Flexagons
Contents of this Page
What is a Flexagon?
How to make a Trihexaflexagon
Flexing a Flexagon
The Trihexaflexagon 
The Tetrahexaflexagon
Higher Flexagons
Tetraflexagons
Flexagons on the Internet
References
My Comments...
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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 top
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. 


If you would like to show the sequence in a diagram, you can make two.
First you can draw a quadrilateral with two diagonals.

You can recognize the sequence 1/34/1/32/1/34/1/32/.


The second diagram shows two triangles with one  common point. 

You can recognize the sequence 1/34/1/32/1/34/1/32/.



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

134/1/32/15/2/   :||
If you pile the number 5 triangles, you get the shape of the tetrahexaflexagon with the same numbers. Go on like above. 

Hexahexaflexagon


.
1/236/2/315/3/124/   :||
If you pile the number 6 triangles, you get the shape of the Pentahexaflexagon with the same numbers. Go on like above. 
Hexahexaflexagon (Variation B)

123/14/3/125/16/5/   :||
Hexahexaflexagon (Variation C)

1256/2/51/23/14/3   :||

You find a detailed description of this flexagon on my German page Hexahexaflexagon.

Heptahexaflexagon


1367/3/61/324/3   :||

If you pile the number 7 triangles, you get the shape of the Hexahexaflexagon with the same numbers. Go on like above. 

Heptahexaflexagon (Variation B)


1257/2/516/5/123/14/3/   :||

If you pile the number 7 triangles, you get the shape of the Hexahexaflexagon (Variation B) with the same numbers. 
Go on like above. 

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 and on my page Hexahexaflexagon (German only).

Flexagons on the Internet top
 

German

Claus Michael Ringel
Flexagone

Randolf Rehfeld
Flexagone

Wikipedia
Flexagon



English

Antonio Carlos M. de Queiroz
Hexaflexagons

David King
Flexagons

Douglas C. George
Flexagon

Ela Schwartz 
Flexagon Fever

Eric W. Weisstein
Flexagon

Erik Demaine
Flexagon

Infinity12
Hexaflexagons

Jill Britton
Let's Make a Flexagon
Foto-TriHexaFlexagon (THF)  (Description of Fernando G. Sörensen's program, program available) 


Program Foto-THF 1.2 (198Kb - fthf12.zip)
Choose "Opciones/Idioma/English (USA)"!

Kathryn Huxtable's
Flexagon Page

Kjartan Poskitt
The Fabulous Flexagons

Les Pook
Flexagons

Martin Gardner
Hexaflexagons
Cambridge University Press, Martin Gardnerís First Book of Mathematical Puzzles and Games (Excerpt)

Lee Stemkoski    (Mathematrix)
Flexagons

NN
Flexifier  (Make your own tetra-tetraflexagon.)

Peter Bradshaw
Flexagon Creator!

Robin Moseley
The Flexagon Portal
 
Scott Sherman
Flexagons
Scott Sherman bietet viele Variationen von Flexagonen aus Dreiecken an. 

Dieses ist das "3 sided isosceles octaflexagon" (Tri-oktaflexagon) als ein Beispiel. 
Wenn man es in der Mitte öffnet, muss man wissen, dass das Achteck in 2/3 der Stellungen nicht eben liegt. 

Wikipedia
Flexagon

www.g4g-com.org
Hexaflexagon

Youtube
Flexagon, Flexigon, hexaflexagon-2   ...

Yutaka Nishiyama
GENERAL SOLUTION FOR MULTIPLE FOLDINGS OF HEXAFLEXAGONS   (.pdf file) 


References   top
(1) Martin Gardner: Mathematical Puzzles & Diversions, New York 1959
(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]
(6) Joseph S. Madachy: Madachy's Mathematical Recreations, Dover Publications Inc., 1979
(7) Les Pook: Serious Fun with Flexagons, Springer-Verlag GmbH, 2009 [ISBN-10: 9048125022] 


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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