Stephen Speicher

PI Day - March 14 (3.14)

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March 14 is not only Albert Einstein's birthday (born March 14, 1879) but it is also PI Day! In celebration of the famed number PI, groups of people (mostly geeks) will be reciting the digits of PI, eating PI pies, discussing PI generating algorithms, and generally having a PI of a good time.

There are many news stories circulating about PI Day, but here is a great little site that is commercializing PI.

Pi_ad_r020a_2006_r001c_=r00.jpg

Any good PI facts or stories?

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I found this link with some interesting stories about how PI was calculated. Of note was Archimedes' attempt to calculate it, without the advantage of decimals or algebra or trig. He used geometry. And Shanks calculated PI to 707 digits in 1873, but it wasn't discovered until 1945 that he made a mistake. (Everyone had PI on their face. :angry: )

History of Pi

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I found thisalso interesting.

With the advent of the electronic computer, there was no stopping the pi busters" (Blatner, 51). John Wrench and Daniel Shanks found 100,000 digits in 1961, and the one-million-mark was surpassed in 1973. In 1976, Eugene Salamin discovered an algorith m that doubles the number of accurate digits with each iteration, as opposed to previous formulas that only added a handful of digits per calculation. (Blatner, 52) Since the discovery of that algorithm, the digits of pi have been rolling in with no end in sight. Over the past twenty years, six men in particular, including two sets of brothers, have led the race: Yoshiaki Tamura, Dr. Yasumasa Kanada, Jonathan and Peter Borwein, and David and Gregory Chudnovsky. Kanada and Tamura worked together on many pi projects, and led the way throughout the 1980s, until the Chudnovskys broke the one-billion-barrier in August 1989. In 1997, Kanada and Takahashi calculated 51.5 billion (3(234) digits in just over 29 hours, at an average rate of nearly 500,000 digits per second! The current record, set in 1999 by Kanada and Takahashi, is 68,719,470,000 digits. (Blatner, 59) There is no knowing where or when the search for pi will end. Certainly, the continued calculations are unnecessary. Just thirty-nine decimal places would be enough to compute the circumference of a circle surrounding the known universe to within the ra dius of a hydrogen atom. (Berggren, 656) Surely, there is no conceivable need for billions of digits. At the present time, the only tangible application for all those digits is to test computers and computer chips for bugs.

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I memorized PI to a few digits 3.14159265...

But...

Have you seen this guy?

His name is Daniel Tammet, he's memorized pi to something like 22000 digits:

http://60minutes.yahoo.com/segment/44/brain_man

It blows my mind that the human mind is capable of that sort of feat. I wonder if he's overloading his circuits with that sort of memory and calculating power.

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The most humorous thing I have ever seen about PI was written by Underwood Dudley in the Journal of Recreational Mathematics, Vol. 9, No. 3, pp. 178-180, 1976-1977.

Dudley is a mathematician who has written, among other things, a book devoted to those who claim to have done the mathematically impossible, appropriately titled Mathematical Cranks. Dudley had previously written a tongue-in-cheek piece in a 1962 edition of Mathematical Magazine, in which he had constructed a least-squares fit to historical data on the calculation of PI and determined that PI was an increasing function of time. He concluded that "schoolchildren in 10201 can look forward with more pleasure than usual to June, for in that month PI_t = 3.20000, and their calculations will be much simpler." Dudley ended his funny article with:

Further, our expression gives the date of the creation -- when PI_t was zero -- as 560,615 B.C., agreeing neither with astronomical theory nor with Arch-bishop Ussher's chronology. Clearly, more research is needed.

And, indeed, leave to Dudley to do the further research, which leads us to his 1977 article in the Journal of Recreational Mathematics.

A previous paper [1] on PI_t, the ratio of the circumference of a circle to its diameter at time t, came to no definite conclusion and called for more research. It is the purpose of this note to incorporate new data which radically change the results in [1]: rather than increasing, it now appears that PI_t is decreasing with time.

The amount of effort expended on determining PI_t has not been large in recent years, but in the last century, many researchers attempted to find its value. It is curious that all of them were under the impression that PI_t was constant; each thought that his value was "correct" and all other values were "wrong." Table 1 gives the results of their calculations, rounded to five decimal places. The data are from DeMorgan [2] and Gould [3]. The line which best fits the data, in the least-squares sense, is

PI_t = 4.59183 - .000773t, (1)

where t is measured in years A.D. In particular, PI_1977 = 3.06361. It is interesting to note that the often-used approximation 3.1415926535... was the value of PI_t in 1876; it is probably only a coincidence that it was the United States' centennial year. Circumferences of circles will be particularly easy to calculate in 2059, when PI_t = 3.

It is disturbing to note that PI_t is decreasing. When PI_t is 1, the circumference of a circle will coincide with its diameter, and thus all circles will collapse, as will all spheres (since they have circular cross-sections), in particular the earth and the sun. It will be, in fact, the end of the world, and from (1) it will occur in 4646 A.D., on August 9, at 4 minutes and 27 seconds before 9 p.m.

This final calculation assumes that the least-squares line gives the value of PI_t precisely. We cannot be sure of that; the standard error of the estimate is .03527, so we can only be 95% sure that PI_t will be its least-squares value plus or minus .07054. Thus the final collapse may occur any time between 4555 and 4738. And that assumes that the slope of the line is exactly -.00077.3. In fact, we can say only that with probability .95, it lies between -.001385 and .000161. If it is the lower value, the end may come as soon as 2542.

Finally, all of the above assumes that PI_t is a linear function of t, which may not be so. It may be asymptotically approaching some value, it may be sinusoidial, it may be something else entirely; we do not yet have enough data, but it is clear that PI_t should be more closely watched.

Dudley-Table.jpg

References

1. U. Dudley, PI_t: 1832-1879, Mathematics Magazine, 35, pp. 153-154, 1962.

2. A. DeMorgan, A Budget of Paradoxes, Chicago, 1915.

3. S. C. Gould, Bibliography on the Polemic Problem. What is the Value of PI, Manchester, N. H., 1888.

Here is a picture of Underwood Dudley in a January 9, 2004 announcement from DePauw university on Dudley receiving a certificate of meritorious service from the Mathematical Association of America.

woody-dudley.jpg

Dudley seems to be a delightful person, and I would truly like to meet the man (certainly before 4646 A.D. when the circumference of a circle will coincide with its diameter, and thus all circles will collapse, including the Earth :angry:).

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March 14 is not only Albert Einstein's birthday (born March 14, 1879) but it is also PI Day! In celebration of the famed number PI, groups of people (mostly geeks) will be reciting the digits of PI, eating PI pies, discussing PI generating algorithms, and generally having a PI of a good time.

I recently took a job as a math tutor @ my school, and Pi Day was the topic of conversation at work today. All the math tutors decided that, since it's our day of celebration, the writing tutors should bring in some Pi Pies in homage. Unfortunately, I'm not there on Wednesdays, so I won't be able to enjoy the fun with my fellow nerds, but somehow I doubt anybody will see any pies (one of the writing tutors didn't even know what pi was! :angry: ).

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Incidentally, the tutoring center opens up @ 4 p.m., which is 16:00 in 24-hour time, so all the math tutors except me will be gathering together with even further precision at 3.1416.

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March 14 is not only Albert Einstein's birthday (born March 14, 1879) but it is also PI Day! In celebration of the famed number PI, groups of people (mostly geeks) will be reciting the digits of PI, eating PI pies, discussing PI generating algorithms, and generally having a PI of a good time.

---------

Any celebration of e day planned?

How many numbers are there that have special names, like PI and e?

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Incidentally, the tutoring center opens up @ 4 p.m., which is 16:00 in 24-hour time, so all the math tutors except me will be gathering together with even further precision at 3.1416.

The tutors seem to be a well-rounded group. :angry:

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How many numbers are there that have special names, like PI and e?
Two more that I'm aware of is Phi (1.618033989...), the Golden Ratio, and Alpha (137) the fine- structure constant, which is the square of the charge of an electron divided by the speed of light times Planck's constant, and represents the probability that an electron will emit or absorb a photon -- the basic physical mechanism of electricity and magnetism.

I read years ago (OMNI magazine I think) that Richard Feynman had said that physicists ought to put that number on a sign in their offices to remind themselves of how much they don't know.(?) The question to answer was: Why is alpha equal to 137?

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Any celebration of e day planned?

I'm afraid that e is a very under-appreciated number. Perhaps we can petition the government to grant it minority status.

How many numbers are there that have special names, like PI and e?

Mathematicians and physicists love to name constants, so there is an endless array of these, though mostly little-known. Here is a site that skims a few off the top.

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Two more that I'm aware of is Phi (1.618033989...), the Golden Ratio ...

One of my favorite numbers. It's amazing how often phi comes up in nature.

... and Alpha (137) the fine- structure constant, which is the square of the charge of an electron divided by the speed of light times Planck's constant, and represents the probability that an electron will emit or absorb a photon -- the basic physical mechanism of electricity and magnetism.

I read years ago (OMNI magazine I think) that Richard Feynman had said that physicists ought to put that number on a sign in their offices to remind themselves of how much they don't know.(?) The question to answer was: Why is alpha equal to 137?

Most people associate the fine structure constant with Feynman because of the meaning he gave to it in quantum electrodynamics, but it was actually introduced into physics long before quantum electrodynamics, by Arnold Sommerfield in 1916. Someone else asked about the fine structure constant on a small thread that I had to delete, so here is a chance to resurrect my deleted post.

The fine structure constant is a dimensionless number that relates several fundemantal constants of physics, including the speed of light, Planck's constant, and the electric charge. The number 137 is actually the inverse of the fine structure constant, and it is close to but not exactly 137. A few years ago there was a flurry of interest in the fine structure constant because some cosmological observations were interpreted as possible evidence that the constant was not really constant, and then the issue was which of the dependent constants was not constant, including the speed of light. Around that time I was asked if "1) whether the speed of light may have differed historically, in the early stages of an evolutionary universe? or 2) whether, instead of being constant, the speed of light actually varies constantly?"

This was my answer then:

*************************

Actually, both 1 and 2.

As to 1): Several years ago there were experimental observations which were taken to suggest that the fine structure constant may be varying over time. (Note that such data remains rather controversial.) Since the fine structure constant is proportional to the square of the electric charge, and inversely proportional to the speed of light, it was surmised that these two constants were candidates for the variance, rather than the remaining one, Planck's constant.

In a paper last summer, P.A.W. Davies et al., "Black holes constrain varying constants," _Nature_, V. 418, pp. 602-603, 8 August 2002, the authors made an argument suggesting that black hole thermodynamics may provide a criterion to constrain these candidates as "varying" constants. In particular, the argument they made favored a change in the speed of light, rather than a change in the electric charge. However, in a recent short note, S. Carlip and S. Vaidya, "Do black holes constrain varying constants," _Nature_, V. 421, p. 498, 30 January 2003, the authors show that "when the entire thermal environment of a black hole is considered, no such conclusion [decreasing c] can be drawn. Basically the authors show that, contrary to the Davies et al. analysis, such black hole thermodynamics places no restriction against an increasing electric charge.

So, as a summary for "1)", I would say that there is highly controversial evidence as to the variability of the fine structure constant, and, though it was initially thought that such evidence would favor a varying speed of light, such a conclusion is no longer warranted.

As to 2): Most all varying speed of light theories have been motivated for solution of cosmological problems, and in the typical theory both covariance and Lorentz invariance is broken. Lorentz invariance follows directly from the principle of relativity and the constancy of the speed of light. One can maintain the indistinguishability of inertial frames, but a varying speed of light leads to what is known as Fock-Lorentz symmetry. See Fock's original work in "The Theory of Space-Time and Gravitation," _Pergamon_, 1964, or more recently Bruce A. Bassett et al., "Geometrodynamics of variable-speed-of-light cosmologies," _Physical Review D_, Volume 62, 103518, 15 November 2000.

It is this breaking of Lorentz invariance which makes these varying speed of light theories so difficult to apply to problems such as black holes. It is interesting to note Bassett et al. develop a Lorentz symmetry which is broken in a "soft" manner, analogous to spontaneous symmetry breaking in particle physics. However, there is some recent work by J. Magueijo who has developed a varying speed of light approach in which it is claimed to be both generally covariant and locally Lorentz invariant, and he does apply this new concept to black holes. See J. Magueijo, "Stars and black holes in varying speed of light theories," _Physical Review D_, Volume 63, 043502, 18 January 2001.

Note that Magueijo has recently written an interesting but somewhat irrevant book geared towards a general audience, on exactly this subject. Joao Magueijo, "Faster Than the Speed of Light: The Story of a Scientific Speculation," _Perseus Publishing_, 2003.

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One of my favorite numbers. It's amazing how often phi comes up in nature.

Most people associate the fine structure constant with Feynman because of the meaning he gave to it in quantum electrodynamics, but it was actually introduced into physics long before quantum electrodynamics, by Arnold Sommerfield in 1916. Someone else asked about the fine structure constant on a small thread that I had to delete, so here is a chance to resurrect my deleted post.

The fine structure constant is a dimensionless number that relates several fundemantal constants of physics, including the speed of light, Planck's constant, and the electric charge. The number 137 is actually the inverse of the fine structure constant, and it is close to but not exactly 137.
Stephen, could you define dimensionless number?

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The fine structure constant is a dimensionless number that relates several fundemantal constants of physics, including the speed of light, Planck's constant, and the electric charge. The number 137 is actually the inverse of the fine structure constant, and it is close to but not exactly 137.

Stephen, could you define dimensionless number?

It just means a quantity which itself has no physical units (no distance, no time, etc.). Most often, the number itself relates together various quantities that themselves do have physical units, but the relationship among the quantities is such that all the units cancel out. So, the fine structure constant relates the speed of light, Planck's constant, and the electric charge in such a way that the units cancel each other out, whatever consistent system of units they are expressed in.

A more common dimensionless quantity that almost everyone has some sense of, is a drag coefficient. Most all know that a sleekly designed car has less air resistance than a car in the shape of a box. The dimensionless quantity that expresses that relationship is known as the drag coefficient.

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...he had constructed a least-squares fit to historical data on the calculation of PI and determined that PI was an increasing function of time.

Is there a Big Bang theory for the origin of pi?

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Is there a Big Bang theory for the origin of pi?

There are three competing theories that I know of: the LM, AC, and SC theories (Lemon Meringue, Apple Crumb, and Strawberry Cream). However, I have limited knowledge in this area, so there might be other flavors theories as well.

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Is there a Big Bang theory for the origin of pi?
There are three competing theories that I know of: the LM, AC, and SC theories (Lemon Meringue, Apple Crumb, and Strawberry Cream). However, I have limited knowledge in this area, so there might be other flavors theories as well.
That seems like a fruity explanation for such a singular event. They don't really get to the crust of the matter.

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That seems like a fruity explanation for such a singular event. They don't really get to the crust of the matter.

They do seem rather half-baked. Someone probably whipped them up just to get some dough.

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