Better than Sex?
June 7, 2004 1:41 AM Subscribe
I played with these as a kid. They have a special place in Mathematics history too:
Centuries of origami had not produced such an elegantly convoluted object. Within days copies of these "flexagons" - or, as this subspecies came to be more precisely known, "hexahexaflexagons" (six sides, six internal faces) - were circulating across the [Princeton] dining hall at lunch and dinner. The steering committee of the flexagon investigation soon comprised Stone, Tukey, a mathematician named Bryant Tuckerman, and their physicist friend [Richard] Feynman. Honing their dexterity with paper and tape, they made hexaflexagons with twelve faces buried amid the folds, then twenty-four, then forty-eight. The number of varieties within each species rose rapidly according to a law that was far from evident. The theory of flexigation flowered, acquiring the flavor, if not quite the substance, of a hybrid of topology, and network theory. Feynman's best contribution was the invention of a diagram, called in retrospect the Feynman diagram, that showed all the possible paths through a hexaflexagon.
Seventeen years later, in 1959, the flexagons reached Scientific American in an article under the byline of Martin Gardner. "Flexagons" launched Gardner's career as a minister to the nation's recreational-mathematics underground, through twenty-five years of "Mathematical Games" columns and more than forty books. His debut article both captured and fed a minor craze. Flexagons were printed as advertising flyers and greeting cards. They inspired dozens of scholarly or semischolarly articles and several books. Among the hundreds of letters the article provoked was one from the Allen B. Du Mont Laboratories in New Jersey that began:
"Sirs:
I was quite taken with the article entitled "Flexagons" in your December issue. It took us only six or seven hours to paste the hexahexaflexagon together in the proper configuration. Since then it has been a source of continuing wonder.
But we have a problem. This morning one of our fellows was sitting flexing the hexahexaflexagon idly when the tip of his necktie became caught in one of the folds. With each successive flex, more of his tie vanished into the flexagon. With the sixth flexing he disappeared entirely.
We have been flexing the thing madly, and can find no trace of him, but we have located a sixteenth configuration of hexahexaflexagon……"
-from James Gleick's Genius
posted by vacapinta at 2:51 AM on June 7, 2004 [1 favorite]
Centuries of origami had not produced such an elegantly convoluted object. Within days copies of these "flexagons" - or, as this subspecies came to be more precisely known, "hexahexaflexagons" (six sides, six internal faces) - were circulating across the [Princeton] dining hall at lunch and dinner. The steering committee of the flexagon investigation soon comprised Stone, Tukey, a mathematician named Bryant Tuckerman, and their physicist friend [Richard] Feynman. Honing their dexterity with paper and tape, they made hexaflexagons with twelve faces buried amid the folds, then twenty-four, then forty-eight. The number of varieties within each species rose rapidly according to a law that was far from evident. The theory of flexigation flowered, acquiring the flavor, if not quite the substance, of a hybrid of topology, and network theory. Feynman's best contribution was the invention of a diagram, called in retrospect the Feynman diagram, that showed all the possible paths through a hexaflexagon.
Seventeen years later, in 1959, the flexagons reached Scientific American in an article under the byline of Martin Gardner. "Flexagons" launched Gardner's career as a minister to the nation's recreational-mathematics underground, through twenty-five years of "Mathematical Games" columns and more than forty books. His debut article both captured and fed a minor craze. Flexagons were printed as advertising flyers and greeting cards. They inspired dozens of scholarly or semischolarly articles and several books. Among the hundreds of letters the article provoked was one from the Allen B. Du Mont Laboratories in New Jersey that began:
"Sirs:
I was quite taken with the article entitled "Flexagons" in your December issue. It took us only six or seven hours to paste the hexahexaflexagon together in the proper configuration. Since then it has been a source of continuing wonder.
But we have a problem. This morning one of our fellows was sitting flexing the hexahexaflexagon idly when the tip of his necktie became caught in one of the folds. With each successive flex, more of his tie vanished into the flexagon. With the sixth flexing he disappeared entirely.
We have been flexing the thing madly, and can find no trace of him, but we have located a sixteenth configuration of hexahexaflexagon……"
-from James Gleick's Genius
posted by vacapinta at 2:51 AM on June 7, 2004 [1 favorite]
I've just built the smallest example. I'm now buggered if I can work out how to flex it! Just like normal origami instructions, I can't seem to translate the 2d diagrams into 3d.
Good link, though.
posted by salmacis at 3:36 AM on June 7, 2004
Good link, though.
posted by salmacis at 3:36 AM on June 7, 2004
There's six equilateral triangles in the hexagon, right, salmacis? Rotate the flexagon so that one of those triangles is at 12 o'clock. There should be a fold at 3o'clock. Put your middle finger behind the fold and push down a bit on either side of the fold with your index finger and thumb. As you start to flex the flexagon, the fold you just worked on, and every other fold around the center, should fold 'up' a bit. Now, push in or down on the fold at 9 o'clock. If the thing doesn't try to flex, rotate the whole flexagon sixty degrees (one fold) and try again.
Thanks, kaibatsu, for posting this - I've astounded friends and had the image of 'incurable geek' cemented to co-worker's opinion of me for *years* using hexa-flexagons. (of course, I also made a regular dodecahedron out of the magnetic sticks'n'balls at a bar the other night.) The highest regular hexa (using straight tape) I can make is a 12-sided one.
posted by notsnot at 5:14 AM on June 7, 2004
Thanks, kaibatsu, for posting this - I've astounded friends and had the image of 'incurable geek' cemented to co-worker's opinion of me for *years* using hexa-flexagons. (of course, I also made a regular dodecahedron out of the magnetic sticks'n'balls at a bar the other night.) The highest regular hexa (using straight tape) I can make is a 12-sided one.
posted by notsnot at 5:14 AM on June 7, 2004
I eventually worked it out myself with the 3 side version. I've now built the 5-side version, and I can't work out how to flex that one... I suspect that as the number of sides increases, so does the necessary accuracy required in construction.
posted by salmacis at 5:18 AM on June 7, 2004
posted by salmacis at 5:18 AM on June 7, 2004
I got into these when I was a kid, through reading anthologies of Martin Gardner's 'Mathematical Recreations,' and had a lot of fun. It's great to be reminded, kaibatsu! I'm off to make one now.
posted by carter at 6:11 AM on June 7, 2004
posted by carter at 6:11 AM on June 7, 2004
Does anyone else remember a commercial variation from the 1970's called, I believe, an "Inverflex"? I bought one from a science museum, it was covered with pictures of outer space, and looked a bit like a cross between a flexagon, a Hoberman sphere, and a doughnut. I've looked on the web and found bubkes.
posted by Stoatfarm at 7:10 AM on June 7, 2004
posted by Stoatfarm at 7:10 AM on June 7, 2004
Have to take the opportutity to plug my free app:
http://hexaflexagon.sourceforge.net/
It lets you take six arbitrary pictures and make a custom hexaflexagon.
posted by Flat Feet Pete at 8:29 AM on June 7, 2004
http://hexaflexagon.sourceforge.net/
It lets you take six arbitrary pictures and make a custom hexaflexagon.
posted by Flat Feet Pete at 8:29 AM on June 7, 2004
*relieved that he has, in fact, not found a link too nerdish for Metafilter, kaibutsu sets the bar higher for next time...*
I really wanted, but don't have the time just yet, to figure out the connection between the flexagons and the Catalan number. Which would likely help the second link's author with his own theories...
posted by kaibutsu at 1:31 PM on June 7, 2004
I really wanted, but don't have the time just yet, to figure out the connection between the flexagons and the Catalan number. Which would likely help the second link's author with his own theories...
posted by kaibutsu at 1:31 PM on June 7, 2004
I've been trying to figure out good ways to do the neccesary paper folding with the absolute minimum tools: Pen and Paper only. It's good to every once in a while remember that yes, Galois was right and Euclid not stupid, and that 60-degree angels are inherently Difficult.
*continues building his Archimedes spiral...*
posted by kaibutsu at 1:36 PM on June 7, 2004
*continues building his Archimedes spiral...*
posted by kaibutsu at 1:36 PM on June 7, 2004
Kaibatsu, you don't need anything but a straight strip of paper. Fold the paper any which way you like. Then fold the paper again at the same point, making sure your first fold lines up with the edge of the paper. Where fold #2 touches the other edge of the strip of paper, make a third fold, such that the length of fold #2 is along the edge of the paper, ad infinitum...within five folds, the difference between your angle and the 60 degrees you want will be minimal. (the error will change sign, and its absolute value will be halved with each sucessive fold.)
posted by notsnot at 3:42 PM on June 7, 2004
posted by notsnot at 3:42 PM on June 7, 2004
Awesome, notsnot.
And so it is demonstrated that the difference between Galois theory and the real world is in the error term!
posted by kaibutsu at 4:57 PM on June 7, 2004
And so it is demonstrated that the difference between Galois theory and the real world is in the error term!
posted by kaibutsu at 4:57 PM on June 7, 2004
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posted by kaibutsu at 1:44 AM on June 7, 2004