Category Archives: Mathematics

Happy birthday, Paul Erdős

A conjecture both deep and profound
is whether the circle is round.
In a paper by Erdős,
written in Kurdish,
a counterexample is found.

Today is the 100th birthday of the inordinately prolific, itinerant Hungarian mathematician Paul Erdős (26 March 1913 – 20 September 1996). During his long and illustrious career, he published just over 1500 papers in collaboration with 511 other mathematicians, on a variety of topics (particularly combinatorics, probability, number theory and analysis). It was perhaps inevitable that someone would analyse the interconnectedness of all these collaborations, and thus was born the Erdős number: the minimal path length between an author and Erdős in the collaboration graph.

So, Erdős has an Erdős number of 0, someone who has written a paper with Erdős but isn’t themselves Paul Erdős has an Erdős number of 1, someone who’s written a paper with one of them, but who hasn’t written a paper with Erdős himself (and isn’t themselves Paul Erdős) has an Erdős number of 2, and so on. This metric seems to have been first proposed by Caspar Goffman (1913–2006) in an article in a 1969 issue of the American Mathematical Monthly.

The American Mathematical Society provides a handy calculator to work out the collaboration distance between any two authors, at least via publications listed in their MathSciNet review database. My own Erdős number is 4, via the following path:

N M Dunfield, S K Friedl, N J Jackson, Twisted Alexander polynomials of hyperbolic knots, Experiment. Math. 21 (2012) 329–352
N M Dunfield, D Ramakrishnan, Increasing the number of fibered faces of arithmetic hyperbolic 3-manifolds, Amer. J. Math. 132 (2010) 53–97
V Kumar Murty, D Ramakrishnan, Period relations and the Tate conjecture for Hilbert modular surfaces. Invent. Math. 89 (1987) 319–345
P Erdős, M Ram Murty, V Kumar Murty, On the enumeration of finite groups, J. Number Theory 25 (1987) 360–378

Further details about the Erdős number can be found here, and biographies of Paul Erdős himself (who was a fascinating person and dedicated mathematician) can be found here, here, or in the book The Man Who Loved Only Numbers by Paul Hoffman.

A related concept is that of the Bacon Number, the minimal distance between oneself and the actor Kevin Bacon in the appropriate collaboration graph. So, for example, Kevin Bacon’s Bacon number is 0, Jack Nicholson’s is 1 (because they both appeared in A Few Good Men (1992)) and so on. The Oracle of Bacon provides a useful interface for finding shortest paths between actors listed in the IMDB database.

Depending on how strict you are about these things, my own Bacon number might be 4.

I was in a short amateur film called G103, as was a chap called Patrick Niknejad, who at the time was studying for a mathematics degree at Warwick, but who has also acted professionally. In particular, he was in 63 episodes of a children’s television series called My Parents Are Aliens which ran from 1999–2006. One of the other regular actors in that series was Tony Gardner, who appeared in Restoration (1995) with Meg Ryan, who appeared in In the Cut (2003) with Kevin Bacon himself.

Whether or not this actually counts depends on whether or not G103 is allowed. I argue that it is, because it has been shown on the big screen more than once (at about three of the Warwick Student Cinema‘s regular All-Nighter screenings, as well as a couple of departmental open days). (Also, this is my blog and I’m in charge. So there.)

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Maths! And Jam!

A while ago, Colin Wright came to Warwick to give a talk on the mathematics of juggling.  Colin has a PhD in combinatorics, and is one of the inventors of the siteswap or Cambridge notation for describing juggling patterns.  This was a highly entertaining and interesting talk, and if you get an opportunity to hear it, I encourage you to do so.

Last month, I went to see Festival of the Spoken Nerd at the Warwick Arts Centre.  This was an evening of amusing and entertaining science-based comedy from Helen Arney (who writes and performs science-related songs), Steve Mould (who does spectacular experiments on stage) and Matt Parker (a stand-up mathematician).  Also excellent, and if you like Radio 4’s Infinite Monkey Cage (which starts a new series today) they’re definitely worth looking out for.

A couple of years back, Matt started up a movement called MathsJam, where people interested in recreational mathematics gather together in pubs around the UK once a month to chat about maths and related things in a friendly and enthusiastic environment.  Colin and Matt, together with a small group of other people also organise an annual conference, partly inspired by the biennial Gathering 4 Gardner conference. I’d been vaguely tempted to go – Colin had mentioned it after his juggling talk, and then I got chatting to Matt after the FotSN show and he also waxed enthusiastic about it, so I decided to go.

And it was absolutely splendid. Roughly a hundred professional and amateur mathematicians (researchers, schoolteachers, engineers, lecturers, and a whole host of other people including at least one archaeologist and a professional magician) gathered together in a hotel near Crewe, and spent the weekend discussing a range of fun aspects of maths. Participants were encouraged to give a short talk (strictly limited to five minutes) on something interesting and vaguely mathematical, and this comprised the bulk of the programme on the Saturday and Sunday. (I gave a very quick talk about Poincaré dodecahedral space.)

Matt had devised a clever method of ensuring everyone stuck to their allotted five minutes: a countdown timer that chimed quietly when there was a minute left, buzzed increasingly urgently after the five minute mark, and then displayed this at the six-minute point.

Tweeting was very much in evidence, and a live feed of the #MathsJam tag was projected up onto the main screen throughout the conference (I’ve collected them all here).

I got to meet lots of splendid people and catch up with a few people I already knew, I learned lots of interesting stuff I didn’t know before, and had my enthusiasm for mathematics reinvigorated. I plan to go again next year.

Emmy Noether (1882-1935)

Emmy NoetherOver the last couple of days I’ve had cause to read a little bit about the life of the German mathematician Emmy Noether, who was responsible for some of the most important work in abstract algebra during the first third of the 20th century; in addition, Noether’s Theorem is important in theoretical physics, where it describes the connection between symmetry and conservation laws.

Born in Erlangen in 1882 into a Jewish family (to the algebraic geometer Max Noether and his wife Ida Kauffmann) she displayed an early talent for languages, and qualified as a teacher of French and English, but decided she wanted to study mathematics at university.  At the time (c.1900) women weren’t permitted to formally enrol at German universities, although with the individual lecturers’ permission they could sit in on lectures.  Nevertheless, she succeeded in persuading Erlangen to admit her as a student in 1904 (after the restrictions were lifted) and successfully completed her PhD in 1907.  For the next eight years she taught at Erlangen, essentially as her father’s unpaid teaching assistant, before Felix Klein and David Hilbert succeeded in arranging a more substantive post at Göttingen (at the time, one of the most prestigious mathematics departments in Europe).  Initially (due largely to complaints from the philosophy department) she taught under Hilbert’s name, but in 1919 she was awarded her habilitation diploma and promoted to the rank of Privatdozent (lecturer).

She worked at Göttingen for the next couple of decades, along the way making some really important discoveries in algebra: amongst other things, she formulated and proved the First, Second and Third Isomorphism Theorems which are a standard component of pretty much every undergraduate group theory textbook; also Noetherian rings are an important class of algebraic structures which she studied and were subsequently named after her.  She supervised 14 doctoral students (according to the Mathematics Genealogy Project) and in 1932 was invited to give a plenary address at the International Congress of Mathematicians.

The following year, she was dismissed from Gottingen along with several other colleagues (including Max Born and Richard Courant) as a result of legislation enacted by the new Nazi government, forbidding Jewish academics from working at German universities.  She reacted with stoicism and continued unofficially lecturing to students at her house. Her colleague Hermann Weyl later remarked that “her courage, her frankness, her unconcern about her own fate, her conciliatory spirit, was in the midst of all the hatred and meanness, despair and sorrow surrounding us, a moral solace”.  Shortly afterwards, she was invited to take up a visiting professorship at Bryn Mawr College, Pennsylvania.  She worked there and at the Institute for Advanced Study in Princeton, for the next year and a half, until her sudden death in April 1935 as a result of complications from surgery.

I’d been aware of some of the details of her life, and obviously I learned the Isomorphism Theorems as an undergraduate (although I only learned of her involvement relatively recently) but reading about her in more detail over the last few days I’ve been struck by how particularly impressive her achievements were given that she was Jewish and a woman working in an environment that was often unsympathetic to both.

Twisted Alexander polynomials of hyperbolic knots

I’ve recently been helping Stefan Friedl and Nathan Dunfield with an interesting project looking at the twisted Alexander polynomials of hyperbolic knots, which has now resulted in two papers [2,3], some software [1], and a number of unanswered questions. I’ve found it all fascinating and have learned a lot of interesting stuff about mathematics (twisted Alexander polynomials, Reidemeister torsion, hyperbolic 3-manifolds, the Mahler measure, etc) and computing (the Sage computer algebra system and the Python programming language).

I plan to look into the details and the background in more depth in the next few posts, but the basic idea is that for any knot {K} we can calculate something called the Alexander polynomial {\Delta_K}. This is a polynomial expression in a single variable {t} which (for various technical reasons) is well-defined up to multiplication by {\pm t^n}, but the reason we care about it is that it’s an isotopy invariant: it can often help us decide whether two knots are equivalent or not.

Knot-theoretic graffiti

Unexpected cubicle topology

More precisely, if you have two knots {K_1} and {K_2} which might be equivalent (that is, one is really just a continuously-deformed copy of the other) then you work out their Alexander polynomials and compare them: if {\Delta_{K_1} \not\equiv \Delta_{K_2}} then {K_1} and {K_2} aren’t equivalent. (Conversely, though, if {\Delta_{K_1} \equiv \Delta_{K_2}} then that doesn’t help, since there are many examples of non-equivalent knots which have equivalent Alexander polynomials.) For example, the trefoil knot (usually denoted {3_1}) has {\Delta_{3_1} = t^2 - t + 1}, but the figure-eight knot {4_1} has {\Delta_{4_1} = t^2 - 3t + 1}, so straight away we know that these are different knots.  (My friend Peter recently found this fact, but not the proof, scribbled on a toilet door.)

On the other hand, the 11–crossing knots {11_{19}} and {11_{25}} have {\Delta_{11_{19}} = \Delta_{11_{25}} = t^8 - 6t^7 + 18t^6 - 33t^5 + 39t^4 - 33t^3 + 18t^2 - 6t + 1}, so although we happen to know by other methods that they’re different, we can’t tell them apart just by using the Alexander polynomial.

But in addition to sometimes being able to tell different knots apart, the Alexander polynomial also contains some geometric information about the knot: it gives us a (not always optimal) lower bound on the genus of the knot, and also a necessary (but not sufficient) condition on whether the knot is fibred.  I’ll explain later what the genus of a knot is, and what it means for a knot to be fibred.

Rather than considering the knot on its own, we need to look at the space surrounding it: on its own a knot is really just a closed loop, a circle, and what makes it interesting is how it’s been embedded in whatever version of 3–dimensional space we happen to be using. In practice, we mostly consider knots embedded in the 3–sphere {S^3}, which can be described in various ways, but a fairly straightforward way of viewing it is like ordinary 3–space {\mathbb{R}^3} joined up at infinity.

During the 1990s, some mathematicians got interested in a generalised version of {\Delta_K}, called twisted Alexander polynomials. The idea here is that we modify {\Delta_K} by incorporating some extra information about the space surrounding the knot. More precisely, we use a representation of the fundamental group of this space. It turns out that if we’re careful and/or lucky about our choice of representation, the twisted Alexander polynomial can often distinguish between knots that the ordinary Alexander polynomial can’t, and can sometimes give us better genus and fibring information too.

Now, due to some really important work by William Thurston during the late 1970s and early 1980s (for which he was awarded a Fields medal in 1982), it turns out that the vast majority of knots (more precisely, all but 32 of the 1701936 knots with 16 or fewer crossings) are hyperbolic. (The exceptions are torus and satellite knots, which are comparatively rare cases.)

What this means is that we can impose a canonical 3–dimensional hyperbolic geometric structure on the complement of our knot {K} (the complement is basically {S^3} with a {K}–shaped hole cut out of it). And the symmetries of this geometric structure give us a similarly canonical representation which we can use to define a twisted Alexander polynomial with. Stefan’s idea was that since the hyperbolic structure is in some sense god-given, and the corresponding “holonomy” representation is similarly divinely-inspired, these twisted Alexander polynomials (which we called {\mathcal{T}_K}) should be pretty special, and hopefully have some interesting and useful properties.

So we set our computers to work calculating {\mathcal{T}_K} for all 313209 hyperbolic knots with 15 or fewer crossings, using the Sage computer algebra system and a hyperbolic topology package called SnapPy (a Python library written by Nathan Dunfield and Mark Culler, based on Jeff WeeksSnapPea software). We found that not only does {\mathcal{T}_K} correctly predict the genus in all 313209 cases, it also correctly detects fibredness too.

There’s an operation called mutation you can do on a knot: broadly speaking, you cut out a segment (a tangle) of the knot, flip it over, and glue it back in. Sometimes (the simplest examples have 11 crossings) you get a different knot as a result. The most famous example pair consists of the Conway and Kinoshita–Terasaka knots; the knots {11_{19}} and {11_{25}} mentioned earlier are another example pair. The ordinary Alexander polynomial {\Delta_K} can’t tell the difference (we say it’s mutation invariant) but the hyperbolic twisted Alexander polynomial {\mathcal{T}_K} often can. (In particular, it can tell the difference between the Conway and Kinoshita–Terasaka knots, and also between {11_{19}} and {11_{25}}.)

References

[1] N M Dunfield, Genus-Comp, software and documentation (2011)

[2] N M Dunfield, S K Friedl, N J Jackson, Twisted Alexander polynomials of hyperbolic knots, preprint (2011) arXiv:1108.3045

[3] S K Friedl, N J Jackson, Approximations to the volume of hyperbolic knots, from: “Twisted topological invariants and topology of low-dimensional manifolds”, (T Morifuji, editor), RIMS Kôkyûroku 1747 (2011) 35–46 arXiv:1102.3742

Three impossible things before teatime

William Blake: The Ancient of Days

William Blake: The Ancient of Days (1794)

Towards the end of the previous post, I mentioned the infamous 1897 bill considered (and nearly passed) by the Indiana state legislature, which amongst other things declared the value of \pi to be 3.2.  The bill itself mentions three great mathematical factoids discovered by the retired physician and amateur mathematician Dr Edward Johnston Goodwin (c.1828-1902), on which he claims copyright but offers as a gift to the State of Indiana for use in education.  This contribution comprises claimed solutions to three ancient problems unsolved since antiquity: the quadrature of the circle, the trisection of an angle, and the doubling of a cube.  Amongst this discussion, we are presented with the assertion that \pi=3.2.

The bill refers to the publication of Goodwin’s work in the American Mathematical Monthly, which is certainly true, although both notes appear in the “Queries and Information” section and are accompanied by the caveat “published by request of the author”.  This latter disclaimer (which doesn’t generally appear in any of the other letters published by the magazine) possibly indicates that the editors were not entirely convinced by Dr Goodwin’s discoveries and wished to make it clear that they weren’t formally endorsing them, whatever Goodwin might have claimed when helping his representative Taylor I Record draft the bill.  (The splendidly-named Record, a farmer and timber merchant, admitted during the debate that he didn’t understand anything of the contents of the bill, but was introducing it at Goodwin’s request.)

So, let’s look at Goodwin’s three discoveries in more detail.

Squaring the circle: This is usually understood to be the problem of constructing, in a finite number of steps, with only a pair of compasses and an unmarked straightedge, a square which has the same area as a given circle.  The problem dates back to the time of the Greek philosopher and mathematician Anaxagoras (c.500-428BC) although Oenopides (c.450BC) is believed to be the first to require only compasses and straightedge; it even puts in an appearance in Aristophanes‘ play The Birds (414BC):

Meton  With the straight ruler I set to work to inscribe a square within this circle; in its center will be the market-place, into which all the straight streets will lead, converging to this center like a star, which, although only orbicular, sends forth its rays in a straight line from all sides.

Goodwin’s method appeared in 1894, in the seventh issue of the first volume [1] and elicited no discussion in later issues (most of the contributing members at that point being embroiled in an ongoing debate over the validity of non-Euclidean geometry), although early in 1896 another article [2] appeared, written by William Heal (member of the London Mathematical Society and Treasurer of Grant County, Indiana), explaining in detail why the problem is unsolvable.  It should be remarked that the problem Heal discusses and Goodwin claims to solve isn’t the classical one: they instead concern themselves with the problem of finding a square with the same perimeter as a given circle.  Where all this falls down has to do with the irrationality, or more specifically the transcendence of \pi.

Irrational numbers (roughly, those which cannot be expressed as a fraction) fall into two categories: algebraic numbers like \sqrt2 and the golden ratio \phi=\frac{1+\sqrt5}{2} which can be expressed as the roots of polynomials with integer coefficients (respectively, x^2-2 and x^2-x-1) and transcendental numbers like \pi and e, which can’t.  The transcendence of \pi follows from a theorem published in 1882 by the German mathematician Ferdinand von Lindemann (1852-1939), and extended in 1885 by his fellow German Karl Weierstrass (1815-1897).

Goodwin’s article is not exactly a paragon of clear and comprehensible prose, and in places it’s really not obvious either what he’s doing or what he thinks he’s doing.  In fact, as well as the assertion that \pi=3.2, he appears to obtain several other values for \pi, as discussed by David Singmaster [4] and Arthur Hallerberg [5].

Trisecting an angle: This is another straightedge-and-compasses problem, and as the name suggests, the idea is to construct an angle \frac\theta3 from a given angle \theta. We can do this for certain angles quite easily (it’s easy to construct an angle of \frac\pi3, and also very easy to bisect an angle, so any angle of the form \frac{\pi}{2^n} can be trisected).

The constructability question was solved in general by the French mathematician Pierre Wantzel (1814-1848), in a paper published in 1837 [6].  Wantzel seems to have been a somewhat disorganised and driven person, and a combination of overwork, recourse to unwise levels of artificial stimulants, and generally just not looking after himself, seems to have contributed to his untimely death, a couple of weeks before his 34th birthday.  His near-contemporary Jean-Claude Saint-Venant (1797-1886) remarked later:

He worked usually during the evening, not going to bed until late at night, then reading, and only sleeping poorly for a few hours, alternately abusing coffee and opium; until he married, he took his meals at odd and irregular hours. [7]

The trisection problem is concerned with solving the triple angle formula \cos(3\theta) = 4\cos^3\theta - 3\cos\theta.  For example, let’s try to trisect the (constructible) angle \frac\pi3, that is, construct the angle \frac\pi9.  We know that \cos\frac\pi3=\frac12 so our triple angle equation becomes (after a little rearrangement) 8\cos^3\frac\pi9 - 6\cos\frac\pi9 -1 = 0.  Writing x=2\cos\frac\pi9 simplifies this to x^3-3x-1=0.  At this point we have to resort to a little bit of Galois theory:

Theorem Let z_1,\ldots,z_n\in\mathbb{C}. Any complex number z which is constructible by straightedge and compasses alone from 0,1,z_1,\ldots,z_n is algebraic with degree 2 over the field \mathbb{F} = \mathbb{Q}[z_1,\ldots,z_n,\bar{z}_1,\ldots,\bar{z}_n]. [8, Section 4.2]

What this means is that in order to construct an angle \frac\pi9 with straightedge and compasses, we need to find a quadratic polynomial ay^2+by+c with rational coefficients a,b,c\in\mathbb{Q} which happens to be a factor of our cubic polynomial x^3-3x-1.  Equivalently, we need to find a single rational root x=\alpha\in\mathbb{Q} of x^3-3x-1.  Now  we make use of another theorem:

Rational Root Theorem Let p(x) = a_nx^n+\cdots+a_0 be a polynomial with integer coefficients a_n,\ldots,a_0\in\mathbb{Z}.  For any rational solution x=\pm\frac{p}{q}, the numerator p must be an integer factor of the constant term a_0 and the denominator q must be an integer factor of the leading coefficient a_n.

So, the only possible rational roots of our polynomial x^3-3x-1 are \alpha=\pm1.  But neither of these are roots, since 1^3-3\cdot1-1 = -3 and (-1)^3-3(-1)-1 = 1.  So we can’t reduce our polynomial x^3-3x-1 as a product of a linear and a quadratic factor with rational coefficients, and hence by the first theorem above, we can’t construct the number x=2\cos\frac\pi9 (or, for that matter, \cos\frac\pi9) with just straightedge and compasses.  So there exist perfectly cromulent (even constructible) angles which can’t be trisected with only straightedge and compasses.

Goodwin’s method, by the way, which he described in a very short letter [9] to the American Mathematical Monthly, is to trisect a chord of the given angle:

(A) The trisection of a right line taken as the chord of any arc of a circle trisects the angle of the arc;

This just doesn’t work.  (Try it.)

Doubling a cube:  This third problem fails for very similar reasons to the trisection problem.  Given an arbitrary cube, we want to construct, using only straightedge and compasses, another cube which has exactly twice the volume of the first one.  Equivalently, we need to be able to construct a cube with sides \sqrt[3]2 times the length of the original one.  This essentially boils down to a very similar polynomial reduction problem to the previous one, but with the polynomial x^3-2 instead of x^3-3x-1.  By the same argument, we need to factorise it into a quadratic and a linear factor, both with rational coefficients, and again this really just requires us to find a single rational root.  But by the Rational Root Theorem, such a root must be either \pm2.  Which it isn’t, since 2^3-2 = 6 and (-2)^3-2 = -10.  So x^3-2 doesn’t factorise into a quadratic and a linear factor with rational coefficients, and hence \sqrt[3]2 can’t be constructed with just straightedge and compasses.

Goodwin’s solution [9] was:

(B) Duplication of the Cube: Doubling the dimensions of a cube octuples its contents, and doubling its contents increases its dimensions twenty-five plus one per cent.

This, again, doesn’t work.  What he seems to be saying is that to double the volume of a cube, you need to multiply the length of each side by 1.26.  This is accurate to three decimal places (\sqrt[3]2 \approx 1.25992105) but the classical problem requires an exact solution constructible by straightedge and compasses, which this isn’t.

I find cases such as Goodwin’s both interesting and poignant.  He was clearly an intelligent and educated man, and his attempts at these three classical problems seems to have been borne from a genuine desire to advance human knowledge and education.  His obituary, printed in the New Harmony, Indiana Times of 27 June 1902 (quoted in the article by Hallerberg [5]) comments on his sincerity and his ultimate disappointment that his work was not appreciated during his lifetime.

References

[1] E J Goodwin, Quadrature of the Circle, Amer. Math. Monthly 1 (1894) 246-247

[2] W E Heal, Quadrature of the Circle, Amer. Math. Monthly 3 (1896) 41-45

[3] F von Lindemann, Uber die zahl \pi, Math. Ann. 20 (1882) 213-225

[4] D Singmaster, The legal values of Pi, Math. Intelligencer 7 (1985) 69-72

[5] A E Hallerberg, Indiana’s squared circle, Math. Mag. 50 (1977) 136-140

[6] P L Wantzel, Recherches sur les moyens de reconnaître si un problème de
Géométrie peut se résoudre avec la règle et le compas
,  J. Math. Pures Appl. (1) 2 (1837) 366-372

[7] J-C Saint-Venant, Biographie: Wantzel, Nouvelles Ann. Math. (1) 7 (1848) 321-331

[8] N Jacobson, Basic Algebra I, W H Freeman (1985)

[9] E J Goodwin, letter, Amer. Math. Monthly 2 (1895) 337

Approximating pi

Every 14 March (or 3/14 in US format) for the last few years I’ve noticed an increasing amount of joviality and excitement about the constant \pi (\approx 3.141592653\ldots).  I assumed for a while that this was one of those internet-originating fads like Talk Like a Pirate Day (19 September) or the related Talk Like a Pilot Day (19 May): a bit of fun but (the adoption of Talk Like a Pirate Day as a religious holiday by the Church of the Flying Spaghetti Monster notwithstanding) essentially a bit of annual silliness.  But it turns out that Pi Day has a slightly more serious heritage: Wikipedia tells us that it was founded in 1988 by Larry Shaw, a physicist working at the Exploratorium, an amazing interactive science museum in San Francisco.  A ceremony is held at 1:59pm on 14 March, culminating in the participants singing happy birthday to Albert Einstein (born on that day in 1879).  In 2009, the US House of Representatives passed a non-binding resolution recognising the date as National Pi Day.

I’m in two minds over whether this sort of thing has a net positive effect on public understanding of mathematics, or whether (like those tedious corporate PR-driven “magic equation” stories that keep turning up in newspapers, and on which Ben Goldacre has eloquently grumbled several times) it merely serves to trivialise the subject and present mathematicians as just interested in numbers and hard sums.  I suppose it depends on how it’s presented: the situation wasn’t entirely helped by the recent news items about a supposed battle between the mathematical establishment and a small group of mavericks and renegades over whether \pi or \tau=2\pi is the more natural constant to use in most circumstances.  But by and large I think Pi Day is probably a good thing: it gets people thinking about mathematics and helps inspire some children to go and find out more (in particular this is the intention of the people behind the whole thing, and also explicitly stated as such in the resolution passed by the House of Representatives).

But today, 22 July, is Pi Approximation Day: a well-known rational approximation to \pi is given by 22/7 (=3.\overline{142857}).  Actually, 22/7 gives a slightly better approximation to the value of \pi than 3.14; an even better one is given by \frac{355}{113} \approx 3.1415929203\ldots but that hasn’t easily translated into a date since the reign of the Roman Emperor Constantius II.

In my EC119 Mathematical Analysis exam this year I set a question requiring the students to calculate the Taylor-MacLaurin series for the function \arcsin x near x=0 up to fifth order and use this (together with the fact that \sin\frac{\pi}{6}=\frac{1}{2}) to obtain a rational approximation to the value of \pi.

I thought this was an interesting bit of a question, but relatively few of them attempted it (which, I guess, goes to show that examiners and students have different ideas about what constitutes an “interesting” exam question).  Anyway, the fifth-order Taylor-MacLaurin expansion for \arcsin x is x = x + \frac12\cdot\frac{x^3}{3} + \frac12\cdot\frac34\cdot\frac{x^5}{5} and so by setting x=\frac12 and multiplying the result by 6 we should get a rational approximation to \pi.  As it happens, we get \frac{2009}{640} = 3.1390625, which is clearly getting there but isn’t as good as 3.14, 22/7 or 355/133.  And although we can get a much better approximation by calculating more terms of the Taylor-MacLaurin series, it’s clear that this isn’t a particularly efficient way of doing things.

Fortunately, there are many better ways of calculating rational or decimal approximations to \pi.  In 1706, the English mathematician John Machin (c.1686-1751) published the identity \frac{\pi}{4} = 4\arctan\frac{1}{5} - \arctan\frac{1}{239} and used it (together with the MacLaurin series for \arctan x) to calculate the first hundred decimal places of \pi.

The Indian mathematician Srinivasa Ramanujan (1887-1920) discovered the formula \frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum^\infty_{k=0} \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}} which converges extremely rapidly.  The best currently known formula (which I thank Arthur Clune for alerting me to) was discovered by the Ukrainian/American mathematicians Gregory and David Chudnovsky in 1989, based on Ramanujan’s work.  This formula \frac{1}{\pi} = 12 \sum^\infty_{k=0} \frac{(-1)^k (6k)! (13591409 + 545140134k)}{(3k)!(k!)^3 640320^{3k + 3/2}}, converges extremely quickly, with each additional term adding another fourteen or fifteen decimal places of accuracy.  They used it in August 2010 to calculate \pi to 5\times 10^{12} decimal places. (5\times 10^{12} is 5 billion or 5 trillion depending on whether you’re using the long or short scales; being English I tend to use the former, and experience a faint sense of guilt on the increasingly common occasions I find myself using the latter.)

An apparently much less accurate approximation to \pi turns up in the Bible.  1 Kings 7:23 says:

And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about.

This verse, repeated almost verbatim in 2 Chronicles 4:2, implies the approximation \pi\approx3.  The other famous approximation to \pi turns up in an infamous 1897 bill considered (but ultimately not passed) by the state legislature of Indiana.  This includes various mathematical observations by the physician and amateur mathematician Edwin J Goodwin (c.1825-1902), such as

Furthermore, it has revealed the ratio of the chord and arc of ninety degrees, which is as seven to eight, and also the ratio of the diagonal and one side of a square which is as ten to seven, disclosing the fourth important fact, that the ratio of the diameter and circumference is as five-fourths to four

This indicates not only a value of 3.2 (=\frac{4}{1.25}) for \pi but also a value of 1.\overline{428571} for \sqrt2.  The bill also includes Goodwin’s “solutions” to the problems of squaring the circle, trisecting an angle and doubling the cube (three ancient unsolved problems known since the middle of the 19th century to be formally unsolvable).  The bill was passed unanimously by the House but due to the efforts of Clarence Waldo, a Professor of Mathematics at Purdue University, enough members of the Senate were persuaded of the errors in Goodwin’s work that the bill was stalled indefinitely in committee and never made it onto the statute book.