Decibels, Amplifier Power and Origami

[The Grand Wazoo]

Here's an interesting diversion to make you think a bit.

I was messing about with some bits of paper preparing the mojo-trinkets required to carry out the 'tweak of destiny' described elsewhere (http://theartofsound.net/forum/showthread.php?t=4199) and I started thinking about the old tale of how you can only fold a piece of paper up so many times – the number varies, between 6 and 8 I think.

Well, I had a go at testing this theory because the tweak of denstiny requires that you fold up a piece of paper until it won't fold any more.

It actually kind of depends on a few things:
Size of paper
Thickness of paper
Sharpness of folds
Your definition of what constitutes a fold

I took a sheet of A4 as that's what most people have to hand.
I understand a sheet of 80 g/m2 paper is about 0.1 mm thick.
I made folds as sharp as possible, pinching them as tightly as possible between fingernails down their length & then squeezing very hard.
My definition of a fold is where the fold holds without opening when you release the two sides of the fold – there is a point where the fold becomes 'a roll' & it doesn't close & hold properly.

For me, the most number of true folds I could make was 5.

So this got me thinking about the concept of exponential expansion & how some people have a problem of relating this to volume levels – we instinctively understand methods of linear expansion (like centimetres or pints). However, the idea of exponential expansion is potentially mindblowing.

I put together a spreadsheet to illustrate this expansion by relating it to the simple game of folding the paper – something we can all understand.

So if you take a 0.1 mm thick piece of paper & fold it once – you get a 0.2 mm thick sheet half the size – simple so far, eh?!

Fold it in half again.

And again.

And again.

Assuming you can fold it more than five or six times (which you can if you have a very big piece of paper – in fact you can go as far as eleven, apparently) it's not long before your number of layers has built up to a huge figure & the thickness is beyond belief (bearing in mind we're dealing with single linear increments of 0.1 mm).

Take a look at this……….

http://img36.imageshack.us/img36/8518/karmanline.jpg

[NRG]

Shouldn't your number of folds start at 0 if the paper thickness is 0.1mm ;)

[The Grand Wazoo]

Sorted out (I think)

[Barry]

Apart from the obvious mistake in the ninth paragraph:

"So if you take a 0.1 mm thick piece of paper & fold it once – you get a 2(sic) mm thick sheet half the size – simple so far, eh?!",

These figures make interesting reading.

A piece of A4 paper is 210mm x 310mm. Assuming the fold occupies no space at all, then the maximum number of times the sheet could be folded would be 8, for then the sheet would be thicker than its length or width (18.5625mm long x 13.125mm wide x 25.6mm thick).

Given a large enough sheet of paper 0.1mm thick, then if it was folded 42 times it would reach past the moon (439,804 Km > lunar distance of 384,403 Km).

As an example of the common misunderstanding of exponential growth, there is an apochryphal tale concering the inventor of the game of chess (actually nobody knows who invented the game, but it is thought to have come from India). According to the tale the local maharajah was so delighted with the newly invented game that he asked the inventor to name his reward, as long as it was within reason. Now the inventor was also a mathematician and knew that the maharajah was somewhat parsimonious, so he asked for the following:

"as many grains of rice that would be placed on the last square of the chess board, if one grain was placed on the first square, two on the second, four on the third square and so on."

The maharajah readily granted his request, thinking it would only amount to a couple of cupfulls. What he didn't know was that there would be 1.845 10^19 grains of rice on the last square! With a typical density of 0.72g/cc and typically 63934 grains in a Kg, the pile of rice on the last square would occupy the same volume as a cube 4.58 miles in each dimension, and would weigh 2.885 10^11 tonnes! This amounts to 3550 annual rice harvests of modern day India (typically 80 million tonnes).

Clever chap wasn't he? - or maybe I should get a life!

Regards

[The Grand Wazoo]

I was reminded of the rice story too.

I don't think you'd need many more than the 42 folds you mention in order to reach the edge of the known universe!
However a piece of paper big enough to do this would require perhaps more deforestation than would be strictly advisable. I'll have a word with some friends in the timber harvesting game & see what they can do!

[The Grand Wazoo]

..........and I'll correct my missing decimal point.

[Themis]

Well, perhaps if I was a black hole, I could fold this paper almost indefinitely... ;)

[The Vinyl Adventure]

isnt 9 the limit? i saw one of those programs where they test theorys and did it on a MASSIVE peice of paper... 9 was the limit

[The Grand Wazoo]

Apparently, it was the subject of some very learned research. There are 2 different formulae to use depending on whether your folds are in 1 or 2 directions (it's harder to fold in 2 directions). They got a very big sheet of paper to 11 folds (which is only 20 centimetres). One important factor is the ratio of the radius of the curves in the previous fold in relation to the unfolded length. I daresay you could find a very light guage of paper too!

[Prince of Darkness]

"as many grains of rice that would be placed on the last square of the chess board, if one grain was placed on the first square, two on the second, four on the third square and so on."

The maharajah readily granted his request, thinking it would only amount to a couple of cupfulls. What he didn't know was that there would be 1.845 10^19 grains of rice on the last square!


Should that not be 9.223 10^18 grains or 2^63? Remember the first square has 1grain or 2^0.
I should definitely get a life!:lolsign:

[Barry]

Should that not be 9.223 10^18 grains or 2^63? Remember the first square has 1 grain or 2^0.
I should definitely get a life!:lolsign:

Hi Kevin,

Yes you're absolutely right - I realised my mistake last night and wondered if anyone would spot it. :doh:

Still a hell of a lot of rice! :lol:

Cheers

[The Grand Wazoo]

I like rice a lot but I think in this case he should've gone for 1/2 and 1/2.

[Krisbee]

Of course, rabbits (and some Australians?) have know about the effects of exponential growth for some time. You could take the Grand Wazoo's table and replace paper folds with generations of your ancesters. One fold would be your parents, two folds your grandparents etc. So going back 30 generations gives you over a 1000 million ancesters. That might get you back to about the time of the battle of Hastings. Going back to biblical times might take a 100 generations and would give you a truly astronomical number of ancestors, equal to 2^100 or 1,267,650,600,228,229,401,496,703,205,376.

But a few questions spring to mind. Are my 1000 million ancesters of 30 generations ago the same as yours? What was the population of the UK in 1066? What was the population of the world in biblical times? Surely there were just not enough people on the planet to make the supposed number of my, and your, ancester from one hundred genrations ago? So who does everyone descend from, and is everyone on this forum related to me?