Slashdot Mirror

mikejuk writes to let us know that today is Pi Day — 3/14 in American date notation. He writes, "This year, it feels as though we aren't celebrating alone. For the first time, it looks as if momentum has built up to the point where people have heard about Pi Day. There are even attempts to sell you Pi-related items as if it was a real holiday. But there is always some one to spoil the party so what ever you do to celebrate don't miss Vi Hart's Anti-Pi Rant video." Thus begins the yearly debate over Pi Day vs. Tau Day (June 28). Phil Plait has a post defending Pi Day's honor, and MIT isn't holding back their Pi Day celebrations. Large-scale celebration of Pi Day began in 1988, mostly through the efforts of physicists Larry Shaw and Ron Hipschmann at the San Francisco Exploratorium. The Exploratorium still runs Pi Day events 26 years later, including Pi-themed processions and pie for dessert. In 2009, Pi Day became semi-official through a vote by the House of Representatives. (They did a better job with Pi than did Indiana, who almost legislated it to be 3.2.)

The best way to celebrate Pi Day is to get together with some friends and talk math over a pie. You could even go for a pizza pie, since a pizza with radius 'z' and height 'a' has volume = pi * z * z * a. If you'd care for a game, head over to the Pi Day Challenge, which features a series of pi-related logic puzzles. Or just spend the day learning about pi.

Cool pi facts: Pi is currently known to about 10 billion decimal places. You can calculate pi using the Fibonacci sequence. A few years ago, Steven Rochen mapped the digits of pi to musical notes and turned it into a violin solo (video). Others have made music from pi as well. Mankind didn't know the first hundred digits of pi until the year 1701. How many digits of pi can you recite? The record for memorization currently stands at 67890 digits. The record for reciting pi while juggling three balls is just under 10,000.

book_reader (Gary Cornell) writes "Wow, what an intriguing title! When I was getting my Ph.D in math, the words 'sex' and 'mathematics' were not juxtaposed all that often, and I suspect we would have been more likely to expect a book titled 'Mathematics and the (lack of) Sex.' But, hey, times change and the author, who is not only a mathematician but also someone who was voted one of Australia's 50 most beautiful people in their equivalent of People magazine -- and remember this is the land of Nicole Kidman -- has a point. As she says, echoing G.H. Hardy's famous comment in 'A Mathematician's Apology': 'Mathematics is the study of patterns: their discovery, their interconnections and their implications.' And what is sexual behavior but the most intriguing pattern of all?" Read on for the rest of Cornell's review. Mathematics and Sex author Clio Cresswell pages 177 publisher Allen & Unwin rating 8/10 reviewer book_reader ISBN 1741141591 summary A very nice introduction to the modelling of inter-personal behavior

The way one studies patterns mathematically is by building models for the behavior being modeled. This is why most of this book is about mathematical models for interpersonal behavior. Well, that together with some amusing anecdotes that make the book a fun read even if you know the literature very well. Still, before I go any further with this review I want to remind everyone that the key question to ask oneself when reading any book that does mathematical modeling of any topic is always the same: are the models built realistic?. Mathematicians can't answer this question: only research by scientists (i.e., experience) can. Einstein probably put it best when he said:

"As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality."

While we do study models for their applicability and their eventual predictive use by and for science, mathematicians can and do also study them for their intrinsic mathematics beauty, and some of the models Cresswell discusses in this book are certainly very pretty (in the mathematical sense of beauty--because the solutions are elegant, though the pun is intended.)

As an example of what this whole subject is like let me tell you about a long-studied model of interpersonal behavior that the author discusses in Chapter 3, a chapter titled "Road Testing the Bed"--I kid you not.

"You have to choose your life mate. The rules we adopt for this model are that you will be presented 100 choices one after another, you may date them, sleep with them, whatever. But, at the end, you must say yea or nay and if you say nay, you will never see them again."

What strategy should you adopt? Well, if you wait to the end, the odds are only 1/100 that the last person is the optimal choice; ditto if you choose the first person. The modeler then asks: what strategy should you adopt for optimum results? A little bit of mathematics involving infinite series gives the answer. You can prove mathematically that the best strategy is to look at (approximately) the first 36.787944117144235 people (rounding it to, say, 37 people) and then you should choose the first person from that point on that is 'better' then the previous 37 people. This increases the odds of your finding the best match from 1% to about 37%- roughly a 37 times improvement. (In the pre-politically correct literature this model was called "The Sultan's Dowry Problem," or "The Secretary Problem"; now, alas, it is usually called simply an example of an "Optimal Stopping Problem." )

Is this a good model for how we behave? Is this a strategy that one can realistically adopt? Certainly, 100 possibilities seems like a lot of choices to have if one is not the current day equivalent of a sultan -- a movie star or an athlete. But the model is intriguing, if not totally realistic and applicable.

Models that spring from modification of the rules of the Sultan problem have always been one of my favorites in this area. This makes Chapter 3 my favorite chapter: it is chock full of goodies with lots of interesting variations of the original problem, and thus even more interesting models. Some may be far more applicable. For example, if you get to play the cad and can keep potential mates 'stockpiled,' then, by stockpiling seven potential mates, there's a strategy that you can use to increase the odds of finding the best one to 96% or so! Or, in another variation of the model, whose solution she refers to as the "twelve bonk rule," there's a result that says that if you simply want to ensure that your choice is better than 90% of the other choices available, simply 'sample' the first 12 possibilities and pick the first person who is better after the first 12. This strategy gives you a 77% possibility of success.

I obviously can't go over all the models she builds, the interesting results she cites, or the interesting observations she makes in a review so let me simply give you some of the high points of the remaining chapters:

Chapter 1 is entitled "Love, sweeeet love" and mostly consists of showing you various differential equations that can model love's attraction and repulsion i.e. variations on standard "prey-predator models." For example, she mentions the following model of attraction:

"The more Romeo loves Juliet, the More Juliet wants to run away ... Romeo gets discouraged and backs off, Juliet finds him strangely attractive. Romeo tends to echo her..."

This model gives rise to a standard and very simple first order differential equation. She then talks about more sophisticated versions of this model including one by Rinaldi that tries to model a famous love poem by Petrarch. (Personally, I think these models are only useful for learning differential equations but don't shed much light on the problem.)

Chapter 2 is called "Marriage and the Happily Ever After" and describes models for behavior in a relationship, including an analysis of how absurd the folk tale is that more sex occurs in the first year of marriage then in all subsequent years combined. Probably the most interesting work she talks about in this chapter are the models by Guttman et al. intended to analyze conversations between lovers to determine if the relationship is on the rocks. In this case the models they build are known to be highly accurate in predicting problems in the relationship.

Chapter 4 is entitled "Dating Services -- are you really being served?" and it has a fascinating analysis of the perils of questionnaires that try to match too many variables (i.e. why those questionnaires don't help that much). As she points out, this is called the "curse of dimensionality" in the literature. The problem is that if you are trying to determine whether two points are very close in n-dimensional space where n is large, you are unlikely to get a whole lot of difference between points and so closeness doesn't really matter much.

Chapter 5 is called "Pairing Up," and shows how Game Theory can (should?) enter into the problem of "choice" preferences. This chapter is a very nice gateway into models that are studied in the greatest depth in economics; there is an incredibly interesting literature on these issues. One should start with Arrow's paradox on voting (that most logical axiom systems for building choice models are actually inconsistent and can't simultaneously be satisfied) and then work up to real problems with how congressional seats are allocated in the United States. Wikipedia has good articles to start with on these models.

Chapter 6 is called "Action Reaction Attraction" and is about ways to model people's attractiveness. This means things like symmetry as a cross cultural model for beauty, and waist-to-hip ratio for females as a cross-cultural model for male choice. Whether these models are correct is an extremely active area of research in anthropology and evolutionary psychology. The jury seems to still be out, but the evidence for their truth is certainly growing.

Chapter 7 is called "Pick a Sex, Any Sex" and is a tantalizing hint of what the mathematics of evolution is all about. In particular this chapter includes a nice discussion of how sex itself can evolve. (It seems paradoxical that the question of how sex itself can evolve is not yet resolved. After all, in a naive "selfish gene" approach to evolution, it would seem seem that asexual methods of reproduction win hands down. But, as usual, the issues are more complex then naive models would predict. For example, who would have thought that parasites might be the reason sex arose? Again, for more details on the science behind the models the author discusses, you can start with a useful Wikipedia article. Ridley's popular science book called the Red Queen (or anything by Maynard Smith) is where to go next.

Chapter 8 is titled "How Ovaries Count and Balls Add Up," and is about models for feedback levels of hormone concentration and circadian rhythms and didn't particular interest me.

Finally, Chapter 9 is called "Orgasm" and I'm not going to summarize it, since that would be telling.

To sum up, is this book perfect? No. I think more mathematically literate people would like appendices which give some indication of the deeper math behind what she discusses. For example, the math that shows why the answer I gave above to the Sultan's choice problem really is approximately 36.787944117144235 - or more correctly n/e, where e is the base of natural logarithms and n is the number of choices one has to go through, is well within the reach of any 2nd year calculus student. The differential equations she introduces in other chapters can be understood by anyone with a good engineering or math background. The game theory and even a proof of Arrow's theorem should be accessible to any literate person etc. As is, though, anyone with even some knowledge of or interest in mathematics will find this book great fun.

You can purchase Mathematics and Sex from bn.com. Slashdot welcomes readers' book reviews -- to see your own review here, read the book review guidelines, then visit the submission page.

stevelinton writes "The UK National Physical Laboratory has a new atomic clock potentially 1000 times more accurate than current cesium clocks: to within 1 second in about 30 billion years! This could lead quite soon to a new definition of the second, and in a while to improved resolution in GPS successor systems. More interestingly, there are theories that some of the universe's fundamental dimensionless constants may have changed by a parts in a million over the last 10 billion years or so. These clocks are so accurate that they should be able to detect these changes over a year or two."

Phoe6 writes "Wired is carrying an Open Letter of Benoît Mandelbrot, the father of the fractal, to the wizards of Wall Street, calling on them to recognize a pattern in the finantial and economic trends in the world. Mandelbrot says, If we can map the human genome, why can't we map how a man loses his livelihood? If millions can contribute a few cycles of their PCs to the search for a signal from outer space, why can't they join a coordinated search for patterns in financial markets?" I'd like to see a debate between Mandelbrot and Friedrich Hayek.

stevelinton writes "I am looking for a tutorial similar to Dave Ragget's excellent HTML tutorial(s), but for XHTML 1.1. I am NOT looking for a "HTML to XHTML" conversion tutorial. I want to teach a class XHTML 1.1 from day 1, without assuming that they know any HTML at all. Does anyone know of such a thing?"

EricR writes "On December 14, 1900, Max Planck presented experimental results in front of the German Physical Society and announced that they could best be explained if energy exists in discrete packets, which he called "quanta." Today is the 100th birthday of Quantum Physics."

An anonymous reader asks: "In The Feynman Lectures on Computation, Richard Feynman poses an interesting little puzzle involving the synchronization of finite state machines acting as generals and soldiers. While he was able to find an answer to the problem, the minimum time solution apparently eluded him, and he ended his description of the puzzle with the following Fermat-like declaration: 'Somebody has actually found a solution with this minimum time. That is very difficult though, and you should not be so ambitious. It is a nice problem, however, and I often spend time on airplanes trying to figure it out. I haven't cracked it yet.' My best attempt performs at about 3N, not quite the minimum time of 2N-2. So I'm asking Slashdot: Has anyone ever come across the minimum time solution to this puzzle? Or maybe someone here can figure it out!"

"Here is the full description of the problem, in Feynman's own words. Please remember that these are finite state machines, so you can't use any methods that involve counting the number of soldiers or assigning a number to each soldier.

Problem 3.4: Before turning to Turing machines, I will introduce you to a nice FSM problem that you might like to think about. It is called the 'Firing Squad' problem. We have an arbitrarily long line of identical finite state machines that I call 'soldiers'. Let us say there are N of them. At one end of the line is a 'general', another FSM. Here is what happens. The general shouts 'Fire'. The puzzle is to get all of the soldiers to fire simultaneously, in the shortest possible time, subject to the following constraints: firstly, time goes in units; secondly, the state of each FSM at time T+1 can only depend on the state of its next-door neighbors at time T; thirdly, the method you come up with must be independent of N, the number of soldiers. At the beginning, each FSM is quiescent. Then the general spits out a pulse, 'fire', and this acts as an input for the soldier immediately next to him. This soldier reacts as in some way, enters a new state, and this in turn affects the soldier next to him and so on down the line. All the soldiers interact in some way, yack yack yack, and at some point they become synchronized and spit out a pulse representing their 'firing'. (The general, incidentally, does nothing on his own initiative after starting things off.)

There are different ways of doing this, and the time between the general issuing his order and the soldiers firing is usually found to be between 3N and 8N. It is possible to prove that the soldiers cannot fire earlier than T=2N-2 since there would not be enough time for all the required information to move around. Somebody has actually found a solution with this minimum time. That is very difficult though, and you should not be so ambitious. It is a nice problem, however, and I often spend time on airplanes trying to figure it out. I haven't cracked it yet."

Judebert writes "Benoit Mandelbrot, the man responsible for much of the interest in fractals, spoke last month at the American Giophysical Union meeting. He explained how he has been using fractals to find order within complex systems in nature, such as coastlines and weather. (I thought he was dead, but apparently he's just been teaching at Yale.) Earth scientists have taken his fractal work to the point of forecasting the size, location, and windspeed of hurricanes at landfall. Their predictions are being made available to FEMA and other government agencies."

The below review was contributed by reader Lozzer, and deals with a book about one of the most fascinating figures in mathematics history (and cryptograhic history in particular). Nearly half a century after his suicide, Alan Turing is still fascinating and relevant on several levels.

Alan Turing: The Enigma author Andrew Hodges pages 587 publisher Walker & Company rating 8.5 reviewer Lozzer ISBN 0802775802 summary A wide-ranging look at one of the most important mathematical minds, before, during and after his role in breaking WWII codes.

I recently finished reading Andrew Hodge's excellent biography of Alan Turing. The second edition was printed in 1992. It included updates based on material declassified by the British Government (Amazon has a 2000 edition, I'm not sure if this is a rewrite or a reprint). Weighing in at nearly 600 pages the book is not for the faint hearted Geek.

For our younger Script Kiddies I'll give a brief overview of Turing's life and what he has to do with computing. He was born in London in 1912 and christened Alan Mathison Turing. After a public school up-bringing he studied maths at King's College, Cambridge from 1931. In 1935 he solved part of one of the great mathematical problems of the time: Hilbert's Second Problem. Godel had solved the first two parts. Turing solved the last part about deciding which mathematical statements were true. His construction for solving this problem was the Turing Machine. This model forms the basis for all modern day computers.

Between this breakthrough and the war, Turing spent a couple of years at Princeton, where he studied under Alonzo Church and John von Neumann, both of whom where pioneers in the computing field.

With the onset of war in 1939 Turing found himself employed as a code breaker at Bletchley Park (which is only a couple of miles from where I live). This is where Turing's theoretical knowledge began to take physical shape. By the end of the war the Colossus had been built. This is sometimes touted as the first computer, though I'll leave that to people with flame retartant underpants. Suffice it to say this "computer" could only be programmed by reconfiguring the hardware.

After the war Turing gravitated to the University of Manchester where he took a role in developing the first prototype computer that was "software" programmable. After that he became a programmer, using the computer to help with mathematical theories. He was convicted of Gross Indecency (Turing was a homosexual) in 1952, and had to suffer a year of oestrogen injections to "cure" him. He committed suicide in 1954.

After that potted history, back to the book. It draws on a lot of sources and manages to bring them together in a very coherent whole. As well as providing a British view of the history of computing it also gives an interesting perspective on the changes in society over the years. The book also conveys Turing's breadth of knowledge and vision well - while most computer users were thinking of mathematical problems he was into AI, chess and other abstract symbolism. He figured out the need for subroutines, was the first to use binary (he noted that routines could change the external notation for human consumption, but continued to use 32 bit numbers entered in reverse order himself). He considered hardware acceleration. The author does well in explaining the scientific portions of the book in a clear and correct fashion. From a Geek perspective the text is possibly a bit dense, with some less interesting chunks (the homosexual aspects of Turing's life, for example, have less impact now than when the book was first published).

I recommend the book if you are interested in some of the wider aspects of Turing's life. For me, being British, having a Cambridge maths degree (ooh shameless), and living near Bletchley brings a lot more of the book to life that it may for most. If you are only interested in Turing's impact on the world of computers there are good online resources for this. Maybe, however, you won't find out why Christopher Strachey was the world's first Hacker."

You can purchase this book at ThinkGeek.

The below review was contributed by reader Lozzer, and deals with a book about one of the most fascinating figures in mathematics history (and cryptograhic history in particular). Nearly half a century after his suicide, Alan Turing is still fascinating and relevant on several levels.

Alan Turing: The Enigma author Andrew Hodges pages 587 publisher Walker & Company rating 8.5 reviewer Lozzer ISBN 0802775802 summary A wide-ranging look at one of the most important mathematical minds, before, during and after his role in breaking WWII codes.

I recently finished reading Andrew Hodge's excellent biography of Alan Turing. The second edition was printed in 1992. It included updates based on material declassified by the British Government (Amazon has a 2000 edition, I'm not sure if this is a rewrite or a reprint). Weighing in at nearly 600 pages the book is not for the faint hearted Geek.

For our younger Script Kiddies I'll give a brief overview of Turing's life and what he has to do with computing. He was born in London in 1912 and christened Alan Mathison Turing. After a public school up-bringing he studied maths at King's College, Cambridge from 1931. In 1935 he solved part of one of the great mathematical problems of the time: Hilbert's Second Problem. Godel had solved the first two parts. Turing solved the last part about deciding which mathematical statements were true. His construction for solving this problem was the Turing Machine. This model forms the basis for all modern day computers.

Between this breakthrough and the war, Turing spent a couple of years at Princeton, where he studied under Alonzo Church and John von Neumann, both of whom where pioneers in the computing field.

With the onset of war in 1939 Turing found himself employed as a code breaker at Bletchley Park (which is only a couple of miles from where I live). This is where Turing's theoretical knowledge began to take physical shape. By the end of the war the Colossus had been built. This is sometimes touted as the first computer, though I'll leave that to people with flame retartant underpants. Suffice it to say this "computer" could only be programmed by reconfiguring the hardware.

After the war Turing gravitated to the University of Manchester where he took a role in developing the first prototype computer that was "software" programmable. After that he became a programmer, using the computer to help with mathematical theories. He was convicted of Gross Indecency (Turing was a homosexual) in 1952, and had to suffer a year of oestrogen injections to "cure" him. He committed suicide in 1954.

After that potted history, back to the book. It draws on a lot of sources and manages to bring them together in a very coherent whole. As well as providing a British view of the history of computing it also gives an interesting perspective on the changes in society over the years. The book also conveys Turing's breadth of knowledge and vision well - while most computer users were thinking of mathematical problems he was into AI, chess and other abstract symbolism. He figured out the need for subroutines, was the first to use binary (he noted that routines could change the external notation for human consumption, but continued to use 32 bit numbers entered in reverse order himself). He considered hardware acceleration. The author does well in explaining the scientific portions of the book in a clear and correct fashion. From a Geek perspective the text is possibly a bit dense, with some less interesting chunks (the homosexual aspects of Turing's life, for example, have less impact now than when the book was first published).

I recommend the book if you are interested in some of the wider aspects of Turing's life. For me, being British, having a Cambridge maths degree (ooh shameless), and living near Bletchley brings a lot more of the book to life that it may for most. If you are only interested in Turing's impact on the world of computers there are good online resources for this. Maybe, however, you won't find out why Christopher Strachey was the world's first Hacker."

You can purchase this book at ThinkGeek.

Caseman writes, "Are you a closet mathematician who wants to come out? British publisher Tony Faber is offering a cool million bucks to the first would-be math head to prove the infamous Goldbach conjecture. Yeah, the one about every even number being the sum of two primes. Impossible, you say? Remember Fermat's Last Theorem? It stood unproven for 350 years until a recent effort yielded a proof. Read The London Times article about the challenge. "

spaceorb writes "According to researchers ... it may be possible to create black holes by creating a vortex of fluid that swirls at velocities comparable to the speed of light. Follow the above link for the theoretical discussion or here for the story on unisci.com." These are optical analogues of black holes, not really gravity wells, but they may advance our understanding of the real thing.

Between 1804 and 1807 Jean Fourier discovered the Fourier Transform: a means of transforming any function into its frequency components. He initially used it to study the propagation of heat in solids. Since then the Fourier transform has found a myriad of applications such as the JPEG, MPEG and MP3 formats... It's even been used to multiply polynomials. The main computational cost of the Fourier transform are the N^2 multiplications it requires. In 1903 Runge noticed that the number of multiplications required could be reduced to N.log(N) by using trigonometric symmetries. In 1965 this was applied in computers by Cooley and Tukey: the fast Fourier transform became popular. Since computers represent numbers in binary, multiplications and divisions by powers of 2 are commonly implemented by shifting bits left and right. Multiplications by constants are easily optimized using the same trick. In 1999 Trac Tran of Johns Hopkins University found an approximation to the DCT which causes very little error, yet uses only 13 shifts and 31 additions for N=8. Given the recursive nature of the FFT, this transform could be used as part of an FFT with N>8. Apparently, he has applied for a patent for this approximation. Do you think this is worth a patent? Do you know of prior art?