# Danica Mckellar eat your heart outJune 22, 2005 3:46 PM   Subscribe

This person folded a piece of paper in half 12 times.
posted by 31d1 (73 comments total)

DAMNIT!
posted by NotMyselfRightNow at 3:50 PM on June 22, 2005

This needs further investigation. The picture of her shows the 'eleventh fold'. Now to my untrained eye, it doesn't look like a twelfth fold is gonna happen.
posted by Frasermoo at 3:52 PM on June 22, 2005

Who would've thought that folding a piece of paper in half a dozen times was so freakin' complicated?? I just tried it and barely got to 6. Good on you, Britney.
posted by maryh at 3:54 PM on June 22, 2005

This is the first time I've seen a photo inserted into a Mathworld article.
posted by vacapinta at 3:54 PM on June 22, 2005

Yeah, I've always kind of figured that if you used a big enough sheet of paper you could do it. A normal sheet just gets too tiny to manipulate. I wonder how big that sheet of paper is when it's un-folded.
posted by salad spork at 4:00 PM on June 22, 2005

I'm more impressed that she discovered the limiting equation than that she was physically able to do it. Very cool.
posted by cali at 4:01 PM on June 22, 2005

Britney is my new hero.
posted by selfmedicating at 4:02 PM on June 22, 2005

According to this, the sheet of paper had to be about 3/4 of a mile long.
posted by 31d1 at 4:03 PM on June 22, 2005

Me brain no unnerstand. Why really, really big paper cannot fold this many times easily? I never knowed this.
posted by Specklet at 4:03 PM on June 22, 2005

Frasermoo : "Now to my untrained eye, it doesn't look like a twelfth fold is gonna happen."

Yeah, it looks like she'd need about six more inches in length, at least...Which means the original paper would have to have been 1024 feet longer than it was.

salad spork : "I wonder how big that sheet of paper is when it's un-folded"

I can provide a minimum estimate, discounting all the curved bits that take up even more length. That paper looks approximately 2 feet long. Which would make the original paper 4096 feet long, or approximately .77 miles long. Of course, if you factor in the distance taken by the curved bits, it would be somewhat larger...perhaps even an entire mile, but I doubt that.
posted by Bugbread at 4:07 PM on June 22, 2005

On postview: Kinda disappointed to be beaten to the punch by 31d1, but kinda happy that my scratch measurements were that close.
posted by Bugbread at 4:10 PM on June 22, 2005

wow ... that's a smart kid
posted by pyramid termite at 4:13 PM on June 22, 2005

The article 31d1 cited above says she used a 4,000-foot roll of special toilet paper. I'll start:

"Hell, I used up one of those rolls this morning -- steak burritos last night!"

"I hear she found it next to one of these."

"She really wiped out that previous record."
posted by brain_drain at 4:25 PM on June 22, 2005

Seems like an honorary doctorate is in order. Or at least admittance to the school of her choice: very bright kid.
The MathWorld article given by vacapinta is much better written than the one in the FPP, so give it a read, too.

I also trust MathWorld, so I doubt this is a hoax or a mistake, Frasermoo. MathWorld understands attribution and such.
posted by teece at 4:28 PM on June 22, 2005

I'll nitpick here a little bit - to me, this doesn't really qualify as "folding" because the intersections between the resting planes aren't angular.

There's a lot about that giant mass of toilet paper that qualifies as "rolled".
posted by Caviar at 4:34 PM on June 22, 2005

Me brain no unnerstand. Why really, really big paper cannot fold this many times easily? I never knowed this.

The size of the paper has no impact on the factor which limits the number of folds. That factor is the thickness of the paper, which is why Britney used gold foil rather than wood pulp derived paper - which definitely can NOT be folded more than 8 times at most.

Every time you make a fold, that fold - the crease part - is being formed over twice the thickness of the stack before you made the fold. So, the first fold involves forming a crease in which the outer surface has to double back 180 degrees over a thickness of twice that of the paper. Next fold, it has to do that over four times the thickness, and so on. This is true no matter what the size of the original piece of paper you start with. There comes a point at which the material being folded is being asked to deform over a thickness of material which is simply too great, and which causes rupture if forced. With wood-based paper, that occurs between 128 and 256 thicknesses of paper, depending on quality. That's 7 or 8 folds.

Repeated doubling causes numbers to increase far more quickly than we intuitively think. Remember the old chess board story, about the guy who asked for rice... one grain on the first square, two on the second, four on the third, all the way to 64. He broke the king. Work out the numbers (2 to the power of 63) and you'll see why.
posted by Decani at 4:35 PM on June 22, 2005

teece: Seems like an honorary doctorate is in order. Or at least admittance to the school of her choice: very bright kid.

From the article:
She now attends U. C. Berkeley.

Go bears!
posted by Coda at 4:36 PM on June 22, 2005

I hate to play devil's advocate, but it seems like it would be well nigh impossible to guarantee that there was not a single break in the paper by the end of the folding process (except by unfolding and checking, and the unfolding process could introduce a break). And it seems to me like any break, especially on high-stress folds, would make the folding process slightly easier -- it'd let the fold compact into the space left by the break.

Not that, even if that's the case, it invalidates her work.
posted by gurple at 4:41 PM on June 22, 2005

On a quiz show here in Japan recently, a mathematician showed that if you could fold a paper 42 times, the stack would be thick enough to reach the moon.
posted by planetkyoto at 4:41 PM on June 22, 2005

I'd fold it.
posted by Kifer85 at 4:43 PM on June 22, 2005

Danica McKellar is still hotter
posted by antifuse at 4:43 PM on June 22, 2005

which is why Britney used gold foil rather than wood pulp derived paper - which definitely can NOT be folded more than 8 times at most

She started with gold foil, but then went on to use paper, and got 11 folds according to the picture. Presumably the formula quoted in the article gives about 4000 feet when you plug in paper thickness and 11 (or 12?) folds. I'd like to see the derivation of that formula. It's stated that:

"In some web pages the limits found by Britney are described as being due to thickness to width ratios of the final folds or attributed to the folder not being strong enough to fold any more times. Both explanations for the mathematical limits are incorrect and misses the actual detailed reason for the physical mathematical limit."

I would have thought the thickness to width ratio was the whole point, so what is the correct explanation? She needs to write a paper.
posted by snoktruix at 4:56 PM on June 22, 2005

Thanks, Decani. I get it now.
posted by Specklet at 4:57 PM on June 22, 2005

a mathematician showed that if you could fold a paper 42 times, the stack would be thick enough to reach the moon.

I've never believed this. It would end up being so thin that it would be practically impossible.

Possibly actually impossible, depending on whether P * 1/(2^42) is smaller than the width of paper's constituent molecules, where P is the original paper length.
posted by weston at 5:17 PM on June 22, 2005

In bed, late into the night, she wishes someone would, for once, compliment her beautiful mind.
posted by The Jesse Helms at 5:33 PM on June 22, 2005

Not to rain on the parade, but this reference notes that it was "Developed with J. R. Gallivan." A little web search turns up James R. Gallivan of Pomona, who works for Raytheon and holds this patent which looks like exactly the sort of background that might be helpful in solving a topology problem of this sort.

Not that I should point fingers, I practically did my daughter's entire clay Sphinx model this year.
posted by srt19170 at 5:46 PM on June 22, 2005

I bet you could do it by "reverse engineering" the folded paper. I guess it would be cheating, but you'd still have the same end result.
posted by cx at 6:19 PM on June 22, 2005

I'd fold it.
posted by Kifer85 at 6:28 PM on June 22, 2005

I heard the paper folding challenge when I was around the same age as Britney and gave up after the fifth or sixth fold. I also couldn't "connect all the dots using just three straight lines" until I was in my thirties.

I'd fold it.
posted by Kifer85 at 6:43 PM CST on June 22

I'd fold it.
posted by Kifer85 at 8:28 PM CST on June 22

That's twice. Ten more times and you folded it 12 times!
posted by Slack-a-gogo at 7:15 PM on June 22, 2005

I'd wad it.
posted by gorgor_balabala at 7:33 PM on June 22, 2005

So would the prez of Harvard probably.
posted by gorgor_balabala at 7:34 PM on June 22, 2005

What Caviar said.

Allowing the edges of each fold to "relax" to create a curve, as is indicated in the picture of the 11th fold, essentially removes the tension of a given fold. The curvature at the later folds is no more radical than the curvature toilet paper experiences when it's sitting on the roll, to begin with.

Still, a nice project, and a good equation.
posted by darkstar at 8:15 PM on June 22, 2005

This is a whole new experience, a step up, really, for toilet paper.
posted by weapons-grade pandemonium at 8:25 PM on June 22, 2005

Now she can get to work on the elusive "sucking a golf ball through a garden hose" problem.
posted by yhbc at 8:31 PM on June 22, 2005

First: in addition to my congratulations, I extend to Britney an offer to come to my house to fold my laundry. The challenge: I like my sheets folded in thirteenths. And I also like Downy.

Second: is it just me, or could only a pretty white girl monopolize the local mall for SEVEN hours with a giant roll of toilet paper that was the better part of a MILE long?
posted by Admiral Haddock at 8:50 PM on June 22, 2005

Second: is it just me, or could only a pretty white girl monopolize the local mall for SEVEN hours with a giant roll of toilet paper that was the better part of a MILE long?

You're just trying to provoke a poop joke, Admiral Haddock. Or should I use your real name -- THE DREADED REAR ADMIRAL!!!!!
posted by ROU_Xenophobe at 9:51 PM on June 22, 2005

This is cool.

She now attends U. C. Berkeley

Well, nobody's perfect.
posted by sellout at 11:45 PM on June 22, 2005

darkstar and caviar, why isn't that the same thing that happens to a folder piece of paper at any level? If you zoom in far enough, isn't the paper curving?

I assume that there is, also, a reason why she did not attempt a 13th fold. Why isn't that reason the same reason why you can't do a seventh or eighth fold of regular paper?
posted by Embryo at 12:46 AM on June 23, 2005

Folded it.
posted by sfslim at 2:49 AM on June 23, 2005

Most importantly - now we've identified the witch - when's the show trial and burining?
posted by SamSugar at 4:07 AM on June 23, 2005

Um.

Decani:
The size of the paper has no impact on the factor which limits the number of folds. That factor is the thickness of the paper

The article:
In some web pages the limits found by Britney are described as being due to thickness to width ratios of the final folds or attributed to the folder not being strong enough to fold any more times. Both explanations for the mathematical limits are incorrect and misses the actual detailed reason for the physical mathematical limit.

That's not to say that I understand.
posted by Marquis at 5:15 AM on June 23, 2005

Slack-a-gogo:
I also couldn't "connect all the dots using just three straight lines" until I was in my thirties.

Oh man, I'm in my thirties, and I still don't get this one. Can you believe that question was on a quiz for an interview I once took? I ended up getting the job, as I tried using creative, inventive, and humorous answers (i.e. "Say to the tester 'Hey, what's that over there?', and while his back is turned, rearrange the dots in such a way that 3 straight lines cross all of them." or "Use a really really fat marker such that it completely obliterates the puzzle.").
posted by thanotopsis at 5:42 AM on June 23, 2005

Embryo : "I assume that there is, also, a reason why she did not attempt a 13th fold. Why isn't that reason the same reason why you can't do a seventh or eighth fold of regular paper?"

Well, looking at the paper at the length of the 11th fold: the paper is about 2 feet long, and she can get one more fold out of it. Let's ignore the curvature taking extra paper in order to get the 13th fold, and assume that she used some length of paper such that instead of being 2 feet long after the 11th fold, it were 2 feet long after the 12th fold. How long would the paper have to be such that you could fold it 12 times and the paper be 2 feet at that point?

8192 feet (1.5 miles or so). My guess is she didn't have access to 1.5 miles of paper (and keep in mind I'm ignoring the amount sucked up in curvature of the bigger folds. If that was taken into account, it could have been closer to 1.8 or so miles).
posted by Bugbread at 5:48 AM on June 23, 2005

thanotopsis : "I also couldn't 'connect all the dots using just three straight lines' until I was in my thirties."

I'm extremely impressed that you got it, though, because, as far as I'm aware, that's impossible. The standard question is "connect these nine dots using just four straight lines".

Unless the questioner was going all non-Euclidean and hardcore, in which case it can be done with 1 line (fold the paper like an accordion, and run your pencil down the outer crease).
posted by Bugbread at 7:07 AM on June 23, 2005

I love MetaFilter.

A high-school girl does what mathematicians have claimed is impossible. She solves the problem they couldn't. She shows her work. They admit it: They were wrong.

Yet certain tenacious Mefites, some of whom haven't even RTFA, know better, and come up with various authoritative explanations why she didn't actually do what she did. Beautiful.
posted by soyjoy at 7:35 AM on June 23, 2005 [1 favorite]

This is the first time I've seen a photo inserted into a Mathworld article.

Vacapinta, something tells me the photo is not so much photographic evidence of a twelve-folded piece of paper but as evidence that yes, hot girls really can exist in mathematics.

Now, I'm not saying the pictures of the cute mathematician who's solved the paper folding problem with her result is totally wankable but . . . OK, that is what I'm saying.

Why focus on mathematical accomplishments when there are boobies involved? I am a shallow bastard!
posted by Anonymous at 8:30 AM on June 23, 2005

All things are possible until they are proved impossible. And even the impossible may only be so, as of now.
- Pearl S. Buck

When a distinguished but elderly scientist states that something is possible, he is almost certainly right. When he states that something is impossible, he is very probably wrong.
- Clarke's First Law

I have learned to use the word impossible with the greatest caution.
- Werner von Braun
posted by spock at 8:46 AM on June 23, 2005

I'm more impressed that she was able to overcome the incredible disadvantage of being saddled with the name "Britney".
posted by spock at 9:33 AM on June 23, 2005

And for extra credit, no less.
posted by absalom at 9:38 AM on June 23, 2005

weston: If you alternate directions when folding, you only need P/(2^21) folds per side. You'd end up with a stack 0.1 micrometers square and 1000000km high if P was originally 25cm.
posted by springload at 9:40 AM on June 23, 2005

The standard question is "connect these nine dots using just four straight lines".

To be pedantic, its 4 straight connected lines, often worded as 4 straight lines without lifting your pencil.

To be ultra pedantic, I'll quote my old math teacher and say that the term "straight line" is redundant - all lines are straight by definition. :)
posted by Bort at 9:50 AM on June 23, 2005

A high-school girl does what mathematicians have claimed is impossible. She solves the problem they couldn't. She shows her work. They admit it: They were wrong.

Out of curiosity, do you have a link justifying that statement other than this from the article: "The task was commonalty known to be impossible. Over the years the problem has been discussed by many people, including mathematicians and has been demonstrated to be impossible on TV."? Because you seem to be jumping to some conclusions.
posted by Bort at 10:01 AM on June 23, 2005

I would have thought the thickness to width ratio was the whole point, so what is the correct explanation?

It's thickness, length, and fold count that matters. And I don't think it has anything to do with deforming or rupturing as stated above. It's simply a matter of each fold uses up a certain amount of paper, which increases with the fold count. How much is used up is a factor of the thickness. So for a certain thickness and a certain fold count, you need to start with a minimum length.
posted by Bort at 10:11 AM on June 23, 2005

Which part needs "justified?" That mathematicians said it couldn't be done? Or that they now admit they were wrong? And in terms of jumping to conclusions... you did follow the links off those pages, right?
posted by soyjoy at 10:12 AM on June 23, 2005

I'm extremely impressed that you got it, though, because, as far as I'm aware, that's impossible. The standard question is "connect these nine dots using just four straight lines".posted by bugbread at 7:07 AM PST on June 23

bugbread - It can be done with three straight (sorry, Bort).
/derail
posted by ObscureReferenceMan at 10:46 AM on June 23, 2005

bugbread: you can connect the nine dots using only three (straight) lines, but the lines would have to be longer than Britney's toilet paper. Hint: the dots are not points.

I want Britney to tell us whether the toilet paper should hang out over the front of the roll, or under at the back.
posted by weapons-grade pandemonium at 10:46 AM on June 23, 2005

Wow. ORM--what are the odds synchronous posts on that after a day?
posted by weapons-grade pandemonium at 10:49 AM on June 23, 2005

She's obviously very bright, and quite attractive as well.

However, I am a bit suspicious of anyone, male or female, that would get a Raytheon researcher to help them with not even a school project, but "extra credit".

Also, if she is a math wunderkind, why does she need "extra credit" in the first place?
posted by Ynoxas at 11:11 AM on June 23, 2005

I used to have this arguement with people all the time. Anyone who has experience with binary trees will see that you can fold the paper as many times as you want if you just have a big enough piece of paper to start with. It may need to be a couple miles long but you can do it. I'm glad this woman has final done the math so I can prove it to others.
posted by Mitheral at 12:50 PM on June 23, 2005

She's obviously very bright, and quite attractive as well.

However, I am a bit suspicious of anyone, male or female, that would get a Raytheon researcher to help them with not even a school project, but "extra credit".

Also, if she is a math wunderkind, why does she need "extra credit" in the first place?

Yeah.. we all know that only those damn ASIANS are overachievers......
posted by Debaser626 at 3:01 PM on June 23, 2005

If she folds one more time she could inadvertently split an atom hidden somewhere in between the sixth and seventh folds that would collapse the matter density separating folds of paper from one another, which would cause a chain reaction in all the other atomic particles, leading to a seismonic rupture of the space time continuum and a massive explosion that would kill us all!

*dramatic chord as lightning flashes just outside a window*
posted by ZachsMind at 4:16 PM on June 23, 2005

Which part needs "justified?" That mathematicians said it couldn't be done? Or that they now admit they were wrong? And in terms of jumping to conclusions... you did follow the links off those pages, right?

I read the main 2 links and a few of the other, what, 2 dozen or so other links and didn't see mathematicians claiming it was impossible. It appears your answer is: no, you cannot provide a link. Thanks.
posted by Bort at 6:04 PM on June 23, 2005

It can be done with three straight (sorry, Bort).

Hmmmm. Are we talking about the same puzzle? 9 dots, lined up 3 by 3. And you can cross them all with 3 lines without lifting the pencil from the paper?
posted by Bort at 6:06 PM on June 23, 2005

"However, I am a bit suspicious of anyone, male or female, that would get a Raytheon researcher to help them with not even a school project, but "extra credit"

I guess I didn't make it obvious enough. He's her dad.
posted by srt19170 at 6:17 PM on June 23, 2005

Bort: I'm sure you're talking about the same puzzle. It has four and three line solutions and is a favorite of lazy staff training facilitators. See the answers for both here.

And if you want to solve it first...remember the wise words of WGP "The dots are not points."
posted by ?! at 6:33 PM on June 23, 2005

Not to rain on the parade, but this reference notes that it was "Developed with J. R. Gallivan." ...Not that I should point fingers, I practically did my daughter's entire clay Sphinx model this year.

I can't help wondering if the same suggestion would be made if she were a nerdy looking asian boy...
Just because he's included as a reference doesn't mean he 'practically did the entire' project. It's entirely possible he actually was just around to help, not the driving force or inspiration.
posted by mdn at 6:43 PM on June 23, 2005

I'm sure you're talking about the same puzzle. It has four and three line solutions and is a favorite of lazy staff training facilitators.

Awesome! Thanks for the link and the hint. I never heard of the three line answer before.
posted by Bort at 7:28 PM on June 23, 2005

weapons-grade pandemonium : "bugbread: you can connect the nine dots using only three (straight) lines, but the lines would have to be longer than Britney's toilet paper. Hint: the dots are not points."

D'oh! That should have occured to me.

What I'm now wondering is if you couldn't solve it with a single straight line, considering the size and curvature of the earth.
posted by Bugbread at 8:45 PM on June 23, 2005

It appears your answer is: no, you cannot provide a link.

It appears that you're trying to be a contentious ass, but as we know from this story, what appears to be so is not always the case.

Given that the "impossibility" was disproven three years ago, one would expect reputable math sites to have updated with the current information, as indeed most of them have (and as I remarked above) but of course there are still remnants here or there... Here are two mathematicians claiming the impossibility of paper folding after 8 times, just to fulfill the exact wording of your, ahem, request.

Watch out for those appearances, bort.
posted by soyjoy at 9:29 PM on June 23, 2005

...sorry: Bort.
posted by soyjoy at 9:32 PM on June 23, 2005

It appears that you're trying to be a contentious ass, but as we know from this story, what appears to be so is not always the case.

Well, I'm sure I appeared to be a contentious ass, as I often due, but "what appears to be so is not always the case." :)

I didn't really want to get into a fight with you, so let me explain what I was after. I wanted to evaluate the mathematicians words for myself - thus the request for the link(s). Typically, when I hear that a mathematician claims something is impossible, that claim is in the form of a proof. I could not imagine the kind of proof that says you can't fold paper in half more than X times, so, if it existed, I was interested in taking a look - and seeing why it was flawed.

It turns out that in the links you provided, these mathematicians are talking in an empirical sense, not a rigorous mathematical proof sense. I'm not claiming that you said one or the other, I just wanted to see.

Let me finish by apologizing. My initial choice of words "jumping to conclusions" provoked you - which was not my intention. Your reply provoked me, which I allowed to cause me to be short, causing us to get into a small argument which I didn't intend. I really just wanted to evaluate the claims for my own edification. I hope there are no hard feelings.
posted by Bort at 5:00 AM on June 24, 2005

Nope. As far as I'm concerned, it's all water under paper under the fold. Cheers.
posted by soyjoy at 9:26 AM on June 24, 2005

...and as Bort and soyjoy clasp hands and walk into the sunset, MetaFilter theme music begins. Fade to black. Closing credits.
posted by UKnowForKids at 10:29 AM on June 24, 2005

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