UKC

While out on the hill yesterday somebody came up with one of those hypothetical questions, as you do.

The question was that if you had a hypothetically large enough piece of paper and folded it 42 times how wide would the final fold be?

the said piece of paper was as large as needed but would be 0.1mm wide.

Anybody know the answer?

In reply to The Lemming: its only physically possible to fold paper in half 8 times

In reply to The Lemming:

2^42 or 4,398,046,511,104

In reply to Gaz lord:


That is not true.

I watched myth busters that physically folded a piece of paper 11 times.

In reply to TRNovice:

elephants?

In reply to The Lemming: hummm well i can only do 7 so im happy with that :P

In reply to Gaz lord: Strangely enough google turns up a paper discussing this very topic:
http://www.informaworld.com/smpp/content~content=a741574786

In reply to TRNovice:

Made the fatal exam flaw of not reading the question, if the width of the paper is 0.1 mm. Then the final width would be

439,804,651,110 mm
439,804,651 m
439,805 km

In reply to Robert Dickson:
'We argue that a piece of paper, of thickness T and width W, can be folded a maximum number N = 0.962 ln(f W/T) times, where f is a factor of order unity, determined by mechanical constraints. N typically takes a value of around six.'

look at the last sentence

'N typically takes a value of around six.'

In reply to TRNovice: hehe units this time another exam flaw sorry im in the thik of mine its all on the brain sorry :P

In reply to TRNovice:

The average distance from the Earth to the Moon is 382,500 km.

In reply to TRNovice:

We guessed it to be something like this. My mate said that the fold would fill the gap between the earth and the moon.

All that from a piece of paper

In reply to The Lemming: how big is this piece of paper? high quality? low quality? weight?

In reply to The Lemming:

As has been pointed out, it would become impossible to fold after only a few goes.

In reply to TRNovice:

Well it takes 2 folds of a square piece of paper to get to another square of half the side. So after 42 folds the side of paper would be 2^-21 (4.8x10^-7) of the length at the start

so if you started with a 1km x 1km square of infinitely thin paper then it would be about 0.5mm wide by the time you finished I reckon

In reply to Steve Culverhouse:

And you would probably have very sore fingers - to say nothing of the paper-cuts!

In reply to The Lemming:

If you were nice and neat, the fold would be 0.1mm wide

Or did you mean that the paper is 0.1mm thick?

In reply to Blue Straggler:

Paper 0.1 mm thick.

Now to get a crease that thick after 42 folds.

How big would the piece of paper need to be?

In reply to The Lemming:

Eh? You want the crease to be the thickness of the paper (0.1mm)?

In reply to Blue Straggler:

Me confused by your question.

The paper that we have to play with is 0.1mm thick.

I have no idea how big the crease would be.

However in this hypothetical question where we managed to make 42 folds spaninh some 400,00KM, how big would the piece of paper need to be?

In reply to The Lemming:

Any size - the area of the paper is immaterial to how high it will stack once folded 42 times (as pointed out above).

In reply to TRNovice: but a 1''x1'' piece of paper wont fold 42 times, it needs to be f*cking big i think. size of U.S.A probably.

In reply to The Lemming:
Assuming some magical properties that allow us to disregard tension and stretching, and assuming a 2-dimensional problem (i.e. the paper's breadth is actually irrelevant as you have not used it in the problem anyway) - to fold the strip one way and then the other (what some people call "concertina-like") if it is 0.1mm thick, then its footprint must be 0.2mm. Thus, the length of the infinitesimally thin strip, is twice as much as the height of the stack.

Rather makes sense, doesn't it - real-world piece of paper 4 inches long and 0.1mm thick, fold it in half, it's 2 inches long and 0.2mm thick now.

Make it three-dimensional, just scale appropriately.

In reply to Gaz lord:

A piece of paper the size of the US, presumably manipulated by Atlas won't fold 42 times any more than the one inch square sheet - as I think you pointed out above. Equally if we state that you can fold some sort of paper 42 times, then I submit that it would be equally possible with a one inch square .

In reply to The Lemming:

You just cannot fold the paper 12 times so the hypothetical argument fails.

Get a piece of toilet paper and fold it in half continuously and count the folds. now do it with a broadsheet newspaper and I bet you cannot do two more folds.

In reply to TRNovice:

Yes, but you can't put a gallon in a pint pot.

In this hypothetical question, how can a 1 inch piece of paper measure just under half a million killomiters after 42 folds?

Surely we're talking about a piece of paper at least a quarter to half the size of our solar system?

In reply to sutty:

I know that there are physical limitations to folding paper but this was a hypothetical question while we walked down off the hill.

Nothing more and nothing less

In reply to The Lemming:

Just done a youtube search on mythbusters and found this.

youtube.com/watch?v=RHcDP_Yew-g&

enjoy

In reply to sutty:

You can fold the paper as many times as you like, as long as you make it thin enough and large enough in area. The article linked to above makes this clear.

To achieve 42 folds, the ratio of the length (assuming square paper) to the thickness would have to be of the order of 10^19. In other words, if the thickness is 0.1mm, the length of the sides has to be around 0.1 light years!

The dependence on this ratio is logarithmic; this means you have to change the ratio a lot to change the number of folds you can make just a little (which is why most "normal" paper can only be folded around seven times).

In reply to The Lemming: Slight hijack with a somewhat easier puzzle along the same lines.

There is the classic puzzle about placing one grain of rice on the first square of a chess board, 2 on the second, 4 on the 3rd and so on until the 64th square. Imagine you instead of grains of rice, you replace them with a £1 coin. Would the resulting pile of coins on the 64th square reach the moon? How high would it be?

In reply to Liam M: get your coat :P

In reply to The Lemming:


Obviously, it depends how thick the paper is doesn't it!

In reply to Liam M:

The issue is not the number of layers of the stack, otherwise you would not be able to stack cut sheets of paper ontop of each other. The proble is after the first fold the radius of the next fold is larger. The extra material required will give a maximum number of folds dependent on the elastic properties of the paper, as each fold will require a greater amount of streach in order to go round the fold.

In reply to Andy S:

Within reason no (a piece of very stiff card won't fold as easily) - it very quickly reaches a case were the aspect ratio (overall thickness/length normal to fold and in plane of paper) becomes sufficiently large such it won't fold any more, largely irrespective of the initial dimensions. Even with a very thin piece of paper the problem still remains (or the paper tears).

It's really quite a difficult problem to define, because physically it becomes impossible and any assumptions can make it trivial.

In reply to The Lemming:

If the resulting tower had a very, very small area - i.e if there wasn't much of a roof garden on top of the skyscraper. In this of course, I'm ignoring such inconvenient issues as the size of a molecule of cellulose (and I suspect you would run into issues long before you got that far - again as pointed out above).

So if we have a piece of paper 1 inch square and 0.1 mm think then it has a volume of 0.000000064516 cubic metres.

If the tower is 439,804,651 m high, then its "roof-garden" has an area of 0.000000000000000146692400448956 metres square. If folded to make a square, each side would be 0.0000000121116638183594 metres. While I'd admit this is rather a small length it is significantly above the Planck Length (0.000000000000000001636393 m) and I guess we can discount any quantum effects .

In reply to Liam M:

Are the coins resting with their circular aspects on top of each other (i.e. in a cylindrical column), or is each balanced on top of the previous one, with its cylindrical aspect facing outwards?

In reply to TRNovice:

If the former, , then each pound coin is about 0.000003125 km high and there are 2^63 of them or about 9,223,372,036,854,780,000 - so the tower would be 28,823,037,615,171 km high - rather higher than the last one, or about 3 light years.

In reply to TRNovice: I'll say the former - a cylindrical column, but when you find the answer you'll realise this detail is of minimal significance

In reply to Liam M:

I wouldn't say that a factor of over 6 is minimal. 18 to 19 light years compared to just 3 seems quite a difference to me.

In reply to TRNovice: Ok maybe it's the engineer in me, but seeing as when I first worked it out I was doing the calculations in my head (e.g. 2^63 ~ 8 * (2^10)^6 ~ 8 * (10^3)^6), and making approximations for the thickness of a pound coin, within one or two orders of magnitude (in metres) felt sufficiently accurate, into which the orientation of the coin is swallowed up in other errors.

In reply to Liam M:

Ah! I guess I'd counter by saying it's the Mathematician in me . Saying which I've probably made a calculation error somewhere (also typical of Mathematicians!).

In reply to TRNovice:

However I did measure the size of a 1 pound coin by stacking them next to a ruler until they lined up with something vaguely sensible - 8 were more or less 25 mm high .

In reply to Robert Dickson: That guy taught me Electric Engineering!

In reply to benstevens:
Did that involve paper folding tutorials?

In reply to benstevens:


That guy is my dad!

In reply to Matt Rees:
Really?
Does he do paper folding tricks at family get-to-gethers?

In reply to karl walton:

Our entire house and all it's contents are actually origami.

In reply to Matt Rees:
Nice!

In reply to The Lemming:
A pisspoor joke:

An economist, a physicist and a mathematician are on a train heading up to the Scottish hills. Shortly after crossing the border, the economist looks out of the window and sees a black sheep:
"Look" he says, "the sheep in Scotland are black."
"You don't know that" says the physicist, "just that some of the sheep in Scotland are black."
The mathematician sighs and says wearily "There is at least one sheep in Scotland, at least half of which is black."

In reply to benwood100:
In other words, if the thickness is 0.1mm, the length of the sides has to be around 0.1 light years!
utter UKC Tosh !
Oh how silly .... this reply is do people have access to this size paper. theory this that and the other...

please get real it cant be done with obtainable paper i.e. A4, A3, newspaper etc.


Another theory...
In theory if lost walk down hill you will eventually get to the sea in the UK - you are no more than 76 miles away from the sea in the UK so its possible..

there are more .... but come one folks theory and reality a bit like what IF ....

In reply to Ian McNeill:

The pound coin question was much more feasible of course...

In reply to The Lemming:

Everyone seems to have assumed that you need to fold the paper in half but at no point dose it say that in the question!

Sam

In reply to sam coward:

Thats true, but if you don't make the assumption that the paper must be folded in half, and you could actually fold the paper in any way you liked, then the final fold could be almost any size, making the question pretty meaningless.

In reply to Matt Rees:

In so far as it wsn't completely meaningless in the first place.