Then the strip fits an A4 page.
(2) Number the triangles as given.
(3) Go over the lines with a ballpoint, so you can fold the paper at
the lines more easily later.
(4) Cut out the strip.
(5) Turn the strip. Number the triangles as shown. Draw the two crosses.
x (left) is behind 3, 2 behind 1 and so on. Later the triangles with a
cross are glued on top of each other. Fold the paper several times, so
that the flexagon will be more flexible.
(6) Fold the strip to form a hook. Then fold at the horizontal line backwards.
Notice that the front face has number 1 and the reverse the number 2, therefore
lay number 3 on 3.
(7) If you have succeeded in creating a hexagon, there will also be
a triangle jutting out from it. It must have a cross on the reverse side.
Glue both triangles with a cross to each other.
The flexagon is finished.
Flexagon turning left
... ... |
You can also produce a trihexaflexagon, if you fold the lower three
triangles backwards and lay the four upper triangles forward on the horizontal
line. Glue the triangles 2 and 3 to each other.
You call this flexagon a flexagon turning left.
If the thumb of the left hand (drawing) points to the strip, the fingers
show you, which way to fold it.
The flexagon above is called right turning, because you must use the
right hand now. |
Right turning Flexagons are regular.
Flexing a Flexagon top
 |
It is difficult to open the flexagon for the first time.
Take hold of two triangles from the top with two fingers. Move the
triangles down. Push two triangles, which form a rhombus, at the opposite
corner with the index finger of your left hand down up to the vertical
3D axis at the same time. Now you can open the hexagon like a flower. The
face number 3 appears.
This process is called "flexing". |
You can use two techniques to flex a flexagon continuously.
If you hold one diagonal of the hexagon horizontally while flexing,
you can open the flexagon on the left and on the right side alternately.
I call this way of flexing "swing".
There is another way of flexing continuously, called the Tuckerman traverse.
You flex, then you go to an adjacent corner, right or left. - The results
are the same.
The Trihexaflexagon top
The trihexaflexagon has 9 segments of paper with triangles on the front
and reverse side. That makes 18 triangles in whole. The triangles for gluing
don't count.
The segments are not distributed regularly. Two lay on another, between
them is a single segment.
The trihexaflexagon has the distribution 1+2+1+2+1+2.
Two triangles of a face are joint together and form a rhombus. The trihexaflexagon
has 3 rhombi on each side.
While folding, the rhombus is folded back and appears at the same place
on the reverse side.
Folding further, the two triangles of the rhombus are laid on one another.
The triangles travel through 60 degrees.
.. ...... ... |
Fix a paper clip to one triangle. If you flex the flexagon in
the way of the swing, the paper clip and so the triangle is moving anti-clockwise.
The triangle with the paper clip is turned three times at the same place
before it is moved further though the flexagon itself is not turned. |
You need 18 flexings for a full round.
If you write down the numbers of the faces appearing, you have 1/2/3/1/2/3/1/2/3...
The character "/" means that you have to change the sides. If you turn
the flexagon from the front to the rear side you have the numbers 1/3/2/1/3/2/1/3/2...
The Tetrahexaflexagon top
The tetrahexaflexagon has four faces and is a bit more complicated
than the trihexaflexagon.
|
Make a strip from triangles as the picture shows. Number the triangles
on the front and on the reverse side as shown. First put triangles 4 to
4 together. Then you have the strip of a trihexaflexagon. Fold it in the
same way as before. Glue the two triangles with a cross after folding. |
If you flex the flexagon in the way of the swing, you get the faces 1/3/2/1/3/2/1...
(You may have to turn the flexagon).
If you want to find face number 4, you have to keep opening it on the
left and on the right side as long as you can. E.g. you get the sequence
1/34/1/32/1/34/1/32/... The numbers 1/34/1/32/ come back regularly. In
1/34/1/32/ there are the numbers 1 and 3 twice, 2 and 4 once.
top
|
If you would like to show the sequence in a diagram, you can make two.
First you can draw a quadrilateral with two diagonals.
The second diagram shows two triangles with one common point.
The second (usual) diagram shows the different parts of face
1 and face 3. In face 3 you are within a triangle, in face 1 you go through
to the next triangle, if you like. |
If you are following the swing techniques, you go in the right diagram
round the whole figure anti-clockwise.
You count the segments of the tetrahexaflexagon. The segments, which
have the faces 2,3 and 4, have the distribution 1+3+1+3+1+3. Only
the hexagon with the face 1 has the distribution 2+2+2+2+2+2. This confirms
the feature of a "through station".
You could think that face 1 is given priority. But if you turn the
flexagon from the front to the reverse side and flex, then 1 and 3 change
their parts. So the symmetry is preserved.
Flexagon with a pattern
If you divide the equilateral triangles of the stripe in three parts with
the help of the centre point and colour them properly, you get a trihexaflexagon
with three nice patterns. The top and the reverse side are the same in
each case.
This version is developped by Krino Hoogestraat from Emden.
Higher Flexagons top
There are expansions to 5,6,...faces, which are called pentahexaflexagon,
hexahexaflexagon,...
Pentahexaflexagon
If you pile the number 5 triangles, you get the shape of the tetrahexaflexagon
with the same numbers. Go on like above.
Hexahexaflexagon
 .
If you pile the number 6 triangles, you get the shape of the Pentahexaflexagon
with the same numbers. Go on like above.
You find a detailed description of this flexagon on my German page
Hexahexaflexagon.
Tetraflexagons top
There are square shaped flexagons, too. I took the following tritetraflexagon
from Gardener's book from 1961, the tetratetraflexagon from David Mitchell's
recommendable book (4).
Tritetraflexagon
... ... |
Pile the squares 3 on 3, 2 on 2 and 1 on 1. Glue the cross squares
on each other. |
Tetratetraflexagon
... ... |
Start now with 4 on 4, then take the following numbers.
The centre is cut along two squares, the cross squares lie in the end.
This is hidden in the drawing. |
Flexing
... ... |
You usually find new faces of the flexagons, if you turn them and open
them like a book in the middle. |
You can find more on my page tetraflexagons.
Flexagons on the Internet
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German
Claus Michael Ringel
Flexagone
Margit Brause
Der Um-Krempler
Randolf Rehfeld
Flexagone
English
Antony S. Conrad and Daniel K. Hartline
FLEXAGONS
Artistbooks
Flexagon Funhouse
Antonio Carlos M. de Queiroz
Hexaflexagons
David King
Flexagons
David Mitchell
Frequently
Asked Questions about Flexagons
Douglas C. George
Flexagon
Ela Schwartz
Flexagon Fever
Eric W. Weisstein
Flexagon
Erik Demaine
Flexagon
Harold V. McIntosh
My
Flexagon Experiences
Infinity12
Hexaflexagons
Jill Britton
Foto-TriHexaFlexagon
(THF) (Description of Fernando G. Sörensen's program)
Kathryn Huxtable's
Flexagon Page
Kjartan Poskitt
The
Fabulous Flexagons
Lee Stemkoski (Mathematrix)
Flexagons
Les Pook
Flexagons
Martin Gardner
Hexaflexagons
Cambridge University Press, Martin Gardner’s First Book of Mathematical
Puzzles and Games (Excerpt)
Robin Moseley
The Flexagon Portal
Scott Sherman
Flexagons |
Scott Sherman offers many varieties of flexagons made from triangles.
This is the 3 sided isosceles octaflexagon ("tri-octaflexagon") as
an example.
If you open it in the centre, you must know that the octagon doesn’t
lay flat in 2/3 of the time. |
Wikipedia
Flexagon
YouTube
http://de.youtube.com/watch?v=SnnvovjcC_o
http://de.youtube.com/watch?v=KVpdt7OSOsg&feature=related
http://de.youtube.com/watch?v=2xn9ffk_LzM&feature=related
Spanish
Fernando G. Sörensen (Argentina)
Flexágonos (Flexagons)
Program Foto-THF 1.2 (198Kb - fthf12.zip)
Choose "Opciones/Idioma/English (USA)"!
References top
(1) Martin Gardner: Mathematical Puzzles & Diversions, New York
1959
I found amusing letters
about flexagons sent to the magazine <Scientific American>.
(2) Martin Gardner: The Second Scientific American Book of Mathematical
Puzzles & Diversions, New York 1961
(3) Martin Gardner: Mathematische Denkspiele, München 1987 (ISBN
3 88034 323 3)
(4) David Mitchell: The Magic of Flexagons, Norfolk England 1998 (ISBN
1 899618287)
(5) Les Pook: Flexagons Inside Out, Cambridge University Press, 2003[ISBN
0 521 52574 8 paperback]
My Comments top
Arthur H.Stone invented the flexagons in autumn 1939.
Flexagons became well known, when Martin Gardner introduced them in
the math corner of the magazine Scientific American in the
end of the 1950s.
The author looked back in his book (1) from 1959. He received more than
100 letters.
Book 3 contains instructions for making a hexatetraflexagon.
It is surprising that the flexagons weren't known in Germany. One reason
probably was that book 1 was translated into German, but the chapter about
flexagons was left out.
It is no accident that I put flexagons at the first place in my homepage.
I hardly don't know another mathematical puzzle of this quality.
Do you know kaleidocycles?
Feedback: Email address on my main page
This
page is also available in German.
URL of
my Homepage:
http://www.mathematische-basteleien.de/
©
1999 Jürgen Köller
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