Posted by tpc at August 17th, 2010
I have no idea who Don Tapscott is and I probably will not read his books, but in the local papers today, he mentioned something on learning in university.
It’s not so much what you know when you graduate that counts as it is your capacity to learn all your life.
Posted by tpc at June 14th, 2010
I’ve always loved Escher’s circle limit. Here’s a neat article attempting to reconstruct it.
Posted by tpc at May 29th, 2010
Picked up a copy of Lewis Carroll in Numberland by Robin Wilson from the library. I must admit that I browsed through it instead of reading it, picking up bits and pieces that I find interesting. For example, it is told of how Queen Victoria was charmed by Alice’s Adventures in Wonderland that she demanded:
Send me the next book Mr Carroll produces …
And the next book that arrived was … An Elementary Treatise on Determinants.
Ok, the story is not true but how cool would it be if it had been. Another fun tidbit is the following game. Starting from the number 1, A and B take turns adding a number from 1 to 10 to the running total. Whoever gets to 100 wins. What is a winning strategy? Working backwards, for B to be certain of winning, he would need to get to 89. That way, A can’t win with one turn but whatever number he picked, B can go for the win. Inductively, to get to 89, B needs to first get to 78, 67, 56, 45, 34, 23, 12 — steps of 11.
Posted by tpc at May 24th, 2010
Martin Gardner passed away last week on 22 May, aged 95. Wikipedia is a good place to read about his contribution in bringing mathematics to the public. My favourite article of Gardner’s is Six Sensational Discoveries that Somehow or Another have Escaped Public Attention, Sci. Amer. 232, 127-131, Apr. 1975. (Also published in Time Travel and Other Mathematical Bewilderments.) Inside, Gardner announces six discoveries among which a counter-example to the four colour theorem. Before you jump off your seat, the article was dated 1st April 1975. Yes, it’s another very clever hoax.
The best among the six is the claim that
is exactly an integer and this fact was found by Ramanujan. The attribution to Ramanujan was clever not because of Ramanujan’s remarkable prowess of calculation but that constant is actually an evaluation of the modular j-invariant
and of course 
has class number one.
Posted by tpc at May 8th, 2010
Being down under results in the privilege of receiving my copy of the Notices of AMS five months late. The December 2009 copy showed up in my mail last week and the best article of all is about Legendre. It seems that despite being such a famous figure in mathematics, the mathematical world had been using a wrong portrait of him until recently. Everything came to light only when a computer search revealed that two different man named Legendre shared the same portrait. Full story available online.
Posted by tpc at July 16th, 2009
There is this nice story that I read about Gorenstein’s chauffeur, google tells me that it came from Sarah Flannery’s book called “in code”. Below is the text lifted off the web.
“In the course of a long lecture tour, the famous American mathematician Daniel Gorenstein was chauffeur driven to various venues around the USA. While Gorenstein lectured his chauffeur sat in the back row of the auditorium. In time the chauffeur became so familiar with the material (her comments: and it was pretty deep stuff) that he joked to Gorenstein, “At this stage, I reckon I could give the lecture myself.”
One evening, Gorenstein was scheduled to speak at a small university where he guessed his face wouldn’t be known. Feeling particularly tired, the master hit upon the idea of the chauffeur to give the lecture while he would rest in the back row. The lecture went off without a hitch, at the end the chauffeur answered without hesitation all the questions, which he heard asked and answered hundreds of times before; all, that is accept for one last question which left him entirely at a loss.
After momentary panic, he composed himself and said., “I believe that question is so simple even my chauffeur could answer it.” The dozing chauffeur shamed the questioner with an immediate reply.”
Anyway, since the title of this post is in plural, there is another chauffeur story. Namely, I was privileged today to be chauffeur to none other than Serre. Can I put that in my vita?
Posted by tpc at June 6th, 2009
I’ve never met him, but he did research in the area of partitions and was a friend of several people I know. One of them - Shaun Cooper - by pure chance found a reference to Richard in the 1996 autobiography of Howard Marks named “Mr Nice”. On the cover it said of Marks “He was Britain’s most wanted man. He has just spent seven years in America’s toughest penitentiary. You’ll like him.”
On page 73, here’s a paragraph of what he had to say.
There were one or two ex-Oxford students attached to the University of Sussex. One was a brilliant mathematics lecturer, Richard Lewis, who would often visit Ilze and me along with Johnny and Gina Martin. Richard came from a relatively wealthy family, owned property in Brighton and London, drank like a fish, smoked everything at hand, thought mathematical profundities, and was a keen and talented chess player. He had heard of Go, was interested in the game, but had never played. I taught him. After a dozen games, he beat me. He still beats me.
There are a few more references to Richard and his wife in the following pages, get the book and read it!
Posted by tpc at March 4th, 2009
The newspaper (26 Feb) reported an economist made this prediction. That’s an alarming figure, but not because the economy is so bad, rather the mathematics is so bad. Can anyone without divine powers get that kind of accuracy? So this Mr Irvin Seah’s job loss model probably takes into account whether the cleaning lady at the coffeeshop downstairs can keep her job, since he is able to predict job losses to the exact number. With such clever economists, no one wonder everyone is bashing the number crunchers. Including, Warren Buffet’s famous or infamous:
Beware of geeks bearing formulas.
This blog is turning out to be something like a mini version of John Allen Paulos’ “A Mathematician reads the newspapers.” By the way, Paulos is apparently in Singapore this week and giving a talk at NTU.
The said article is available here if you care to read it.
Posted by tpc at January 7th, 2009
Pronouncing names has always been a problem and I think it is embarrassing and just not right to mispronounce names, if one can help it. I always cringe when I hear Euler pronounced as “U-ler” instead of “Oil-ler”. Thankfully we now have a guide, via the Notices of AMS.
Posted by tpc at April 30th, 2008
I have no idea who is Heaviside until I started to teach this course which included Laplace transforms. That got me really interested and I checked out P. Nahin’s biography from the library. Perusing the borrowing slip, the book was last borrowed in Sep 2006, prior to that Mar 1990. 16 long years.
Meanwhile, in the preface a quote attributed to Lazarus Long:
Anyone who cannot cope with mathematics is not fully human. At best he is a tolerable subhuman who has learned to wear shoes, bathe, and not make messes in the house.
Posted by tpc at April 15th, 2007
My favourite sum:
To Euler,
The master of infinity, who summed with impunity!
PS: A nice article on infinity, explaining concepts up to Cantor’s work.
via http://www.mathed.org via amazon.
Posted by tpc at March 26th, 2007
Ah, the wonders of youtube. We can now view videos of Tom Lehrer performing some of his musical compositions. It’s great old fashioned music.
He sang the following songs with lyrics complied on this site by Graeme Cree.
1) The Derivative Song
2) There’s a Delta for every Epsilon
3) The Professor
4) Sociology
5) That’s Mathematics
Dig around, there’s also a video somewhere of the song New Math.
Posted by tpc at March 15th, 2007
Picked up this book from and library. I particularly liked this paragraph:
Intuition, in the Star Wars sense, may be the most important and most ignored part of our machine-oriented world. When do you look beyond the rational, the routine, to see what is really around you? Trust your feelings, your “instincts.” In science, you need to know when your equipment isn’t working properly. We’re obsessed with high technology, yet a few bad sensors can cause the meltdown of a megawatt nuclear generator, and a small chunk of foam insulation can destroy a billion-dollar space shuttle and its crew. Skywalker, strong in the Force, knew when to use his computer and when to ignore it. Do we?
And how do you develop this intuition in maths? By immersing yourself in it, by working through pages and pages of calculations and certainly not through using fancy-schmancy calculators!
Posted by tpc at November 12th, 2006
Here’s a quote attributed to Robin Wilson, from gooseania
I once gave a talk on classical Greek mathematics, in honour of Casablanca it was called Here’s Looking at Euclid.
I literally laughed out loud. (I make it a point to never use that particular internet acronym, for no particular reason though.)
Posted by tpc at September 5th, 2006
Mathematicians are like Frenchmen: whatever you say to them they translate into their own language and forthwith it is something entirely different
Johann Wolfgang von Goethe
Posted by tpc at April 24th, 2006
The following is a quote from Mario Livio’s The Golden Ratio, which in turn quotes Tobias Dantzig’s Number.
A squire was determined to shoot a crow which made its nest in the watch-tower of his estate. Repeatedly he had tried to surprise the bird, but in vain: at the approach of man the crow would leave its nest. From a distant tree it would watchfully wait until the man had left the tower and then return to its nest. One day the squire hit upon a ruse: two men entered the tower, one remained within, the other came out and went on. But the bird was not deceived: it kept away until the man within came out. The experiment was repeated on the succeeding days with two, three, then four men, yet without success. Finally, five men were sent: as before, all entered the tower, and one remained while the other four came out and went away. Here the crow lost count. Unable to distiguish between four and five it promptly returned to its nest.
Posted by tpc at April 9th, 2006
Honestly, I know next to nothing about statistics. It’s no wonder that I have not heard of this book, and most of the actors in the stories within. This is a good book that tells the tales of the origin of statistical studies and should be a must-read for all statistics majors. One major flaw was that he was writing too much for a layman, and most of the details are hidden behind some general explanation. While this is alright for a brief history of the subject, I sincerely believe that anyone who is willing to read a 300 page book on statistics, would like to see a deeper discussion. Another gripe is that the author seem to come across as one who is very critical of those who are only interested in pure theory/mathematics. That aside, the book is filled with gems.
I particularly like this delightful quote attributed to R. A. Fisher
A scientific career is peculiar in some ways. Its raison d’etre is the increase of natural knowledge. Occasionally, therefore, an increase of natural knowledge occurs. But this is tactless, and feelings are hurt. For in some small degree it is inevitable that views previously expounded are shown to be either obsolete or false. Most people, I think, can recognize this and take it in good part if what they have been teaching for ten years or so comes to need a little revision; but some undoubtedly take it hard, as a blow to their amour propre, or even as an invasion of the territory they have come to think of as exclusively their own, and they must react with the same ferocity as we can see in the robins and chaffinches these spring days when they resent an intrusion into their little territories. I do not think anything can be done about it. It is inherent in the nature of our profession; but a young scientist may be warned and advised that when he has a jewel to offer for the enrichment of mankind some certainly will wish to turn and rend him.
Posted by tpc at February 13th, 2006
By John Stillwell. I picked this up from the new arrivals counter at the library. The hyperbolic tessellation on the front cover was invitation enough for me.
The book is about teaching geometry to undergrads from four different perspectives: 1) axiomatic a la Euclid, 2) linear algebra, 3) Projective and 4) transformations. It’s quite interesting and comes with lots of illustrations, although personally, I would prefer a book that delves more deeply into each aspect.
One main focus of the book is the cross-ratio which was discussed at length in the second half of the book. There is this wonderful quote.
At this point I can hear someone asking, “What is the geometric significance of the cross-ratio?” Although I first encountered cross-ratios as a senior in high school, and have dealt with them many times since then, I must say frankly that I cannot visualize a cross-ratio geometrically. If you like, it is magic. Here is this algebraic quantity whose significance is impossible to understand, and yet it turns out to do something very useful. It works. You might say it was a triumph of algebra to invent this quantity that turns out to be so valuable and could not be imagined geometrically. Or if you are a geometer at heart, you may say it is an invention of the devil and hate it all your life.
- Robin Hartshorne, Geometry: Euclid and Beyond p 341.
Posted by tpc at January 27th, 2006
I was quite happy to pick up a copy of Escher prints for $7 at borders the other day. I guess most people are suitably impressed by the work of Escher. For me, it’s especially so because of the four circle limits, their geometric significance and relationship to modular forms. Go to the official page -> picture gallery -> 1955-1972.
Here’s an interesting anecdote from Douglas Hofstadter (Yes, Mr Godel, Escher, Bach. ) in the article “Mystery, Classicism, Elegance: An Endless Chase After Magic“, from the book MC Escher’s Legacy. It seems that even Mick Jagger was interested in Escher’s work and wrote him to get permission to put a particular print on his upcoming album. He began “Dear Maurits …”
Of course, Escher was not interested and in his reply (to Jagger’s assistant) he ended with
By the way, please tell Mr Jagger I am not Maurits to him, but
very sincerely,
M.C. Escher
Posted by tpc at April 24th, 2005
The most difficult part of teaching trigonometry to kids (more specifically, pre-university and below) is proving trigonometric identities. Typically, the stronger students would relish the challenge, while the weaker would despair. Some of them would even question what is the point of all these. Another potential problem is that students are told to accept certain identities and use them. One chief culprit would be the addition formula:
.
In all my years of formal mathematical education, I do not recall ever seeing that identity being proved. I sort of figured it out for myself, while learning complex analysis, which to me is the best proof. One version is given here. Ah, the beauty of complex analysis!
The shortest path between two truths in the real domain passes through the complex domain.
-Jacques Hadamard (1865-1963)
More info on the above quote given here.
Of course, a purely geometric proof exists and involves drawing one triangle on top of another. A more interesting second proof uses Ptolemy’s theorem.
