Trailblazing Hexa-Dodeca-Flexagons?
Go to Flexagon Discovery: The Shape-Shifting 12-Gon
for details on the hexa-dodeca-flexagon,
a newly discovered hexagonal flexagon
with 12 pats and 6 faces that is folded
from a straight strip of paper. This flexagon
displays some unique characteristics, including
these: it flexes into a new, flat shape for
some of its flexes and it naturally mixes
leaves from different faces to form hybrid faces.
This Page Revised 13 Apr 2005
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By Vernon Gutenkunst (B.S. in Chemistry, Army Musician, B.A. in Interdepartmental Sociology)
*****What is a Flexagon?
Here is one of many similar definitions. Flexagons: flat one-piece one-sided paper band constructions capable of generating different faces or designs when manipulated or flexed. Or, I suppose you could go with Webster's Unabridged Dictionary's definition (second edition 1998). "A three dimensional figure having polygonal faces that is constructed from a folded sheet of paper in such a way that different faces are exposed when the figure is flexed along its folds." For me, I just think that Flexagons are phenomena with an infinite supply of highly diverse fascinations. Even More Definitions!
*****What are some uses for flexagons?
People of almost all ages and all educational levels have found uses for flexagons. Flexagons have been used as decorations, greeting cards, bookmarks, drink coasters, toys, marketing tools, origami creations, etc.
I'm sure that many other applications have been either used or have yet to be tried. One decorative thing that I want to add to my home are flexagon mobiles. Whenever I would want to have a different looking mobile, I could change the faces while the mobiles are in place.
I bet that you have some ideas for other flexagon applications, either simple or very complex. To make it a little easier for you to get started, I have provided some information plus some materials that you can download in order to help you get started. (See Faces/Downloads)
What an excellent way to pass the time while riding in a car, train, or plane. Kids can be kept occupied at day care centers and in waiting rooms. Babysitters might be able to spend more time on the phone, time with friends that come to visit and maybe even some more time on homework. Flexagons can be used at any time and any place to make the time go fast
(i.e.. waiting rooms, airports, standing in line). They can be easily kept out of sight while listening to boring lectures, speeches, and sermons without getting caught!
Flexagons can be made using extremely cheap materials
(i.e.. paper, pencil, glue). There is almost no limit as to how sophisticated one can get with both existing technology (computer graphics) and available materials (coasters have been made out of wood). Making and flexing flexagons aren't the only mental challenges available. Extremely complex mathematics has been applied in areas such as topology and group theory. Maybe someone will try to be added to the Guinness Book of Records for the largest functional flexagon!
+++++My Flounderings With Flexagons (Potential new 'twist' to Star Trek's "The Trouble With Tribbles?")
These magical mathematical mavericks were the topic of my first science project in 1967-68. I had a lot of fun with flexagons at that time and even though I was quite certain that I did not know what I was doing, I managed to advance all the way to the state science fair.
My interest did not stop at that state science fair, but has occasionally erupted during the past thirty plus years. My wide variety of experiences has now opened my eyes to previously unsuspected applications of my newly found version of a trihexaflexagon. Little did I know that my reflections or re'flex'ions during the summer of 1999 would lead to such a find.
It is increasingly becoming quite certain that I am presenting several new flexagon phenomena and specifically trihexaflexagon phenomena at this time. My new version of a flexagon may be added to the present large field of flexagons where it now appears that many new and widely varied applications may be explored. Imagination abounds where patterns, characteristics, and qualities of flexagons may be applied to magic squares, psychology, philosophy, topology, palindromes, complex art, and much much more.
* Projected additions and renovations: more flexagon designs, create printer friendly portions, update links, more references, refine text, add material, optimize sizes for all flexagon strip downloads, and pretty much anything else that my flexagons will give me time enough to do.
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