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How Long is a Piece of String?

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6824.  Thu Apr 15, 2004 8:14 pm Reply with quote

Remember the Monty Python sketch about aging northerners competing to have had the most deprived childhood? Taken straight from Leacock.(also in Literary Lapses - the chapter called Self-made Men).

6835.  Fri Apr 16, 2004 10:49 am Reply with quote

It wasn't a boarding house, Jenny, it was a guest house. Admittedly, some of our guests stayed for months, but it wasn't their home as such. And we never adjusted the bills depending on our state of tension or amiability with the guests. What, never? No, never. What, never? Well, hardly ever. And not very much, in any case.

7221.  Thu May 13, 2004 4:05 pm Reply with quote

I did like this bit:

JumpingJack wrote:

This answer isn't of itself very interesting, but what is interesting is that 0.000595 miles happens to be 3.1416 feet...the value of Pi!

7222.  Thu May 13, 2004 4:15 pm Reply with quote

Well, actually I do, but I cheated:

The big question: how long is a piece of string theory?

The inherent uncertainty of mathematics means we will never fully understand our world, writes Paul Davies.

The world about us looks so bewilderingly complex, it seems impossible that humans could ever understand it completely. But dig deeper, and the richness and variety of nature are found to stem from just a handful of underlying mathematical principles. So rapid has been the advance of science in elucidating this hidden subtext of nature that many scientists, especially theoretical physicists, believe we are on the verge of formulating a "theory of everything".

When Stephen Hawking accepted the Lucasian Chair of Mathematics at Cambridge University in 1980 he chose as the title of his inaugural lecture: "Is the end in sight for theoretical physics?" What he meant was that physicists could glimpse the outlines of a final theory, in which all the laws of nature would be melded into a single, elegant mathematical scheme, perhaps so simple and compact it could be emblazoned on your T-shirt. Now Hawking has done something of a U-turn by claiming in a lecture at Cambridge last month that we will never be able to grasp in totality how the universe is put together.

The quest for a final theory began 2500 years ago. The Greek philosophers Leucippus and Democritus suggested that however complicated the world might seem to human eyes, it was fundamentally simple. If only we could look on a small enough scale of size, we would see that everything is made up of just a handful of basic building blocks, which the Greeks called atoms. It was then a matter of identifying these elementary particles, and classifying them, for all to be explained.

Today we know atoms are not the elementary particles the Greek philosophers supposed, but composite bodies with bits inside. However, this hasn't scuppered the essential idea that a bottom level of structure exists on a small enough scale. Physicists have been busy peering into the innards of atoms to expose what they hope is the definitive set of truly primitive entities from which everything in the universe is built. The best guess is that the ultimate building blocks of matter are not particles at all, but little loops of vibrating string about 20 powers of 10 smaller than an atomic nucleus.

String theory has been enormously beguiling, and occupies the attention of physicists and mathematicians. It promises to describe correctly not only the inventory of familiar particles but the forces that act between them, like electromagnetism and gravity. It could even explain the existence of space and time, too.

Though string theorists are upbeat about achieving the much sought-after theory of everything, others remain sceptical about the entire enterprise. A bone of contention has always surrounded the word "everything". Understanding the basic building blocks of physical reality wouldn't help explain how life originated, or why people fall in love. Only if these things are dismissed as insignificant embellishments on the basic scheme would the physicist's version of a final theory amount to a true theory of everything.

Then there is a deeper question of whether a finite mind can ever fully grasp all of reality. By common consent, the most secure branch of human knowledge is mathematics. It rests on rational foundations, and its results flow seamlessly from sequences of precise definitions and logical deductions. Who could doubt that 1+1=2, for example? But in the 1930s the Austrian philosopher Kurt Godel stunned mathematicians by proving beyond doubt that the grand and elaborate edifice of mathematics was built on sand. It turns out that mathematical systems rich enough to contain arithmetic are shot through with logical contradictions. Any given mathematical statement (eg, 11 is a prime number) must either be true or false, right?

Wrong! Godel showed that however elaborate mathematics becomes, there will always exist some statements (not the above ones though) that can never be proved true or false. They are fundamentally undecidable. Hence mathematics will always be incomplete and in a sense uncertain.

Because physical theories are cast in the language of mathematics, they are subject to the limitations of Godel's theorem. Many physicists have remarked that this will preclude a truly complete theory of everything. Now it seems Hawking has joined their ranks.

So does this mean physicists should give up string theory and other attempts at unifying the laws of nature, if their efforts are doomed to failure? Certainly not, for the same reason that we don't give up teaching and researching mathematics because of Godel's theorem. What these logical conundrums tell us is there are limits to what can be known using the rational method of inquiry. It means that however heroic our efforts may be at understanding the world about us, there will remain some element of mystery at the end of the universe.

Paul Davies is professor of natural philosophy at the Australian Centre for Astrobiology at Macquarie University

7225.  Thu May 13, 2004 5:16 pm Reply with quote

I heart string theory, though I obviously don't understand it the way a physicist would. But the idea that you, me, and the computer I'm writing this on are all built of tiny packets of vibrating energy, and largely composed of space, is a fascinating one. The Hindu idea of the world as maya, or illusion, fits rather neatly in there too.

7237.  Fri May 14, 2004 1:45 pm Reply with quote

Jenny -

Eh voilà!

7483.  Sat Jun 05, 2004 5:37 am Reply with quote

SCIENTIFIC and medical journals, with their august panels of peer reviewers and fact checkers, are not the sort of places many mistakes are to be expected. Yet Emili García-Berthou and Carles Alcaraz, two researchers at the University of Girona in Spain, have found that 38% of a sample of papers in Nature, and a quarter of those sampled in the British Medical Journal (BMJ)—two of the world's most respected journals—contained one or more statistical errors. Not all of these errors led to erroneous conclusions, but the authors of the study, which has just been published in BMC Medical Research Methodology, another journal, reckon that 4% of the errors may have caused non-significant findings to be misrepresented as being significant.
Dr García-Berthou and Dr Alcaraz investigated 32 papers from editions of Nature published in 2001, and 12 from the BMJ in the same year. They examined the numbers within each, to see whether the data presented actually led to the statistical conclusion the authors drew, and also whether there was anything fishy about the numbers themselves. Appropriately, they used a statistical technique to do their checking. If a set of data are "unedited", the last digits in the numbers recorded will tend to have the values 0-9 at random, since these digits represent small values, and are thus the ones that are hardest to measure. If those numbers are rounded carelessly, however, 4s and 9s (which tend to get rounded up to the nearest half or whole number) will be rarer than they should be. The two researchers duly discovered that 4s and 9s were, indeed, rarer than chance would predict in many of the papers under scrutiny.

7484.  Sat Jun 05, 2004 6:31 am Reply with quote

Thought so. Just can't trust anyone these days, can you!

7741.  Mon Jun 28, 2004 12:38 pm Reply with quote

String b's length cannot be worked out using pi because it is stretched tightly over the 6-inch sticks so is a polygon rather than a circle. However, assuming the length of the string is known by whoever tied it around the earth and the sticks are equidistant then both pi and the diameter of the earth can be calculated by applying a bit of trigonometry and using the fact that the sum of the angles between the string and the sticks is 360 degrees.

This method of adding sides to a regular polygon to evaluate pi was used by both Archimedes, who found the figure to be between 3 + 1/7 and 3 + 10/71 and by Ludolph Van Ceulen (1540 - 1610), who spent most of his life working out pi to 35 decimal places and had it engraved on his tombstone.

Yasumasa Kanada calculated pi to 6,442,450,000 decimal places in 1995 but only the first 39 are needed to work out the circumference of a circle the size of the known universe to the accuracy of the radius of a proton. Way to go Ludolph!

Stats and that: Here.

7743.  Mon Jun 28, 2004 4:27 pm Reply with quote

Nice. But, with reference to stat number 44:
44) In 1897 the State House of Representatives of Indiana unanimously passed a bill setting pi equal to 16/(sqrt 3), which approximately equals 9.2376!

I think it may be inaccurate - the bill was proposed by some loon, but rejected out of hand by the sensible Hoosier Reps. (Also, I'd always heard that the value proposed was 3, which seems more likely although I can't vouch for it).

7749.  Wed Jun 30, 2004 2:55 pm Reply with quote

I read a book about String theory about four years ago. It was pellucidly clear while I was reading it, but twenty minutes later I couldn't get my mind round it at all. Very like Calculus.

7750.  Wed Jun 30, 2004 4:04 pm Reply with quote

Or supper in a Chinese restaurant.

598902.  Thu Aug 13, 2009 8:23 pm Reply with quote

BobTheScientist wrote:
Narrrr. Of course it is. Because the difference in the diameters is one foot, so the difference in the circumferences is 1*PI feet. Let's not get carried away with conversion between units:

Circumf = PI * Diameter.

The pyloned string's diameter is 1 foot more than the snug string.
What's the difference in the length?

PI * EqDiameter+1ft - PI * EqDiameter.

Factor out the PI:

PI * (EqDiameter+1ft - EqDiameter) = PI * 1ft. = 3.1415926ft

length of pyloned string (original question) is 24902 miles and 3.1415926ft
I had heard this puzzle several years ago, I could still prove it, but not to a non math person so I needed "better" proof. The item to consider is that the diameter of the sphere doesn't matter-- basketball, beachball, golfball!!!

605802.  Mon Aug 31, 2009 5:35 am Reply with quote

garrick92 wrote:
Assume the earth is a perfect sphere, and you tie a piece of string (which we'll call a) right round the equator. That string will be a given length -- 24,902 miles.

Isn't it easier to use metric? The distance from the pole to the equator on a line passing through Paris was originally used to define the metre; the metre being a ten millionth of that distance. So the circumference is 40,000 Km.

605819.  Mon Aug 31, 2009 5:55 am Reply with quote

I see everyone used an approximation for pi. In 1913 the famous mathematician Ramanujan gave an approximation for pi as the forth root of the fraction 2143/22 which is 3.1415926525826461252... pi itself is 3.1415926535897932384626433832795.... What is quite interesting is that there is a rule and compass construction for this, so you can square the circle accurately to within 8 decimal places. If you use his approximation and the diameter of the Earth, the string will be too short by about 12.8 millimeters (half an inch in old money).


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