Bag of Beans

The Poincaré Conjecture: In Search of the Shape of the Universe
by Donal O’Shea.
Walker & Company, 2007.

In The Poincaré Conjecture, Donal O’Shea explains a conjecture in topology from 1904 that had remained unsolved for nearly a century. Aside from its importance in topology, the conjecture also has implications on determining the shape of our own universe. It is also one of the seven Millennium Prize problems listed by the Clay Institute in 2000, with a one million dollar reward for a correct solution. It was finally solved in 2002 by Grigory Perelman and since then his solution has been accepted. He may be eligible for the Millennium Prize but does not appear to be interested. In 2006, he was awarded the Fields medal—the highest honor for mathematicians and which also carries a monetary reward—for his work but he declined the award.

In this book, O’Shea takes us through the history of the conjecture and the attempts at solving it, and also takes some time to give us the historical context along the way by describing the social and political climate surrounding each mathematician that has sought to prove the conjecture. He does a good job of providing relatively clear and simple explanations of the complex ideas in topology and non-Euclidean geometry involved, but the book does move at a fairly brisk pace (minus the notes at the end the main text is only 200 pages long) so some work is still required to follow along, but I never felt completely lost. This book contains a nice mix of mathematical ideas and history for a general audience, and it managed to keep my interest throughout.

Rating: 8/10

Links:

• Amazon.ca
• LibraryThing

Related:

• Book: The Möbius Strip

Evil Mad Scientist Laboratories has an ingenious how-to on creating a clay model of a fractal known as a Sierpinski triangle.

One of our favorite shapes is the Sierpinski triangle. In one sense, a mere mathematical abstraction, on the other, a pattern that naturally emerges in real life from several different simple algorithms. On paper, one can play the Chaos Game to generate the shape (or cheat and just use the java applet).

You can also generate a Sierpinski triangle in what is perhaps a more obvious way: by exploiting its fractal self-similarity.

• Link

The Möbius Strip: Dr. August Möbius’s Marvelous Band in Mathematics, Games, Literature, Art, Technology, and Cosmology
by Clifford Pickover.
Thunder’s Mouth Press, 2006

As the title implies, The Möbius Strip explores the strange characteristics of Möbius strips and other related one-sided or single-surface constructs, like Klein bottles and real projective planes. It delves into a wide range of topics like topology and higher-dimensional mathematics, as well as introducing places where they might naturally be found in chemistry and cosmology. While there are a small number of places where formulas are presented, they are not completely essential to enjoying the book. This is a worthwhile read on a truly mind-bending topic.

Rating: 7/10

Links:

• Amazon.ca
• LibraryThing

William Wu has quite a large collection of tech-interview style riddles and puzzles.

This is an archive of problems I’ve been collecting since the Spring of 2002. They come from many places, including word of mouth, college courses, books, and job interviews for hi-tech positions. Many are even written by members of our own forum community. These carefully chosen puzzles will demand you to think in creative ways you perhaps normally would not. In fact, some will seem outright impossible at first … but once you crack them, the epiphany can be truly rapturous!

• Link (via Neatorama)

This is a creative way to create fractal images using Christmas ornaments.

It’s a little easier to see the images of the camera lens in this photo. Look at the three Cantor sets formed by the number of reflections of the camera lens on the three spheres. Math in action!

• Link
• Photoset on flickr

This is a disturbingly funny call between George Vaccaro and Verizon, trying to teach them basic math.

From Slashdot:

Blogger George Vaccaro recently had a problem with his Verizon based on an unfortunate miscommunication of currency. The crux of the matter was that he was quoted .002 cents per kilobyte for data during a trip to Canada but was charged .002 dollars. Normally this would have been an easy fix, however several humorous calls later the Verizon reps still were unable to discern the difference between the two rates. You really have to hear it to believe it. Kudos George, you have the patience of a saint.

• George’s blog
• YouTube audio link
• Digg post

Following the astounding claim that a professor has “solved” the problem of division by zero, and inexplicably reported by BBC as seemingly legitimate news (imagine my head exploding at this point), here’s an article describing the obvious problems with the claim.

Tons of folks have been writing to me this morning about the BBC story about an idiot math teacher who claims to have solved the problem of dividing by zero. This is an absolutely infuriating story, which does an excellent job of demonstrating what total innumerate idiots reporters are.

What this guy has done is invent a new number, which he calls “nullity”. This number is not on the number line, can’t be compared to other numbers by less than or greater than, etc. In other words, he’s given a name to the basic mathematical concept of “undefined”, and proclaimed that this is somehow a solution to a deep and important problem.

• Link
• BBC report

This may not be that practical, but it is an interesting way to do multiplication visually by using intersecting lines.

• Video (via reddit)

You would think this is pretty obvious yet people keep throwing their money into these things.

Someone’s done the math and my hunch was right. A New York Times article titled The Word on Warranties: Don’t Bother expains why:

• They’re actually a profit center for retailers. The margins on electronics are very thin, but they extended warranty margins are as high as 80%.
• It’s a sucker’s bet: you’re betting against the bathtub curve and you’re also betting against the trend of falling prices in the belief that the cost of repair will exceed the cost of replacement.
• A Consumer Reports study shows that only 10% of digital cameras fail in their first five years. For the extended warranty to be valuable, it would have to be less than 10% of the purchase price, yet extended warranties often cost as much as 20% of the item’s price. Besides, if you had a five-year old digital camera, you’d probably want to replace it, not repair it.
• Last year, suckers spent $16 billion on extended warranties.

• Link

Supposedly, this is one of the hardest Sudoku puzzles to solve.

It’s times like these that the internet makes me most happy. When respectable online publications publish entertaining and interesting stories about the world’s hardest Sudoku puzzle, you’d at least expect them to give you a link to it, or a picture of it. Oh, yes, you guessed it. They don’t. But I couldn’t find it anywhere! So I undustriously hunted, and eventually found the secret formula hidden away in an ASCII-like tomb of Sudoku knowledge.

• Link (via reddit)

If you think the Brainfuck programming language is twisted, how about a language with only two symbols?

For todays dose of pathological programming, we’re going to hit the worlds simplest language. A Turing-complete programming language with exactly two characters, no variables, and no numbers. It’s called Iota. And rather than bothering with the rather annoying Iota compiler, we’ll just use an even more twisted language called Lazy-K, which can run Iota programs, Unlambda programs, as well as its own syntax.

• Link (via reddit)

I can’t even imagine how you would design a folding sequence that takes 40 hours to complete.

These days patterns requiring more than 100 steps are common. Some of that competitive acceleration is due to Lang, who transformed the art by writing a computer program that can generate the blueprint for ultracomplex origami sculptures. Even with digital assistance, figuring out the sequence of folds that will create a beetle and all its ornaments is a mathematical problem of staggering complexity. Still, the reigning champion of intricate origami is a 23-year-old Japanese savant named Satoshi Kamiya. Unaided by software, he recently produced what is considered the pinnacle of the field, an eight-inch-tall Eastern dragon with eyes, teeth, a curly tongue, sinuous whiskers, a barbed tail, and a thousand overlapping scales. The folding alone took 40 hours, spread out over several months.

• Link (via Digg)

Damn Interesting has an entertaining article on how a girl in high-school proved that you can fold a piece of paper in half twelve times. She did it by calculating exactly how much paper she would need, and then tracking down an industrial-sized roll of toilet paper that met the requirements.

I have heard it stated as fact that one cannot fold a paper in half more than eight times, because the doubled and re-doubled paper quickly becomes too thick. This reasoning had always seemed pretty odd to me – what was this magical property of hardened tree pulp that caused it to stop folding after so long? Thanks to a student named Britney Gallivan, it turns out that this impossibility is just as mythical as not being able to lick your elbows. By developing her own formulae for paper folding, she calculated how much paper one needs to achieve any number of folds (and herself managed to break world records by folding a piece of paper twelve times).

• Link

This article presents a significant optimization for Conway’s Game of Life.

Making a slow program fast can lead to both joy and frustration. Frequently, the best you can do is a low-level trick to double or maybe quadruple the speed of a program; for instance, many readers may have implemented John Conway’s “Game of Life” using bit-level operations for a significant speedup. But sometimes a whole new approach, combining just a few ideas, yields amazing improvements. A simple algorithm called “HashLife,” invented by William Gosper (”Exploiting Regularities in Large Cellular Spaces,” Physica 10D, 1984), combines quadtrees and memoization to yield astronomical speedup to the Game of Life. In this article, I evolve the simplest Life implementation into this algorithm, explain how it works, and run some universes for trillions of generations as they grow to billions of cells.

• Link (via reddit)

Freedom to Tinker has an interesting post on how HDCP could be broken.

Every new HDCP device is given two things: a secret vector, and an addition rule. The secret vector is a sequence of 40 secret numbers that the device is not supposed to reveal to anybody. The addition rule, which is not a secret, describes a way of adding up numbers selected from a vector. Both the secret vector and the addition rule are assigned by HDCP’s central authority. (I like to imagine that the central authority occupies an undersea command center worthy of Doctor Evil, but it’s probably just a nondescript office suite in Burbank.)

• Link (via Boing Boing)

Infoverse has an entertaining proposal to replace the decimal numbering system with the octal system. They’ve come up with new set of digits and a new system for time-measurement.

do you think our conventional decimal number system is perfect? if not…this is for you: a new octal_based mathematics_system for the information age.

• Link (via Digg)

This article argues that Minesweeper belongs to the class of NP-Complete problems (see “Complexity Classes P and NP” in Wikipedia to brush up on P and NP).

My result in the Mathematical Intelligencer states that a decision problem which I like to call “the Minesweeper Consistency Problem” and which is exactly equivalent to the problem of playing the minesweeper game, is yet another one of these NP-complete problems.

For the current discussion, it suffices that the problem of simply detecting which squares are or are not mines is equivalent to the Minesweeper Consistency Problem, and the fact that it is NP-complete means, for Minesweeper fans, that their favourite game can be seen as being right at the cutting edge of mathematical research. There are two possible viewpoints one might take on this.

The more “sober” viewpoint is that the NP-completeness of Minesweeper shows that Minesweeper really is a rather good game. The fact that it is NP-complete means that it is very difficult to spot when it is possible to clear a square safely in full knowledge that that square is clear, and when some guessing is required. In fact, even if you are told in advance that guessing is not required, it may still be difficult to decide what squares to clear. In some sense, when you play the game you cannot be expected to do much better than the hundreds of very good mathematicians who have worked on the P=NP? question for many many years.

• Link (via reddit)

Happy Pi day! “Written in the USA date format, March 14 is an unofficial celebration for Pi Day derived from the common three-digit approximation for the number Ï€: 3.14.”

• Digg it.

I came across this oddity while reading The Curious Incident of the Dog in the Night-time. From Wikipedia, the problem can be stated as follows:

Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?

Common sense might tell you that because there are two unopened doors left, you have a 50-50 chance of picking the one with the car so it doesn’t matter if you switch doors. But in fact if you always switch doors, you will win a car 2 out of 3 times. To understand this you can look at the possible outcomes:

If you stay, you will only have a 1 in 3 chance of winning a car. But if you switch doors, you have a 2 in 3 chance.

Here is another programming challenge site. This one is specifically geared towards mathematical problems. You can use any programming language you like.

The problems are rated according to how many people have already solved a particular problem. The more people that have solved a problem, the less points that problem is worth. The scores are calculated dynamically, so your overall score can change depending on what the problems are currently worth.

I’m currently using Ruby for these problems. I still prefer Python, but Ruby does have some nice features.