Happy Pi Approximation Day!
Published: July 22, 2009
Pi, as everyone but samuraipandapoetry will recall, is the ratio between a circle’s circumference and its diameter. If the diameter of a circle is 1, than its circumference is π. Its value is traditionally represented as 3.14159… Some people celebrate the 22nd of July for its approximation of π (22/7). Others celebrate March 14 (3.14), although the deluded followers of the March date refer to it simply as Pi Day. But all values for π are approximations, because π is irrational. The ancients did not know this - and it wasn’t finally proven until the 18th century - but π cannot be expressed as any exact fraction. (This despite the Indiana House of Representatives passing a law stating that π = 3.2.) Pi extends forever and ever, and where π goes, its devotees follow.
In physics, π appears in the cosmological constant, Heisenberg’s uncertainty principle, and Einstein’s field equation for general relativity. Pi comes at us in unexpected ways, too, often having nothing to do with circles. It arises in the distribution of prime numbers, and in the probability that a pin dropped on a set of parallel lines will intersect one of them. The ratio of the actual length of a river and its “as the crow flies” length is, on average, π.
An Egyptian named Ahmed provided a fairly accurate estimation for π around 1650 B.C. He approached the problem as one of “squaring the circle,” that is, constructing a square with the exact same area as a circle. The “constructability” problem would bedevil mathematicians for three and a half millennia until it was proved unsolvable. Antiphon and Bryson of Heraclea in the 5th century B.C. worked on calculating the area of a circle and invented the “principle of exhaustion.” Calculating the area of a polygon is trivial, so Antiphon repeatedly doubled the sides of a hexagon inscribed in a circle, so that it came closer and closer to approximating the circle. Bryson’s insight was to add another polygon, this one inscribed around the outside of the circle, and to treat the solution as one of lower and upper bounds.
Two hundred years later, Archimedes (of “Eureka!” fame) devised the first known theoretical method for calculating π. Building on Antiphon and Bryson’s exhaustion method, he focused on the perimeters of polygons instead of their areas, thereby estimating the circumferences of circles. Using 2 96-sided polygons, he calculated π as 3.1419, which is within three ten-thousandths of its actual value.
In 263 A.D. Liu Hui in China used polygons of 3,072 sides to come up with an estimate of 3.1416. Later scientists would use polygons of 24,576 sides and even 393,216,216 sides to calculate π accurately to 10 decimal places. In the 16thcentury, the Dutchman Adriaen Romanus used a polygon with over 100 million sides to achieve 15 digits, and the German Ludolf van Ceulen used polygons with 32 billion sides each to calculate π to 20 places. van Ceulen later reached 35 decimal digits, and today in Germany π is sometimes called the Ludolfian number.
By this time, the exhaustion method had been pretty much exhausted. Brute force gave way to more elegant methods such as infinite product series and the Gregory-Leibniz arctangent series. These are “converging” methods, where subsequent terms of a formula hone in on a true value. One drawback of these methods is that they can be maddeningly slow. The original Gregory-Leibniz series required 300 terms to extract a meager 2 decimal places of π, and would have required more terms than there are particles in the known universe to achieve 100 decimal places (by which time you would probably run out of paper anyway). Nonetheless, once refined, this method forced π to yield over 500 of its precious digits by the nineteenth century.
The next major breakthrough for calculating π came with the invention of electronic computers and yet more efficient algorithms. Suddenly, thousands of digits were being calculated, and records were being overturned more and more rapidly. In 1973, 1 million decimal places of π were computed. By the 1980s, records of 16 million and then over 201 million digits had been set. In 1989, David and Gregory Chudnovsky found 480 million digits using a home supercomputer they custom built from off-the-shelf parts that took up most of their apartment. Later that year, they reached 1 billion digits. When a competitor reached 6 billion digits 6 years later, the Chudnovskys turned around and attained 8 billion digits the year after that. In 1997, a Japanese team calculated 51.5 billion digits of π. The same team set the current record of just over 1 trillion digits.
Make no mistake: 30 decimal places would suffice to measure a circle the size of the observable universe to within the width of an atom. Calculating decimal places beyond that has no practical value. The strenuous, sometimes life-long exertions of number theorists throughout history to eke out the next digit of π, and then the one after that, have served the goal of finding patterns in the unending river of numbers. The mystery is the message. But as far as anyone has been able to determine, the digits are more or less random.
If mathematicians are the knights errant of π, those who attempt to memorize π are its troubadours. Simon Plouffle memorized over 4,300 digits in the 1970s. Rajan Mahadevan set a world record in 1983 by memorizing 31,811 digits. In 1995, Hiroyuki Goto memorized 42,000 digits. The Guinness Book of World Records recognizes Lu Chao for reciting 67,890 digits of π from memory.
I’ll leave you with this thought. Some people believe that folks born on this day are unusually good-looking, talented, and witty, and that they are exceptionally skillful lovers as well. [Um, I’m going to need to review your notes on this. – Ed.]
Source: David Blatner, The Joy of π, Walker & Company, NY, 1997 (except where otherwise indicated).
Image of pi/pie via http://en.wikipedia.org/wiki/Pi
Image of Archimedes via http://www.math.nyu.edu
Image of π poster via http://www.unihedron.com/projects/pi
So, is the gist of this post “Happy Birthday, Rene!”?
Also, that cheesecake looks delicious.
I’m sorry I was lead to believe there would pie and punch at this meeting? That’s what the poster said.
And it’s George Sanders Day on TCM – it doesn’t get any better than this.
Ha, that’s excellent. And I forgot about Indiana pulling that…Indiana is hilarious.
I’ve often been fond of insisting that mathematics is a shoddy hoax, otherwise pi wouldn’t exist. Mathematicians sometimes have a conniption fit when I say this. But I just keep saying, “No, sorry. You’re wrong. Math doesn’t work.” They go all red in the face and a fried egg appears on their foreheads.
Also: Arpy Bird Day, Sance!
Kudos to anyone who can make me care about this number with all it’s enigmatic decimal places.
Monk, are you there? Rene’s got his Dirty Old Man hat on — and it’s a beret.
Oh, oh, do e next!
I can recite pi to 21 decimal places, a legacy of having math geek friends when I was a kid. But this morning, I had trouble remembering my online banking password.
If you have pie, you don’t need to be a terribly skillful lover in my book. Happy birthday!
3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 then I forget.
Oh wait, Pi = 3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679
8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196
4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273
7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094
3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912
9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132
0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235
4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999837 2978049951 0597317328 1609631859
5024459455 3469083026 4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303
5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787 6611195909 2164201989
3809525720 1065485863 2788659361 5338182796 8230301952 0353018529 6899577362 2599413891 2497217752 8347913151
5574857242 4541506959 5082953311 6861727855 8890750983 8175463746 4939319255 0604009277 0167113900 9848824012
8583616035 6370766010 4710181942 9555961989 4676783744 9448255379 7747268471 0404753464 6208046684 2590694912
9331367702 8989152104 7521620569 6602405803 8150193511 2533824300 3558764024 7496473263 9141992726 0426992279
6782354781 6360093417 2164121992 4586315030 2861829745 5570674983 8505494588 5869269956 9092721079 7509302955
3211653449 8720275596 0236480665 4991198818 3479775356 6369807426 5425278625 5181841757 4672890977 7727938000
8164706001 6145249192 1732172147 7235014144 1973568548 1613611573 5255213347 5741849468 4385233239 0739414333
4547762416 8625189835 6948556209 9219222184 2725502542 5688767179 0494601653 4668049886 2723279178 6085784383
8279679766 8145410095 3883786360 9506800642 2512520511 7392984896 0841284886 2694560424 1965285022 2106611863
0674427862 2039194945 0471237137 8696095636 4371917287 4677646575 7396241389 0865832645 9958133904 7802759009
9465764078 9512694683 9835259570 9825822620 5224894077 2671947826 8482601476 9909026401 3639443745 5305068203
4962524517 4939965143 1429809190 6592509372 2169646151 5709858387 4105978859 5977297549 8930161753 9284681382
6868386894 2774155991 8559252459 5395943104 9972524680 8459872736 4469584865 3836736222 6260991246 0805124388
4390451244 1365497627 8079771569 1435997700 1296160894 4169486855 5848406353 4220722258 2848864815 8456028506
0168427394 5226746767 8895252138 5225499546 6672782398 6456596116 3548862305 7745649803 5593634568 1743241125, should I continue?
Calculating the probability that you’ll lodge that grape in my cleavage when throwing it from a three-foot distance: π
Happy birthday, RS. Mine’s in four days, so let’s try to get together for a drink soonest.
@Bookish: We’re both Leos!
3.1415926535 897 NOM NOM NOM.
@MP: I wouldn’t have it any other way, she of the tresses.
Chien, dear, you need a new hobby.
@Smitros: Was this supposed to be a LOLCat reference, like “Ur Numberz Iz Hurtin’ Mai Hed!”
This is hurting my Sarah Lawrence head actually…my parents paid a lot of money so I wouldn’t have to learn this.
@Nina: I can top that. I took the Math for Social Sciences majors in college (“Math, It’s Form & Impact” was the course name) and it was that same summer I met my husband . . . a Mathematics Ph.D. student. I’ve had a headache for the last 20 years!
@Mama Penguino:
I was thinking more about the pun possibilities, but da numberz dey hurtz mai hed 2.
Christ. That’s Math, ITS Form and Impact. I guess maybe I should have taken Apostrophes 101?
Is this what that movie was about with the teenager and the apple pie and the sex with the pie?
@Chien please stop posting the all the contacts from my phone!
MP: How do you talk you him? I had a math PhD boss and I ran amok – in my head. Sometimes in the atrium. Other times, I took a valium & laid on the floor.
@Strawberry: I think my Visa number’s in there.
@Monkeyrash well at least he remember my birthday in there, so I can’t be too upset
Oh my Gd. I’m so stupid.
Oh, I just came in here to say happy birthday to pi (pie?) enthusiast Rene Sance but now all the numbers are making me woozy and are spinning all around like The Electric Company on (more) acid.
Who takes the copy books? Vaq or Wence? I wrote everything down, I think I got it all right. Can I have my graham crackers now?
Was your answer 25, Monkey?
28 is also acceptable, if you went to a cheap school.
Wence, yes?
-The Housekeeper and the Professor, pg. 115
Check this out: Housekeeper & Professor page 78:
I prefer pi.
Vaq: HA!
Or should I say “A-HA!” to make a matching palindrome?
What are the odds of my finding that on Pi approx day just as Rene posts on Pi approx day and it being also his birthday? Who, among us, is willing to do the math?
Wicked cool…I am a sucker for geek-bait like this. Thanks and happy b-day RS!
@Nina: I close my eyes and think of England.
Cherry π:
Let’s try this again.
Cherry π:
[Smitros...use the embed code...that'll throw the whole thingie right in here.]
“The Guinness Book of World Records recognizes Lu Chao for reciting 67,890 digits of π from memory.” Do you think that maybe the rest of us are just being lazy using 3.14? Also, how do we know that Lu Chao is really remembering pi and not just doing mathematical calculations in his head. Scammer.
What the fuck are you guys talking about?
SPP – right?
Now back to my assignment — it’s due tomorrow! Like one of Lawyergay’s nightmares. Samuraipandapoetry is going to have his paws full this week.
When a group of math nerds calculates pi to a trillion digits, how do the other math nerds know the digits are correct?
The pi symbol on the cheesecake is off-center. This bothers me. Also, it is not a pie.
@Captain: I think either Rene or Nina can back me up here. Math nerds love nothing more than to prove one another wrong. I have witnessed Mr. Penguino work on another math guy’s theory or theorum or whateverthehell for weeks. If the math nerd is at all involved in academia, then they’ll write to the AMA or Math Monthly or whateverthehell and get their smack-down of the other guy’s work published. It’s self-perpetuating. I’ve heard Mr. P say that 90% of the articles in the Math journals are B.S., anyway, so I feel confident that if even one numeral in the trillion is off, someone will catch it.
I thought pi day was the 14th of March. o_O?
Thank you all for your kind comments and wishes!
@Chosa: Didn’t make it past the 4th sentence, I see. That’s OK, we’re glad to have you here anyway! Everyone, say hello to Chosa. She’s here all the way from Australia, and you can find her blog here.
@RS: Hi, Chosa!