In geometry, flexagons are flat models, usually constructed by folding strips of paper, that can be flexed or folded in certain ways to reveal faces besides the two that were originally on the back and front.
Flexagons are usually square or rectangular (tetraflexagons) or hexagonal (hexaflexagons). A prefix can be added to the name to indicate the number of faces that the model can display, including the two faces (back and front) that are visible before flexing. For example, a hexaflexagon with a total of six faces is called a hexahexaflexagon.
In hexaflexagon theory (that is, concerning flexagons with six sides), flexagons are usually defined in terms of pats.[1][2]
Two flexagons are equivalent if one can be transformed to the other by a series of pinches and rotations. Flexagon equivalence is an equivalence relation.[1]
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Hexaflexagons
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PAPER FLEXAGON ENGLISH 20MB
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브롤스타즈 종이접기 신기한 플렉사곤 접기 origami flexagon
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How to make a paper moving FLEXAGON
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Origami flexagon
Transcription
So say you just moved from England to the US and you've got your old school supplies from England and your new school supplies from the US and it's your first day of school and you get to class and find that your new American paper doesn't fit in your old English binder. The paper is too wide, and hangs out. So you cut off the extra and end up with all these strips of paper. And to keep yourself amused during your math class you start playing with them. And by you, I mean Arthur H. Stone in 1939. Anyway, there's lots of cool things you can do with a strip of paper. You can fold it into shapes. And more shapes. Maybe spiral it around snugly like this. Maybe make it into a square. Maybe wrap it into a hexagon with a nice symmetric sort of cycle to the flappy parts. In fact, there's enough space here to keep wrapping the strip, and then your hexagon is pretty stable. And you're like, "I don't know, hexagons aren't too exciting, but I guess it has symmetry or something." Maybe you could kinda fold it so the flappy parts are down and the unflappy parts are up. That's symmetric, and it collapses down into these three triangles, which collapse down into one triangle, and collapsible hexagons are, you suppose, cool enough to at least amuse you a little bit during your class. And then, since hexagons have six-way symmetry, you decide to try this three-way fold the other way, with flappy parts up, and are collapsing it down when suddenly the inside of your hexagon decides to open right up. What? You close it back up and undo it. Everything seems the same as before, the center is not open-uppable. But when you fold it that way again, it, like, flips inside-out. Weird. This time, instead of going backwards, you try doing it again. And again. And again. And again. And you want to make one that's a little less messy, so you try again with another strip and tape it nicely into a twisty-foldy loop. You decide that it would be cool to color the sides, so you get out a highlighter and make one yellow. Now you can flip from yellow side to white side. Yellow side, white side, yellow side, white side Hmm. White side? What? Where did the yellow side go? So you go back, and this time you color the white side green, and find that your paper has three sides. Yellow, white, and green. Now this thing is definitely cool. Therefore, you need to name it. And since it's shaped like a hexagon and you flex it and flex rhymes with hex, hexaflexagon it is. That night, you can't sleep because you keep thinking about hexaflexagons. And the next day, as soon as you get to your math class you pull out your paper strips. You had made this sort of spirally folded paper that folds into again, the shape of a piece of paper, and you decide to take that And use it like a strip of paper to make a hexaflexagon. Which would totally work, but it feels sturdier with the extra paper. And you color the three sides and are like, Orange, yellow, pink. And you're sort of trying to pay attention to class. Math, yeah. Orange, yellow, pink. Orange, yellow, white? Wait a second. Okay, so you color that one green. And now it's orange, yellow, green. Orange, yellow, green. Who knows where the pink side went? Oh, there it is. Now it's back to orange, yellow, pink. Orange, yellow, pink. Hmm. Blue. Yellow, pink, blue. Yellow, pink, blue. Yellow, pink, huh. With the old flexagon, you could only flex it one way, flappy way up. But now there's more flaps. So maybe you can fold it both ways. Yes, one goes from pink to blue, but the other, from pink to orange. And now, one way goes from orange to yellow, but the other way goes from orange to...neon yellow. During lunch you want to show this off to one of your new friends, Bryant Tuckerman. You start with the original, simple, three-faced hexaflexagon, which you call the trihexaflexagon. And he's like, whoa! and wants to learn how to make one. And you're like, it's easy! Just start with a paper strip, fold it into equilateral triangles, and you'll need nine of them, and you fold them around into this cycle and make sure it's all symmetric. The flat parts are diamonds, and if they're not, then you're doing it wrong. And then you just tape the first triangle to the last along the edge, and you're good. But Tuckerman doesn't have tape. After all, it was invented only 10 years ago. So he cuts out ten triangles instead of nine, and then glues the first to the last. Then you show him how to flex it by pinching around a flappy part and pushing in on the opposite side to make it flat and trianglly, and then opening from the center. You decide to start a flexagon commitee together to explore the mysteries of flexagation. But that will have to wait until next time.
History
Discovery and introduction of the hexaflexagon
The discovery of the first flexagon, a trihexaflexagon, is credited to the British mathematician Arthur H. Stone, while a student at Princeton University in the United States in 1939. His new American paper would not fit in his English binder so he cut off the ends of the paper and began folding them into different shapes.[3] One of these formed a trihexaflexagon. Stone's colleagues Bryant Tuckerman, Richard Feynman, and John Tukey became interested in the idea and formed the Princeton Flexagon Committee. Tuckerman worked out a topological method, called the Tuckerman traverse, for revealing all the faces of a flexagon.[4] Tuckerman traverses are shown as a diagram that maps each face of the flexagon to each other face. In doing so, he realized that each face does not always appear in the same state.
Flexagons were introduced to the general public by Martin Gardner in the December 1956 issue of Scientific American in an article so well-received that it launched Gardner's "Mathematical Games" column which then ran in that magazine for the next twenty-five years.[3][5] In 1974, the magician Doug Henning included a construct-your-own hexaflexagon with the original cast recording of his Broadway show The Magic Show.
Attempted commercial development
In 1955, Russell Rogers and Leonard D'Andrea of Homestead Park, Pennsylvania applied for a patent, and in 1959 they were granted U.S. Patent number 2,883,195 for the hexahexaflexagon, under the title "Changeable Amusement Devices and the Like."
Their patent imagined possible applications of the device "as a toy, as an advertising display device, or as an educational geometric device."[6] A few such novelties were produced by the Herbick & Held Printing Company, the printing company in Pittsburgh where Rogers worked, but the device, marketed as the "Hexmo", failed to catch on.
Varieties
Tetraflexagons
Tritetraflexagon
The tritetraflexagon is the simplest tetraflexagon (flexagon with square sides). The "tri" in the name means it has three faces, two of which are visible at any given time if the flexagon is pressed flat. The construction of the tritetraflexagon is similar to the mechanism used in the traditional Jacob's Ladder children's toy, in Rubik's Magic and in the magic wallet trick or the Himber wallet.
The tritetraflexagon has two dead ends, where you cannot flex forward. To get to another face you must either flex backwards or flip the flexagon over.
Hexatetraflexagon
A more complicated cyclic hexatetraflexagon requires no gluing. A cyclic hexatetraflexagon does not have any "dead ends", but the person making it can keep folding it until they reach the starting position. If the sides are colored in the process, the states can be seen more clearly.
Contrary to the tritetraflexagon, the hexatetraflexagon has no dead ends, and does not ever need to be flexed backwards.
Hexaflexagons
Hexaflexagons come in great variety, distinguished by the number of faces that can be achieved by flexing the assembled figure. (Note that the word hexaflexagons [with no prefixes] can sometimes refer to an ordinary hexahexaflexagon, with six sides instead of other numbers.)
Trihexaflexagon
A hexaflexagon with three faces is the simplest of the hexaflexagons to make and to manage, and is made from a single strip of paper, divided into nine equilateral triangles. (Some patterns provide ten triangles, two of which are glued together in the final assembly.)
To assemble, the strip is folded every third triangle, connecting back to itself after three inversions in the manner of the international recycling symbol. This makes a Möbius strip whose single edge forms a trefoil knot.
Hexahexaflexagon
This hexaflexagon has six faces. It is made up of nineteen triangles folded from a strip of paper.
Once folded, faces 1, 2, and 3 are easier to find than faces 4, 5, and 6.
An easy way to expose all six faces is using the Tuckerman traverse, named after Bryant Tuckerman, one of the first to investigate the properties of hexaflexagons. The Tuckerman traverse involves the repeated flexing by pinching one corner and flex from exactly the same corner every time. If the corner refuses to open, move to an adjacent corner and keep flexing. This procedure brings you to a 12-face cycle. During this procedure, however, 1, 2, and 3 show up three times as frequently as 4, 5, and 6. The cycle proceeds as follows:
- 1 → 3 → 6 → 1 → 3 → 2 → 4 → 3 → 2 → 1 → 5 → 2
And then back to 1 again.
Each color/face can also be exposed in more than one way. In figure 6, for example, each blue triangle has at the center its corner decorated with a wedge, but it is also possible, for example, to make the ones decorated with Y's come to the center. There are 18 such possible configurations for triangles with different colors, and they can be seen by flexing the hexahexaflexagon in all possible ways in theory, but only 15 can be flexed by the ordinary hexahexaflexagon. The 3 extra configurations are impossible due to the arrangement of the 4, 5, and 6 tiles at the back flap. (The 60-degree angles in the rhombi formed by the adjacent 4, 5, or 6 tiles will only appear on the sides and never will appear at the center because it would require one to cut the strip, which is topologically forbidden.)
Hexahexaflexagons can be constructed from different shaped nets of eighteen equilateral triangles. One hexahexaflexagon, constructed from an irregular paper strip, is almost identical to the one shown above, except that all 18 configurations can be flexed on this version.
Other hexaflexagons
While the most commonly seen hexaflexagons have either three or six faces, variations exist with any number of faces. Straight strips produce hexaflexagons with a multiple of three number of faces. Other numbers are obtained from nonstraight strips, that are just straight strips with some joints folded, eliminating some faces. Many strips can be folded in different ways, producing different hexaflexagons, with different folding maps.
Higher order flexagons
Right octaflexagon and right dodecaflexagon
In these more recently discovered flexagons, each square or equilateral triangular face of a conventional flexagon is further divided into two right triangles, permitting additional flexing modes.[7] The division of the square faces of tetraflexagons into right isosceles triangles yields the octaflexagons,[8] and the division of the triangular faces of the hexaflexagons into 30-60-90 right triangles yields the dodecaflexagons.[9]
Pentaflexagon and right decaflexagon
In its flat state, the pentaflexagon looks much like the Chrysler logo: a regular pentagon divided from the center into five isosceles triangles, with angles 72–54–54. Because of its fivefold symmetry, the pentaflexagon cannot be folded in half. However, a complex series of flexes results in its transformation from displaying sides one and two on the front and back, to displaying its previously hidden sides three and four.[10]
By further dividing the 72-54-54 triangles of the pentaflexagon into 36-54-90 right triangles produces one variation of the 10-sided decaflexagon.[11]
Generalized isosceles n-flexagon
The pentaflexagon is one of an infinite sequence of flexagons based on dividing a regular n-gon into n isosceles triangles. Other flexagons include the heptaflexagon,[12] the isosceles octaflexagon,[13] the enneaflexagon,[14] and others.
Nonplanar pentaflexagon and nonplanar heptaflexagon
Harold V. McIntosh also describes "nonplanar" flexagons (i.e., ones which cannot be flexed so they lie flat); ones folded from pentagons called pentaflexagons,[15] and from heptagons called heptaflexagons.[16] These should be distinguished from the "ordinary" pentaflexagons and heptaflexagons described above, which are made out of isosceles triangles, and they can be made to lie flat.
In popular culture
Flexagons are also a popular book structure used by artist's book creators such as Julie Chen (Life Cycle) and Edward H. Hutchins (Album and Voces de México). Instructions for making tetra-tetra-flexagon and cross-flexagons are included in Making Handmade Books: 100+ Bindings, Structures and Forms by Alisa Golden.[17]
A high-order hexaflexagon was used as a plot element in Piers Anthony's novel 0X, in which a flex was analogous to the travel between alternate universes.[18]
Vi Hart, a well-known recreational mathematician and public educator, gained attention for her video on hexaflexagons.
See also
References
- ^ a b Oakley, C. O.; Wisner, R. J. (March 1957). "Flexagons". The American Mathematical Monthly. 64 (3). Mathematical Association of America: 143–154. doi:10.2307/2310544. JSTOR 2310544.
- ^ Anderson, Thomas; McLean, T. Bruce; Pajoohesh, Homeira; Smith, Chasen (January 2010). "The combinatorics of all regular flexagons". European Journal of Combinatorics. 31 (1): 72–80. doi:10.1016/j.ejc.2009.01.005.
- ^ a b Gardner, Martin (December 1956). "Flexagons". Scientific American. Vol. 195, no. 6. pp. 162–168. doi:10.1038/scientificamerican1256-162. JSTOR 24941843. OCLC 4657622161.
- ^ Gardner, Martin (1988). Hexaflexagons and Other Mathematical Diversions: The First Scientific American Book of Puzzles and Games. University of Chicago Press. ISBN 0-226-28254-6.
- ^ Mulcahy, Colm (October 21, 2014). "The Top 10 Martin Gardner Scientific American Articles". Scientific American.
- ^ Rogers, Russell E.; Andrea, Leonard D. L. (April 21, 1959). "Changeable amusement devices and the like" (PDF). Freepatentsonline.com. U.S. Patent 2883195. Archived (PDF) from the original on June 14, 2011. Retrieved January 13, 2011.
- ^ Schwartz, Ann (2005). "Flexagon Discovery: The Shape-Shifting 12-Gon". Eighthsquare.com. Retrieved October 26, 2012.
- ^ Sherman, Scott (2007). "Octaflexagon". Loki3.com. Retrieved October 26, 2012.
- ^ Sherman, Scott (2007). "Dodecaflexagon". Loki3.com. Retrieved October 26, 2012.
- ^ Sherman, Scott (2007). "Pentaflexagon". Loki3.com. Retrieved October 26, 2012.
- ^ Sherman, Scott (2007). "Decaflexagon". Loki3.com. Retrieved October 26, 2012.
- ^ Sherman, Scott (2007). "Heptaflexagon". Loki3.com. Retrieved October 26, 2012.
- ^ Sherman, Scott (2007). "Octaflexagon: Isosceles Octaflexagon". Loki3.com. Retrieved October 26, 2012.
- ^ Sherman, Scott (2007). "Enneaflexagon: Isosceles Enneaflexagon". Loki3.com. Retrieved October 26, 2012.
- ^ McIntosh, Harold V. (August 24, 2000). "Pentagonal Flexagons". Cinvestav.mx. Universidad Autónoma de Puebla. Retrieved October 26, 2012.
- ^ McIntosh, Harold V. (March 11, 2000). "Heptagonal Flexagons". Cinvestav.mx. Universidad Autónoma de Puebla. Retrieved October 26, 2012.
- ^ Golden, Alisa J. (2011). Making Handmade Books: 100+ Bindings, Structures & Forms. Lark Crafts. pp. 130, 132–133. ISBN 978-1-60059-587-5.
- ^ Collings, Michael R. (1984). Piers Anthony. Starmont Reader's Guide #20. Borgo Press. pp. 47–48. ISBN 0-89370-058-4.
Bibliography
- Martin Gardner wrote an excellent introduction to hexaflexagons in the December 1956 Mathematical Games column in Scientific American. It also appears in:
- The "Scientific American" Book of Mathematical Puzzles and Diversions. Simon & Schuster. 1959.
- Hexaflexagons and Other Mathematical Diversions: The First "Scientific American" Book of Puzzles and Games. University of Chicago Press. 1988. ISBN 0-226-28254-6.
- The Colossal Book of Mathematics. W. W. Norton & Co. 2001. ISBN 0-393-02023-1.
- Hexaflexagons, Probability Paradoxes, and the Tower of Hanoi: Martin Gardner's First Book of Mathematical Puzzles and Games. Cambridge University Press. 2008. ISBN 978-0-521-73525-4.
- Gardner, Martin (January 2012). "Hexaflexagons". The College Mathematics Journal. 43 (1): 2–5. doi:10.4169/college.math.j.43.1.002. JSTOR 10.4169/college.math.j.43.1.002. S2CID 218544330. The issue also contains another article by Pook, and one by Iacob, McLean, and Hua.
- Jones, Madeline (1966). The Mysterious Flexagons: An Introduction to a Fascinating New Concept in Paper Folding. Crown Publishers.
- Mitchell, David (2000). The Magic of Flexagons – Paper curiosities to cut out and make. Tarquin. ISBN 1-899618-28-7.
- Pook, Les (2006). Flexagons Inside Out. Cambridge University Press. ISBN 0-521-81970-9.
- Pook, Les (2009). Serious Fun with Flexagons, A Compendium and Guide. Springer. ISBN 978-90-481-2502-9.
External links
- My Flexagon Experiences by Harold V. McIntosh – contains historical information and theory
- The Flexagon Portal – Robin Moseley's site has patterns for a large variety of flexagons.
- Flexagons
- Flexagons – Scott Sherman's site, with variety of flexagons of different shapes.
- MathWorld's page on tetraflexagons, including three nets
- Flexagons – 1962 paper by Antony S. Conrad and Daniel K. Hartline (RIAS)
- MathWorld entry on Hexaflexagons
- Yutaka Nishiyama (2010). "General Solution for Multiple Foldings of Hexaflexagons" IJPAM, Vol. 58, No. 1, 113–124. "19 faces of Flexagons"
- Vi Hart's video on Hexaflexagons part 1 part 2