Welcome to the 71st edition of Carnival of Mathematics!
I know that an improvisation on the theme of 71 is eagerly anticipated at this point. But to be honest there's no way I can improve on what is offered by the charming where no fewer than forty-five 71-isms (and two pending) are to be found.
Actually, this fact about 71 is not yet one that appears in Prime Curios (unless it is one of the two that are pending of course).
Carnival time means another month has passed, so this prefatory spot may extend itself by reflecting on October's mathematical news stories. The biggest one was a sad one: the death of Benoît Mandelbrot, which I will come back to below. We also lost Alf van der Poorten the distinguished number theorist, famous for his role in the proof of Apéry's sensational 1978 result that ζ(3) is irrational. Linked to another sad event but defiantly upbeat was the worldwide celebration Gathering for Gardner which took place on 21st, which would have been Martin Gardner's 96th birthday. And more cause for celebration: the 20th marked the first United Nations World Statistics Day.
Carnival of Mathematics itself is a wonderful way of looking back over the past month and seeing what the mathematical world has been thinking about. It's nice to imagine that those reading edition number 1271, in a hundred years time, will be able to form a clear impression of what we were up to; much more sharply than we can for the mathematical world of the early nineteen-hundreds.
So what is the word on the mathematical street? Well, by coincidence the first two submissions to arrive for Carnival no. 71 both related to a hot topic from the early fifteen-hundreds: squaring the circle. Approximations to π go back thousands of years but there is some doubt whether ruler and compass constructions which equate to algebraic numbers were really of interest to the Greeks. FŽanor, in JOST A MON: Squaring the Circle posted at JOST A MON, has a nice little anecdote about Leonardo da Vinci having a go at it; Mark Eichenlaub presents Viete's Formula and Spinning Pizza posted at Arcsecond. Now Viete's Formula tells us that π is algebraic (constructed using ordinary arithmetic plus n-th roots) if you go to the limit, a very remarkable idea. Mark's posting moves on to a more general formula of Euler and some delightful trigonometry calculated to bring any classroom alive. If I might plug another book, by the way, Abstract Algebra and Famous Impossibilities, by Arthur Jones, Sidney A. Morris and Kenneth R. Pearson, is an absolute gem.
Irrational numbers figure in Katie Sorene's submission 9 Most Mathematically Interesting Buildings in the World posted at Tripbase, as one might expect, with π in the great pyramids and φ in the Parthenon. I am reminded of the cautionary words (2MB download) of George Markowsky on looking for numbers in cultural artefacts. In some cases he is surely rightly, though always politely, scornful. I can't help bringing in, at this point, the submission of Dave Richeson: Exceptional MathReviews posted at Division by Zero; scorn, especially when it is not at all polite, is always good fun to read (provided you are not the target!)
Well, whatever your views, Katie Sorene's images are lovely to look at and the whole thing is thought-provoking. Guillermo P. Bautista Jr. too, is thought-provoking about the implications of the irrationality and transcendence of π in his philosophical contribution Is mathematics an exact science? posted at Mathematics and Multimedia. He is in good company asking this question: it led Imre Lakatos, meeting Karl Popper in 1960 immediately to import the latter's ideas into mathematics, the result being Proofs and Refutations: The Logic of Mathematical Discovery, published after Lakatos' tragic early death in 1974 but based on the first three chapters of his 1961 PhD thesis.
We keep half an eye on mathematical philosophy as Richard Elwes presents Richard Elwes – Concrete Incompleteness 1 posted at Richard Elwes' blog. Gödel's theorems belong to mathematical logic and most working mathematicians are happy to keep it that way. But the logicians have been pushing away at the boundary: since the 70s there have been intuitively appealing mathematical results which are provable but not within the mathematical systems used to state them (unless these systems are inconsistent). Now there are examples where the mathematical system concerned is pretty much the whole of set theory. Leading edge stuff!
Gödel's first incompleteness theorem was brought closer to every-day scientific concerns (much closer if we agree with Roger Penrose's books about consciousness), along the different route of undecidability, with Alan Turing providing the breakthroughs. This work is likely to be a highlight of the Turing centenary celebrations in 2012. GrrlScientist submits a fine piece of science journalism reporting, in How the leopard got its spots posted at Punctuated Equilibrium, on an extension of another Turing classic, this time from mathematical biology.
My next submission is going to take us towards probability theory via the great Paul Erdős. This is not such a jump, actually: the story that Turing won his Cambridge fellowship for discovering the central limit theorem has become well-known and attests equally to a surprising aspect of his talent of which many are ignorant and to an area of surprising ignorance of the British mathematicial community at the time. By the way, Turing has an Erdős number of 5 (at most), although perhaps this is a low degree of separation which we would take for granted.
Alexander Bogomolny, whose name is synonymous, at least to me, with the magisterial cut-the-knot, tells us about a nice challenge in A. Soiferís Book, P. Erdosí Conjecture, B. Grunbaumís Counterexample posted at CTK Insights: find for yourself a counterexample which eluded the great Erdős! Well, Erdős was not infallible, if you believe the (possibly apocryphal) Monty Hall story; but his absolute signature was the delightfully simple solution to a seemingly insoluble problem; so much so that 'proof from the book' has entered the language, certainly of mathematics. So I think this must have been a result he was very pleased to see.
Monty Hall is a particularly famous example of just how elusive questions of probability can be. This issue is very well illustrated by John Cook in Probability that a number is prime — The Endeavour posted at The Endeavour. If we need lots of very large prime numbers then we can generate them quickly provided we accept a small probability of error - of compositeness. These are the so-called 'industrial grade' primes (the term is attributed to Henri Cohen who, by the way, collaborated with Alf van der Poorten on the ζ(3) irrationality proof) and they are genuinely important in industrial-strength cryptographic systems. The question is, how do we assign probabilities to one-off events such as a given number being prime? John finds in this an interesting contrast between the Bayesian and frequentist schools of thought.
Relative frequency can often provide a way to re-pose probability problems so that they become more intuitive. When the number of trial outcomes is small, Venn diagrams make this approach helpfully pictorial. Steve Mould's Venn Vs Euler: The Diagrams | SHIFT_beep, posted at SHIFT_beep, made me curious to think how Euler diagrams, in which impossible outcomes or joint outcomes are not represented, might simplify things still further.
Venn diagrams are probably most widely used as a teaching tool and there are many other ways to help introduce students to set theory or probability. One example comes unexpectedly in Gianluigi Filippelli's submission Eyeballing mathematics - Science Backstage posted at Science Backstage: online geometrical games come with frequency plots showing where your performance falls in the distribution—what a great way to illustrate probability density: get your whole class to play, and see for themselves that their chance of ending in the tail of the distribution is small!
Mathematical analysis is notoriously at the hard end of this pedagogic spectrum; the concepts are elusive and formalising them is a technical business. Sam Shah confronts this with obvious enthusiasm and integrity in Where do I go from here? posted at Continuous Everywhere but Differentiable Nowhere. Anyone who has their own hard-won experience in this area should read his story and give him a shout. My personal hero in this respect is David M. Bressoud, whose A Radical Approach to Real Analysis acknowledges the conceptual difficulties by retracing the historical path followed in their resolution.
Talking of pedagogic heroes I was very pleased to hear from Peter Rowlett's Travels in a Mathematical World! His wonderful series of student-targeted podcasts under that name for the Institute of Mathematics and its Applications has concluded since he moved on to other things. In his post These are your important living mathematicians he presents the results of a survey of who matters in mathematics today. Paul Halmos famously provided a ranking system for mathematicians, in which third-rate meant something like 'outstanding, only just not in the first or second rank' (I forget the details—it'll be somewhere in the good bibliography provided in this ZALA Films page). G.H. Hardy proposed, even more famously, that anything less than first-rate wasn't worth bothering about and this contention has a respectable history. Still, I don't think anybody would argue that the names offered to Peter are ones to conjure with.
By the way, I have posted my own little slideshow tribute to Mandelbrot, comprising whatever nice fractal images I could find and get permission to use in a hurry. If anybody would like to offer any more I will add them and try and make something more worthwhile.