Talk:Flexagon

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[edit] Early comments

this really needs a diagram. Kingturtle 08:12, 5 Mar 2004 (UTC)

Provided a diagram of the tritetraflexagon. I'd like to do ones for the larger tetraflexagons too, but it'd just be duplication of the MathWorld page. --AlexChurchill 11:18, Sep 7, 2004 (UTC)

I would just like to point out that the so-called Tuckerman Traverse was also independently worked out by me, myself, when I was 11 years old, without any help from anyone. Bloody poncy mathematicians with their "look at us, we're so clever" attitude. A child could work that stuff out, and did. So there. Bonalaw

In reply to the last comment:
No one says that one needs to study maths for some 10 years and get those cert. to be good in maths. Even the most difficult unsolved maths problem didn't relate to anything advanced, it's just persistance and the faith that matter.
I remember from somewhere a physican said something like 'I was thinking a physics problem hard and have no idea, so I go out to have a walk and relax and I saw some children on the street, I asked them about that question, and, to my surprise, they answered it correctly straight away.'
I think the point of mentioning Tuckerman is to say that he was the first recorded person to disover it. It's a very common thing in mathematics of all branches for someone to discover something for themself and then discover someone else has done it earlier. Now, if anyone has a claim they discovered the Tuckerman traverse earlier than 1940, and published information on it somewhere, that might be worth mentioning here. --AlexChurchill 11:18, Sep 7, 2004 (UTC)
Yeah, sorry, just my deadpan sense of humour. Doesn't come across very well in type, I'm afraid. (Though I really did work it out independently, that bit is true.) I was rather surprised to find the concept being named for its discoverer though, given its obviousness. Has anyone discovered that water is wet yet? I'll have that one. Bonalaw 12:47, 13 Sep 2004 (UTC)
I guess a lot of enthusiastic young readers of Martin Gardner's column also worked out the theory for hexaflexagons, part of which became known as the Tuckerman traverse. I can recall doing likewise in my teenage years. DFH 19:56, 25 July 2006 (UTC)

[edit] Tukey & Feynman "not published"?

The statement that the complete theory of flexagons developed by Tukey & Feynman was "not published" requires a citation. There are lots of theoretical works on flexagons that have been published since they worked on them, and it seems incredible that none of their early work has since surfaced. DFH 08:36, 25 July 2006 (UTC)

[edit] Tuckerman traverse

I think a new page is needed for the Tuckerman traverse and related theory, so that this article doesn't become too unwieldy as an introduction to the subject. DFH 19:52, 25 July 2006 (UTC)

[edit] Flexagons Inside Out

I found this description of the book by Les Pook, (DFH 14:28, 26 July 2006 (UTC))

Description: (182 pages) Photocopy and make flexagon nets plus explanation of maths at recreational level.Flexagons are paper models that can be flexed in different ways to change their shape. They are easy to make, and work in surprising ways. This book explains the maths behind flexagons and includes instructions to make them. Flexagons will appeal to anyone interested in puzzles or recreational maths.Flexagons are paper models that can be flexed in different ways to display different faces. They are easy to make, and work in surprising ways. This book contains numerous diagrams that the reader can photocopy and use to construct a variety of fascinating flexagons. Alongside this, the author also explains the mathematics behind these amazing creations. The technical details would require a mathematical background but the models can be made and used by anyone. Flexagons bring maths to life and will appeal to anyone interested in puzzles or recreational maths.Flexagons are hinged polygons that have the intriguing property of displaying different pairs of faces when they are flexed. Workable paper models of flexagons are easy to make and entertaining to manipulate. Flexagons have a surprisingly complex mathematical structure and just how a flexagon works is not obvious on casual examination of a paper model. Flexagons may be appreciated at three different levels. Firstly as toys or puzzles, secondly as a recreational mathematics topic and finally as the subject of serious mathematical study. This book is written for anyone interested in puzzles or recreational maths. No previous knowledge of flexagons is assumed, and the only pre-requisite is some knowledge of elementary geometry. An attractive feature of the book is a collection of nets, with assembly instructions, for a wide range of paper models of flexagons. These are printed full size and laid out so they can be photocopied.1. Making and flexing flexagons; 2. Early history of flexagons; 3. Geometry of flexagons; 4. Hexaflexagons; 5. Hexaflexagon variations; 6. Square flexagons; 7. Introduction to convex polygon flexagons; 8. Typical convex polygon flexagons; 9. Ring flexagons; 10. Distorted polygon flexagons; 11. Flexahedra.
  • An excellent resource for anyone with little previous knowledge to understand the basics, but with enough detail to satisfy the interest of all but the most ardent mathmos. Eureka
  • Pook's book summarizes a great deal of what is known about flexagons of all shapes and types, and contains much new material. An excellent purchase for someone who already knows something about flexagons and wants to know more. Ethan Berkove, Lafayette College.
  • This interesting book contains a wide collection of nets for making paper models of flexagons. Zentralblatt.

[edit] Images of irregular strips required to make flexagons

You can get images from The Colossal Book of Mathematics by Martin Gardner, which I possess. I unfortuantely doesn't have a scanner. If nyone would be kind enough to provide such images, it would be greatly appreciated.--24.149.204.116 14:51, 1 August 2006 (UTC)

[edit] Procedure for Creation of Hexaflexagon Strips

Flexagons of every order may be created using the following procedure.

It is known that all flexagons are eventually folded into a hexagon. In order to generate the development, all we do is reverse the process.

When folded, the hexagon has free edges indicated by thick lines in the figure (right). Unfolding at a triangle adjacent to either of these free edges generates a new triangle. The next free edge is found from this partial development by counting three triangles from the fold. This is similarly unfolded and the final free edge is again three triangles away from the previous fold. The three folds increases the number of triangles from six to nine. A tenth is added on the end to provide an overlap to stick the flexagon together, but the resulting straight strip is recognisable as the development of the trihexaflexagon.

Taking this a stage further, the free edge is initially taken at an arbitrary point along the strip. The strip is unfolded. Again, counting three triangles along the strip a 'free edge' of the higher order development is found, this is unfolded. Note: the direction of the unfolding is always the same; in this case the first unfolding moves the right hand section of the strip up, the second moves the left end down. Looking along the strip from the right hand end, the development is unwound in a clockwise direction. Anti-clockwise is also correct, but the direction of 'unwinding' cannot change half way through the process.

The process is repeated three times yielding the complete development of the next highest order flexagon (in this case the tetrahexaflexagon).

By repeating this process, developments of all possible hexaflexagons can be produced. Different variants arise from different choices of start position, and the ancestry of the development.

I have not included this in the article because I cannot find a reference for it. Gordon Vigurs 12:09, 10 December 2006 (UTC)

[edit] Some help and feedback

In my online book thedreamingofvictoria.com I have used a trihexaflexagon in the narrative, and have added pages to demonstrate the making of a trihexaflexagon, and how a maze can be added to a trihexaflexagon's surface. There are photos and a video page. I would really appreciate any comments, especially if I can make things clearer, or have made a mistake. TIA. David Horsley 222.144.134.247 08:57, 27 December 2006 (UTC)

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