eon

Posted by tpc at September 3rd, 2010

I used to be a member of MAA but found that it did nothing much for me except I get to issues of American Mathematical Monthly. I would classify that as good to have - something to read when I have spare time but most of the articles are not quite my cup of tea. So I decided to stop my membership after a year. The simple fact is this, as a mathematician not based in US, I do not get most of the benefits of belonging to MAA (likewise AMS.) I don’t get to attend their meetings. So why should I pay normal rates? I’m now considering whether or not to continue my membership of AMS at reciprocity rates.

However, I’ve recently added MAA to my twitter account and I have to say that they do quite a good job of pumping feeds, so much so that I have to pick and choose what I read. For example, there is a very neat article on topology and car-shades.

Posted by tpc at July 20th, 2010

Recently I had the horrendous realization that I’ve lost whatever little origami skills that I once possessed. I was trying to entertain a bored child on the plane who speaks a little English, and I knew no German save “guten tag”. I thought I would fold a paper crane and I couldn’t!

Anyway, I attended an extremely good talk by Robert Lang about Origami and the connection with mathematics. A fifteen minute version is available here. (The first and last minutes are about the same as the talk, but I didn’t view the 13 minutes in between.)

Posted by tpc at July 12th, 2010

Just attended a nice talk by Erik Demaine on Origami and puzzles. He demonstrated some remarkable stuff and ended with a cute picture hanging puzzle - how to hang a puzzle with two nails such that by removing either one, the picture drops. A brief description is at the end of the following paper
http://erikdemaine.org/papers/FUN2004i_TheoryComputSys/paper.pdf
and a preprint dated 2004 that seem to be still unavailable is announced. I would love to have a look at that illusive paper.

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 27th, 2010

How cool is that!
http://www.bathsheba.com/math/klein/klein_x1.html
$78 though.

Posted by tpc at May 11th, 2010

Learned a cool trick today. The finite projective plane of order n has
points,
lines,
points on each line,
lines passing each point.

The best known example is that of a fano plane which is of order 2. Now for order 3, there are a total of 13 lines, 13 points and 4 points on each line. It turns out you can use a standard deck of cards to represent this. You can divide the deck into 13 lines of 4 cards each, with each face value (regardless of suit) representing the same point.
Now if
,
each line is

The fun thing is to distribute the each line of 4 cards to different persons. Then you can randomly call out two face value and be sure that exactly one person has both of these cards - two points line on a unique line. Moreover, any two line intersect at a unique point - means that any two persons should have exactly one card in common.

Posted by tpc at March 19th, 2010

The first of the 7 Clay milliennium prize has been awarded. Sure it was a controversial one, but I guess it was fitting that the winner was one who didn’t want the million bucks. Money is not the ultimate prize as most would presume. Why does my mind conjure up a image of clay pigeon being shot down?

Posted by tpc at September 16th, 2009

From boing boing, exactly what the title says.

Posted by tpc at September 28th, 2008

Any rigid body displacement where a point is fixed is equivalent to a rotation. I saw this neat proof from Don Koks’ Explorations in Mathematical Physics.

By the hypothesis

Hence is orthogonal and But should vary continuously from the identity transformation, allowing us to conclude So


since we are working in 3D. We are forced to conclude
for some .
Thus has an eigenvector with eigenvalue 1. The transformation is thus the rotation about the direction .

Posted by tpc at January 17th, 2007

I gave a talk before on the golden ratio, so when I came across this blog entry
Ipod and the golden ratio via Gooseania, I quickly saved this link in case I have to give the same talk again.

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 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.

Posted by tpc at February 6th, 2005

I’ve never seen the proof of this theorem, although it is mentioned in every advanced calculus and geometry/topology course that I’ve taken. What I’ve learnt recently is that it cannot be generalised to higher dimensions, and the Alexander’s Horned Sphere is a counter example. I particularly like the picture of Conway.
See the following link for more details.