Thursday, 16 March 2017

Pi Day

I'm a bit late for Pi Day, but hey, research takes time. It has become trendy to celebrate the 14th of March as 'Pi Day', because in American notation the date looks like '3/14'. These are the first three digits of the mathematical constant π, or pi, the ratio between a circle's circumference and its diameter. (Assuming we're talking about Euclidean geometry. Which we are.)

Outside the US, some people like to celebrate the 22nd of July instead -- because in everyone else's notation, that looks like '22/7', and 22/7 is a very good approximation for π.

A few incidental bits of trivia about π:
  • π is an irrational number. This means it cannot be expressed as a ratio of two integers. Put another way: the circumference and diameter of a circle are incommensurable. Or put yet another way: if you write out π in decimal notation, it will never ever repeat.
  • π is also a transcendental number. This means that it cannot be expressed as the solution to a polynomial equation with integer coefficients. That is: given an equation axn + bxn-1 + cxn-2 ... + zx0= 0, where a, b, c ... are integers, a transcendental number is any number that x cannot be.
  • π is widely suspected to be a normal number. This is not known for sure. A normal number is, roughly, one whose decimal expansion shows no patterns, where every digit is equally likely, and every finite sequence of digits is equally likely. This sounds pretty limiting; at present no one really has any idea how to prove that a given number is normal with 100% certainty. But if you look at it statistically, almost all real numbers are irrational; almost all irrational numbers are transcendental; and almost all transcendental numbers are normal. If you randomly pick a number on the real number line, the probability that it will be normal is 1. So, pretty good odds that π is normal, then.
  • If you know π to 39 decimal places -- 3.14159 26535 89793 23846 26433 83279 50288 4197 -- then you know it precisely enough to measure a circle the size of the observable universe to a precision finer than the width of an atom.
So much for interesting trivia of the day. What about myths? Give me modern myths about antiquity!



OK, here's one myth.
The first recorded algorithm for rigorously calculating the value of π was a geometrical approach using polygons, devised around 250 BC by the Greek mathematician Archimedes.
-- Wikipedia, 'Pi'
It is true that Archimedes used this method to calculate π. But it is not true that he devised the method. He just did it with a bit more precision than anyone had done previously. He made an advance, but it was an incremental advance, not something revolutionary. You can find Archimedes' full exposition in a surviving work, the Measurement of the circle.

The 'exhaustion method'. If you draw regular polygons inside and outside the circle, then the more sides the polygons have, the more closely they approximate the actual circumference of the circle. (source: Wikimedia.org)

The illustration shows how the exhaustion method works. Using 96-sided polygons, Archimedes narrowed down the value of π to between 3 10/71 and 3 10/70 -- that is, he found that π is somewhere between 3.1408... and 3.1429...

But the method was already in use 200 years earlier. Antiphon of Athens (ca. 480-411 BCE), Bryson of Heraclea Pontica (ca. 400 to after 340 BCE), and Eudoxus of Cnidus (ca. 391-338 BCE) had all used a similar method to calculate π long before Archimedes came along.

Antiphon, the earliest of the bunch, only used inscribed polygons -- that is, he only drew one shape, inside the circle, but not outside. As a result he only had one bound for the value of π. We don't know much about Eudoxus' effort. We do know that Bryson guessed (wrongly) that π would be given by the arithmetic mean of the inner and outer perimeters; and that Antiphon and Bryson were working on the area of the circle, not its perimeter. It was Eudoxus who showed that the area and perimeter were linked by the square of the radius.

The New Pauly encyclopaedia reports (subscription needed) that it was Eudoxus, not Archimedes, whose influence led to the widespread use of exhaustion for all problems involving infinitesimals. Archimedes' work on π was just a refinement of Eudoxus.



Here's another myth, from a Time article published on 'Pi Day' this year.
However, not too many generations after [Archimedes'] lifetime, the world experienced a "real decline in math," according to John Conway, mathematics professor emeritus at Princeton University who once won the school's Pi Day pie-eating contest. "Math and science in general went into a great decline from roughly the year zero to the year 1,000, and then the Arabs developed lots of math after that, like trigonometry."
Oooh, do I detect a note of a renowned world expert saying something a little bit silly about another field? I think I do!

What, no love for all those Alexandrian mathematicians of the Roman era? No love for Heron, whose Metrica has recently been published in a new French translation? Or Menelaus, whose work on spherical geometry was foundational for Arabic, Hebrew, and western astronomers for over a thousand years? Not to mention Diophantus, whose work laid down the parameters for the modern study of polynomials, and whose notation foreshadowed the development of algebra?

And then there are many other figures who are, admittedly, lesser, but still made important contributions: Sporus of Poros, who demolished earlier mathematicians' reliance on a curve called the 'quadratrix' in problems to do with squaring the circle; Ptolemy, who in the early 100s CE gained the world record for closest approximation of π (3 + 8/60 + 30/3600, = 3.141666...); and commentators like Pappus, Theon, Hypatia, Proclus, and Eutocius, whose work on Euclid, Ptolemy, and Archimedes were colossally useful in helping later mathematicians to understand the impenetrable language of their predecessors.

I guess it is fair to speak of a decline in Greek mathematics -- but Archimedes was not the be-all and end-all. If there was a decline, it was after the time of Diophantus. Archimedes has a curiously inflated reputation. I suppose that's because there are lots of good stories about him: the story of his death ray; the dramatic story of his death that we find in Plutarch and Valerius Maximus; the story of the bathtub and the running around naked shouting 'Hēurēka!'; and the story of the Cattle Problem, whose solution involves a number with over 200,000 digits (ca. 7.76 × 10206544). Everything about him sounds tremendously exciting. But hey, let's not forget later giants like Hipparchus, Menelaus, and Diophantus, all right?

5 comments:

  1. Not to mention that John Conway's comment about 'year zero' shows a certain lack of understanding about the difference between numbers in maths and in chronology.

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    1. Certainly true - I just wasn't going to be too picky about that. (But in fact there are some fields where it /is/ customary to refer to a year zero! In archaeoastronomy, for example: so 0 = 1 BCE, -1 = 2 BCE, -2 = 3 BCE, etc. I've seen confusion arise from that convention more than once...)

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  2. Very interesting Peter!

    Among mathematicians, Archimedes is definitely considered a giant who was ahead of his time. I've heard several times that he "almost invented calculus". I'm not sure what this really means.

    Another question (for my upcoming inaugural talk, perhaps you'd like to come?) -- is the history of the liar paradox given in https://en.wikipedia.org/wiki/Liar_paradox correct? Thanks!
    -- Noam

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    1. Noam! It's been a while, hasn't it? Good to hear of your impending inaugural. Yes, I'd love to come!

      Archimedes' "almost inventing calculus" is a reference to his work with infinitesimals, using the "method of exhaustion" (https://en.wikipedia.org/wiki/Method_of_exhaustion). The polygon method I mentioned above is an example of the principle. You might say it had the potential for exhaustion to be to calculus as geometry is to algebra. But that potential wasn't achieved: the ancient Greeks never developed the idea of limits.

      We don't have nearly as much surviving of Archimedes' predecessors, but it does sound rather like the popularity of the technique owes more to Eudoxus. Who was no slouch (http://dx.doi.org/10.1163/1574-9347_bnp_e404350).

      Having said that, Archimedes may have come up with a much more precise measurement of pi, to four or even five significant figures. This isn't in his /Measurement of the circle/, but in Heron of Alexandria's /Metrica/. Unfortunately the text of the numerals is problematic, so it's impossible to be sure what Archimedes actually wrote: but the revised figures Heron quotes involve numerators and denominators with four or five digits each, so it was apparently a much more precise calculation. The numbers that Heron quotes are incorrect, though, so it's either a textual corruption or else Archimedes made a boo-boo.

      Of course nothing I've said should be taken as a slight on Archimedes himself! He was a genius, but he stood on the shoulders of other geniuses. My concern is about slighting all those /other/ geniuses. When you celebrate Archimedes and /only/ Archimedes, that leads to things like people inferring that the Antikythera mechanism must have been Archimedes' design ... just because he's famous.

      The liar paradox: yes, that seems to be accurate. With two minor provisos: when the article calls Epimenides "semi-mythical", there's nothing "semi" about it. There were however real literary works attributed to the mythical Epimenides. As for Euboulides/Eubulides, apparently there's some doubt over the exact wording of his paradox -- the New Pauly encyclopaedia quotes Cicero for the wording: "If you say that you are lying and say it truly, are you lying or telling the truth?" (/Academics/ 2.96, https://archive.org/stream/academicscicero00cicegoog#page/n72/mode/2up). Elsewhere, Cicero also attributes the "heap paradox" to him -- "With which additional grain will a number of individual grains become a heap?" I haven't checked the reference to Jerome, but I'd assume that relates to Paul quoting the "All Cretans are liars" line in one of his letters in the New Testament, but that's a quotation from the playwright Menander, not from Epimenides. I can't answer for the later, non-Greco-Roman itesm, I'm afraid.

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  3. Thanks again Peter! Yep, it's been a while... hope all is well with you. Would be great if you could come, it's the 4th of April, 6pm, Hunter building. Cheers -
    -- Noam

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