OT: The five trillionth digit of Pi is...... 2!



#2

New record, interesting read.

http://www.numberworld.org/misc_runs/pi-5t/details.html


#3

I'm working on the sixth trillion digit using my trusty hp-12C ARM-based calculator. My initial guess is that it will be somewhere between 0 and 9, I'll you posted.


Edited: 6 Aug 2010, 2:48 p.m. after one or more responses were posted


#4

Personally, I am immigrating to another universe where PI is exactly 3 ... no decimals!!! I hope life there will be smooth in every other aspect!!!

I will send a post card!

:-)

Namir


#5

Quote:
I hope life there will be smooth in every other aspect!!!

Do you think bikes with wheels like that will run smoothly?

     __ 
/ \
\__/

;-)

Gerson.


#6

So you HAVE been there already!!! Shucks!! I thought I will be the first earthling there ..... not my day!!!

:-)

Namir

#7

Quote:

Do you think bikes with wheels like that will run smoothly?


On the other hand, squaring the circle would be possible.


#8

Might work very well! Remember, we are talking about ANOTHER universe, were basic laws can be very different.

#9

Why not just redefining pi to 3 so you don't have to change universe?

Alabama's Pi

This Alabama's Pi story is fake (Indiana's not, as we know :-)


#10

Why define it to 3? The correct value of Pi is exactly 3.125, and you can earn yourself 300,000 Swedish crowns if you can prove it wrong.


#11

I, like most of us, would not take the time to pore over this mess, but would confidently bet real money that there is a fallacy buried deep in there, but this man, who ranks up there with the best charlatans, con men and conspiracy theorists (and Uri Geller?) is counting on our collective common sense not to pursue this, so that he may retain his "fifteen minutes" of fame. Thanks for digging out this drivel. =)

#12

Quote:
Personally, I am immigrating to another universe where PI is exactly 3

You DO live in a place where pi=3; planet earth! Go to the North Pole with a long rope and a stake. Stake the rope down, and pull it tight exactly 30-degrees down from the pole. Draw a BIG circle at this parallel. The circumference of the circle is EXACTLY 3 times twice the "radius" (your rope). Pi does equal three in the correct geometry, (or more correctly, in special cases of the correct geometry.)

Now, how long does your "radius" need to be so that pi = e? or even 1?

Cheers


#13

Quote:
Now, how long does your "radius" need to be so that pi = e? or even 1?

I came up with the formulas (I hope I haven't made any mistakes)

(1)   piL=180*sin(L)/L     
and
(2) RL=pi*r*L/180

where

L = latitude in degrees
piL = pi at latitude L
RL = "radius" or length of the rope
r = radius of the sphere (or the Earth radius in this case)

Thus, solving 180*SIN(L)/L=EXP(1) for L on the 33s, for instance, we get

L = 56.6160486813 degrees, which implies RL = 0.9183233r

Likewise, solving 180*SIN(L)/L=1 for L we get

L = 132.567200683, which implies RL = 2.313734r

Notice we can use the ->RAD function (degrees to radians conversion) to compute RL


Edited: 6 Aug 2010, 10:18 p.m.


#14

Nicely done. I think I came up with a little formula for what "pi" is for any angle (looks similar to what you have also):

"pi" = pi sin(a) / a

for 0<a<pi

where the unquoted pi is the usual 3.1415.....2 :)

CHUCK

Edited: 7 Aug 2010, 3:53 p.m.


#15

Yes, it is better to work in radians instead of degrees as the answer becomes straightforward when solving P=pi*SIN(A)/A for A in radians mode. It is interesting to notice P ranges from pi at North Pole (which is understandable as the surface tends to flatness when the "radius" tends to zero) to 0 at South Pole, which is not intuitive to me. (*)

Anyway, I guess Namir would prefer a universe where pi is always 3 or at least a rational number :)

Gerson.

------------


(*) On second thought the latter should be easier to understand than the first, the result is just 0 over something which is obviously zero.


Edited: 7 Aug 2010, 9:02 p.m.


#16

That's right ... where PI is ALWAYS 3. THEN I can (easily) boast on the hp museum web site (on THAT universe) that I have calculate PI to gazillion and kajilliam decimal places with an HP-5s.

Always a reason for my madness.

Namir

Edited: 8 Aug 2010, 10:17 a.m.


#17

Moving back on topic, pi should ALWAYS appear on the keypad of a calculator as a shifted function of the 3-key, where it can be easily found — not on the CHS key (11C), the EEX key (15C), the RCL key (32S), the roll-down key (42S), the SPC key (50g), etc.


#18

I rather like it as a shifted zero key like on the 41. The zero looks most like a circle on the keyboard so this is my "perfect" placement.

#19

Quote:
Moving back on topic, pi should ALWAYS appear on the keypad of a calculator as a shifted function of the 3-key, where it can be easily found — not on the CHS key (11C), the EEX key (15C), the RCL key (32S), the roll-down key (42S), the SPC key (50g), etc.

Nah... the only important thing is that 'pi' be grouped on the keyboard with the other data-entry keys. Both the HP-11C and HP-15C meet that test, but the HP-15C's location is slightly better because the EEX key and the blue shift key are closer together than is the CHS key to the yellow shift key.

Why would it be important that the shifted '3' key be used, just because pi is approximately 3?

-- KS

Edited: 11 Aug 2010, 1:53 a.m.


#20

Karl,

You're right: the most important thing is that pi be grouped with the other data entry keys. However, I think that, ideally, pi should be in the same position on every HP calculator, just so it is easier to find (especially if the keypad is crowded). I'd go further and say that other common keys (e.g. STO and RCL) ought to have a consistent place on the keypad as well.

Pi is approximately 3, so I can't think of a more appropriate place on the keypad for it.

Andrew

#21

I recall an interesting idea in Carl Sagan's fictional "Contact". After many trillions of digits of PI were calculated, they discovered a non-random pattern of digits. Essentially the digits formed some sort of message.

Of course,for some alien intelligence to be able to insert a message in the digits of PI, they would have to be able to manipulate the universe itself.


#22

The idea that some ultra-advanced group or entity could embed a message in pi is absurd.

Even God cannot manipulate the value of an abstract mathematical constant.

Legislatures may try it, but apparently they think they are above God!


#23

Quote:
Even God cannot manipulate the value of an abstract mathematical constant.

Perhaps your god is too small. Mine created abstract mathematical constants. (It was a side effect of creating everything else).
#24

not sure about 6 trillion, but you could use your 12c to calculate the nth digit of pi without having to calculate the previous n-1 digits.

article: http://pictor.math.uqam.ca/~plouffe/Simon/articlepi.html
from, http://pictor.math.uqam.ca/~plouffe/

i did a 41c version once, but it was very slow. also, i think the method has been improved since then.

:-)

#25

Quote:
The five trillionth digit of Pi is...... 2!

I checked this out on my HP-41CX and found it to be 5, but I counted the first 3 :-)

Now seriously, thanks for the interesting link.

#26

...is 2!

#27

Since this is a record, presumably that means no one else has done this.

How do we know they are right? :-)


#28

BBP algorithms have been proven mathematical correct. They can be used to find specific Pi digits without the overhead of computing all digits.

The record holders used BBP to validate some digits, but not all (it'd take far too long). It's reasonable to assume that if you got the last 100 or so correct then the path to get there must also be correct. Got faith? :-)

I'm sure there is a computable probability that there is a wrong digit somewhere. However, the percentage of people that care round to zero.

I read somewhere that to compute the circumference of the universe to the precision of a photon you would only need 22 or 23 digits of Pi.


#29

Quote:
I read somewhere that to compute the circumference of the universe to the precision of a photon you would only need 22 or 23 digits of Pi.

Not quite - but close if your units are meters rather than photon (wavelengths).

Age of the universe - about 13.7e9 years. In the big bang theory, one might interpret this as (very roughly!) the radius of the universe in light years.

One light year = number of seconds in a year (3.2e7) times the speed of light (3e5 m/sec) = 9.5e12 meters.

If circumference is 2*pi*r, then the circumference is about 8e23 meters. Thus, 23 or 24 digits would suffice for full accuracy in meters if we knew everything else that accurately - about what you read.

However, the wavelength of light (in the middle of the optical spectrum) is around 5000 angstroms, or 500 nanometers (500e-9 meters) in proper SI units. So, you need to multiply the circumference just given by another 1/5e-7 = 2e6 . So, you would need more like 30 digits in wavelength (or photon) units.


#30

Dave,

may I correct a bit?

Quote:
One light year = number of seconds in a year (3.2e7) times the speed of light (3e5 m/sec) = 9.5e12 meters.

Speed of light is 3e5 km/s = 3e8 m/s so 1 Ly = 9.5e15 m. Thus the circumference of the universe is about 8.1e26 m so 27 digits are necessary for 1 m precision. BTW, a photon may be any electromagnetic radiation particle, so it may have an arbitrary wavelength. Assuming - as you did - visible green photons of 500 nm, we would need 33 digits for such "photon accuracy".

As my late professor in theoretical physics used to say: "For sake of simplicity, we may set pi = 1." But he would never have dropped an order of magnitude nor even three. d8-)

Edited: 12 Aug 2010, 11:03 a.m.


#31

Quote:
Speed of light is 3e5 km/s = 3e8 m/s so 1 Ly = 9.5e15 m

Of course you are right - I know that, but that's what happens when you start calculating at 10 PM, after a couple glasses of wine!

Quote:
Thus the circumference of the universe is about 8.1e26 m so 27 digits are necessary for 1 m precision.

Yep. And per ths discussion continuing below, if you want it in units of protons, then (as you point out), the proton size (a somewhat fuzzy concept in itself!) is around 1e-15 meters, so another 15 digits are needed, or about 42 in all.

Edited: 12 Aug 2010, 2:06 p.m.


#32

Quote:
about 42 in all.

Hmmm, maybe this is why it is the Answer to the Ultimate Question of Life the Universe and Everything


#33

So those of us who already have (HP) 42s are all set?


#34

Of course we are. Though Free42 will do as well. You didn't expect a deviating answer, did you?

#35

Quote:
Not quite - but close if your units are meters rather than photon (wavelengths).

Ok, I'll dig up the reference...

Back. You're right. I misquoted. The actual statement:

Quote:
If one were to find the circumference of a circle the size of the known universe, requiring that the circumference be accurate to within the radius of one proton, how many decimal places of Pi would need to be used?

The actual number stated was 39 decimal digits (Pi Unleashed, 1998).

#36

Egan,

Quote:
If one were to find the circumference of a circle the size of the known universe, requiring that the circumference be accurate to within the radius of one proton, how many decimal places of Pi would need to be used?

OK, that changes the game significantly. Assume a proton has a diameter of 1e-15 m (being its Compton wavelength), then the circumference as calculated above corresponds to about 1e42 protons. So 43 digits would be sufficient.

Edited: 12 Aug 2010, 11:04 a.m.


#37

A quick search has values ranging from 39 to 47. Some 39's are using a hydrogen atom vs. a proton.


#38

Don't search, think! Diameter of an atom is about 1e-10 m, so five orders of magnitude larger than a proton. So, we'll end with 1e37 atoms.


#39

I think you missed my point. Other calculations are using various other constants. Opinions and calculations range from 39 to 47. Your solution falls in the middle. It was my way of saying that others generally agree.


#40

Fram a (sort of) practical point of view, maybe we only need 50 digits of pi. But for a deeper meaning of life, the universe and everything we need to keep calculating. In Contact Carl Sagan told us that there's important stuff encoded in the expansion of pi. :)


#41

Quote:
Carl Sagan told us that there's important stuff encoded in the expansion of pi. :)

See Message #21 above.

#42

Sorry about that, I got lost in this giant thread and missed that post.

#43

I hope I got your point. IMHO it's just more than a bit disturbing that by "using various other constants ... opinions and calculations range from 39 to 47" - that's 8 orders of magnitude or a factor of 100 millions. Due to some coarse assumptions I made, though each of them is well within a confidence intervall of +/- 33%, I may be off by a factor of 3 but not more.

Assuming we all live in the same universe with the same universal laws of physics and thus the same physical constants, a range "from 39 to 47" points to very different boundary conditions, wild guessing, massive errors, or a combination of these. It may have happened in my calculation - though it is correct IMHO ;) - but in any case such big differences must be explained for the sake of science.


#44

Quote:
... a range "from 39 to 47" points to very different boundary conditions, wild guessing, massive errors, or a combination of these.

And do you really think that when science attempts to define something as vast (presumably) and complex (ditto) as the universe, that it is anything else other than the above?

#45

The universe is actually amazingly well-defined!

The age is 13.7e9 years, to about a per cent or two, and the Hubble constant, which measures the expansion rate due to the Big Bang (BB), is good to a couple of per cent. Measurement of the fluctations in the "3 degree" background radiation agree to an amazing extent with the predictions of BB cosmology.

I'm not sure I want to get into an argument about god, creationism, evolution, etc., but my opinion is that everything we can measure is in COMPLETE agreement with a Big Bang cosmology that occurred 13.7e9 years ago and the accompanying evolution of the universe and life, in a system completely devoid of any requirement for divine intervention!!!!!!!


#46

Dave, not to pick on you particularly, or to knock science, but I sometimes find it amazing how utterly arrogant (so it seems to me) present-day scientists can be. Do they not realize that each generation before them also believed that the "universe" was well-defined? Do they not have any concept of the limits of human knowledge?

Now I am certainly no scientist, and claim no cosmological knowledge, but it seems to me that many theories about the nature of the universe are based on HUGE extrapolations. Somewhat akin to an ant measuring vibrations passing through his anthill in central Africa, and developing complex theories about automobile traffic in downtown Manhattan. Within 2% error.

Indeed, there are now many competing theories on the origin of the universe. Some atheists in particular are uncomfortable with Big Bang, because it implies a beginning, and opens the question of causation.


#47

AFAIK the basic approach in science is as follows in plain English:

  1. You observe something.
  2. You try to explain it as simple as possible, and consistent with the observations explained already.
  3. You check: Does the explanation meet the observation?
    • If true then you're done. Return to step 1 if you like and focus on something else.
    • Else your explanation was too simple. Return to step 2, taking into account even older explanations may be wrong.
This means in our case:
  1. Egan reported 8 orders of magnitude deviations between different results for our challenge.
  2. I tried to find out about the reasons - no answer yet, so we are stuck. Instead, you tell us something off topic.
BTW, there may well have been repeated big bangs, i.e. a pulsing universe, but that has no influence on the solution of our simple problem.

HTH

Edited: 14 Aug 2010, 8:22 a.m.


#48

Quote:
Instead, you tell us something off topic.

Walter, it's only off-topic if you believe that science has the ability - right now - to apply the "scientific method", which you kindly reminded me of, and make observations of sufficient accuracy to define the universe to an order of magnitude or less.

I was responding, albeit philosophically, to your statement

Quote:
a range "from 39 to 47" points to very different boundary conditions, wild guessing, massive errors, or a combination of these.

I am suggesting, that just maybe, scientists 100 years from now may look back and say exactly that, just as we look back 400 years and say the same thing.

#49

Martin,

Quote:
I was responding, albeit philosophically, to your statement
Quote:
a range "from 39 to 47" points to very different boundary conditions, wild guessing, massive errors, or a combination of these.

I am suggesting, that just maybe, scientists 100 years from now may look back and say exactly that, just as we look back 400 years and say the same thing.

Nobody knows what will be in 100 years. If, however, people will then find significantly different results, I sincerely hope they will check them and will not rest earlier than their reasons are known and thus these problems are settled.

Martin, I share your guess "we" will probably know better in 100 years. OTOH I didn't compare ancient results with those of today - that would have been unfair IMO - and I hope our successors will treat us just the same way. But they will rightfully shake their heads if they will find us unable solving today's problems. If we can't even settle such simple topics like deviating results of different authors today in a forum of educated experts, how shall mankind cope with the real challenges of our planet? Let's go - or there may be no human being left wondering in 100 years.

#50

Martin, to answer your question, let's go back a second. I wrote (and the first half of my sentence is important):

Quote:
Assuming we all live in the same universe with the same universal laws of physics and thus the same physical constants, a range "from 39 to 47" points to very different boundary conditions, wild guessing, massive errors, or a combination of these.
And you responded:
Quote:
And do you really think that when science attempts to define something as vast (presumably) and complex (ditto) as the universe, that it is anything else other than the above?
Answer: What did the other folks do differently? I agree with Dave: The universe as we know it is pretty well defined. So I don't see any reason yet for such big deviations in results (except confusing protons and atoms). Since there must be reasons (no sorcery in physics), however, we should know them at least IMHO to settle this problem, eeh challenge. For the advancement of science or whatever you like better.

#51

Quote:
The universe as we know it is pretty well defined.

Emphasis added. See my response to Dave Shaffer.

#52

Now you missed my point. I hate to quote myself, but:

Quote:
What did the other folks do differently? ... I don't see any reason yet for such big deviations in results ... Since there must be reasons, however, we should know them ...

Boiled down a bit - please find the complete text above.

#53

Quote:
Now you missed my point.

Actually, I got your point, although I did not explicitly acknowledge it. Rather my response is more on the philosophical side.
#54

Egan, thanks for your original post, and I've certainly enjoyed the entire thread..

But, I hope I am not the only one who is struck by the _irony_ that the guys who calculated the five trillionth digit of pi were using hardware that has 6.17E-4 the processors of the computer you, the original poster, have the keys to..

If you wanted to, could you not own that record in a few hours??


#55

Quote:
If you wanted to, could you not own that record in a few hours??

Dunno. The last system we built has over 30,000 cores, over 30 terabytes of RAM, and a multi-petabyte parallel file system. It also requires about a megawatt of power and has a MTBF of 24 hours. When I benchmarked it for the top500.org list last summer it was the 16th fastest system in the world (#1 in Canada).

However, systems like this need applications and data sets that can be broken up and run in parallel to realize any speed advantages. Although many of the current PC Pi programs better utilize multiple cores in a share memory system, I doubt that they can scale up to a 30K core distributed memory system. Something new would have to be written or something old in a different way. E.g. BBP.

BBP is a great algorithm that can compute any series of Pi digits without the overhead of computing all of Pi. So, I should be able to assign each of the 30K cores a slice of Pi. Furthermore, since BBP runs in O(n2) time you can speed up the computation by giving the cores that represent the beginning of Pi more digits and the cores representing the end of Pi fewer digits (IOW, equal time slices). BBP is not the best algorithm for computing all the digits of Pi, but it may be one of the simplest to run on a distributed memory machine. IOW, its a great algorithm to through hardware at.

Another option would be to use any series that computes Pi and split that up and use a binary tree to sum it all up. That would probably be much faster. However, it would slowly bottleneck on the sum and would require a shared file system. It would be fun to see and try. Google for "distributed Pi". One call PiHex used BBP as described above. Can this be done in hours on a 30K core system? I do not know. I'd have to run a few benchmarks first.

As for the keys, well, after we finish building a system and benchmarking it, we turn the keys over. In some cases we get access to run a few benchmarks, but real work takes priority and someone has to pay the power bill. On this most recent system we automatically power off unused resources to save power. So, cycle harvesting is not an option.

These beasts take a long time to architect, sell, build, and test. This cycle takes about two years. Perhaps next year I'll have a bigger system to play with and give it a shot.


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