As noted in the last page of the pdf, I suggest calling the alternate constant 2 π=6.283… `1 turn’, so that 90 degrees is `a quarter turn’, just as we would say in natural language. The main point is that the historical choice of the value of π obscures the benefit of radian measure. It is easy to see that 1/4 turn is more natural than 90° , but π/2 seems almost as arbitrary. It is apparent that we can’t eliminate π but it is to be aware of its pitfalls, and introduce an alternative for those who might wish to use one.

— Bob Palais

Palais then goes on to define a “newpi” symbol through a TeX macro, which resembles the fusion of two π:

The aforementioned article has been in print since 2001, and very little has changed on this front since then. The ideas it put forth are an amusing opinion that many of us tend to agree with, but 2π has not been adopted by the mathematical community.

Today Michael Hartl announced “The Tau Manifesto” on what he calls Tau Day (6/28 for 6.28…). In this document, Hartl echoes the good points that Palais made and builds upon them to construct a strong case in favor of adopting a circle constant which is the ratio of a circle’s circumference to its radius, not its diameter. Inspired by Palais’ use of the word “turn”, he proposes calling this constant τ (tau).

As Hartl argues, this symbol already exists (unlike the odd symbol that Palais introduced), it’s still generally available in mathematics, and it resembles π.

This new constant would not only be an improvement from a pedagogical standpoint (as shown in the diagram above), but would also “improve” several formulas. For example, Euler’s identity:

Or:

Which would become neater as:

This makes sense intuitively (a rotation in the complex plane by one turn is 1).

(The Tau Manifesto addresses the issue of how this too can relate to the “five most important numbers in mathematics” with a slight rearrangement.)

What are your thoughts on this? As mathematics evolves, is it time to start using “Let τ = 2π” as a means of adopting a better circle constant?

With endorsements from star programmers and its unique user interface (UI), this site is very community oriented and aimed at highlighting the best answers and questions through a voting system. As such it has quickly become the number one programming forum, overshadowing the dreaded – and much shadier – Experts-Exchange.

However, what not everyone may know is that Joel and Jeff have made the platform available to others for a non-negligible monthly fee, at StackExchange. Over the course of the last few months several communities have grown out of this service, some more successfully than others.

The most successful of these sites is called MathOverflow. It currently has the largest number of active participants and voters on its posts. The site offers the same UI as StackOverflow, but adds in support for LaTeX formulas. Here at Math-Blog.com we feel it’s important to promote such nice initiatives.

From the look of things (such as the existing questions and answers), it’s clear that there is some serious mathematics going on at the site. If you’re looking for interesting discussions on topics at a graduate/research level, head over to MathOverflow – you won’t be disappointed. However, please keep in mind that the aim of the site is to target those with an interest in advanced math. If you’re seeking a more basic (read undergraduate) level of math assistance, you should probably look elsewhere.

That said, what are some the other graduate level forums and mailing lists that you find useful when it comes to your mathematical research? One of my favorites is the Number Theory List.

PS: I hope you like our new look. Be sure to drop by Math Blog this coming Monday, as we’ll be publishing a really interesting post that I’m sure you’ll love. In order not to miss it, you can also grab our feed.

The author of the site, describes the project as follows:

Project Euler is a series of challenging mathematical/computer programming problems that will require more than just mathematical insights to solve. Although mathematics will help you arrive at elegant and efficient methods, the use of a computer and programming skills will be required to solve most problems.

When you first register, you will be presented with 252 problems available to be solved. This number changes frequently, however, as new problems are routinely created and added to the list. The solutions to some problems veer towards the obvious, while others require advanced mathematics to tackle. Likewise, a few can be solved with paper and pencil and a good dose of insight. Most however, will require the aid of a computer program written in the language of your choice.

A sample problem (187)

Each program you execute should output your result within a minute. That’s part of the challenge, even though there is no way for the site to enforce this rule.

When you think you’ve found the correct result for a given problem, you can fill in the result field for the problem, and see if you got it right. If you have, your answer will be added to the list of problems you solved and will count towards your overall score, while bumping up your position on the participant scoreboards. There are several scoreboards, depending on what level you are at, which in turn is determined by the number of problems you have solved so far.

More importantly, solving a problem will gain you access to the respective forum (for that problem), where you’ll be able to see solutions and discussions from other participants, who may have used different languages and solution strategies.

Project Euler is appealing for a variety of reasons. In particular:

• It allows you to explore areas of math that you might not be familiar with. Solving certain problems may require you do a fair amount of research and in turn help you to learn more about certain branches of mathematics;
• If you are a math major who is trying to hone their programming skills, you’ll find Project Euler to be challenging, fun, and an excellent opportunity for improvement. Having to implement several small programs and being able to compare your style with those of other people in the forum, will definitely end up improving your knowledge of many algorithms and your skills as a programmer;
• The site’s one minute rule has an important consequence. The computational complexity of your programs really ends up mattering. While some simple problems can in fact be brute forced, the majority of them require you to write faster algorithms and take advantage of mathematical insight so as to improve the performance of your program. For certain problems a naive approach would take more than a lifetime on current PCs;
• Finally, if you are a programmer who’s scoping out a new programming language, be it Python, Ruby, Scala, Haskell or Erlang, you’ll find a great ally in Project Euler. Having to write hundreds of programs in a given language, will naturally increase your familiarity with that language. And again, comparing your newcomer coding style with those of more experience participants, will no doubt contribute to your advancement within the given language you’re focusing on.

Project Euler truly deserves five stars. I recommend it highly to anyone who is unafraid of writing a few lines of code in order to solve interesting math problems.

The most interesting feature of this initiative is the open letter that prefixes each paper, in which the author illustrates what they believe is the reason behind their work’s rejection and asserts their paper’s case. Scientific papers, like notorious novels, do sometime get undeservedly rejected before their true value is acknowledged. The hard part is distinguishing the hidden gems.

Contrary to what one may have expected though, this edition clearly debunks the notion that Rejecta Mathematica would be a breeding ground for crackpots. The six papers included in this first edition all appear to be interesting and well researched.

Is it the revenge of the rejected? Perhaps, but this unusual publication will still help to enrich our beloved discipline in its own way.

At first glance we’re peering at a service that is able to provide objective information about a wide subset of human knowledge. But the idea behind Wolfram|Alpha is much more ambitious, as demonstrated by the first screencast to have been released (narrated by Stephen Wolfram).

Taking advantage of the popular and powerful mathematical software Wolfram Mathematica, this new engine is able to perform calculations on information requested by its users. Therefore there are three components at work: the ability to correctly interpret queries (in English) from the users, the ability to elaborate on the data source so as to reply with coherent results to somewhat complex questions, and finally the not-so-simple task of maintaining an up-to-date and accurate knowledge base for a very wide spectrum of human knowledge.

This is the theory behind – or at least the ultimate aim of – the service. And as such, unlike what has been reported elsewhere, Wolfram|Alpha should immediately be viewed as an addition to what Google already offers (not as a replacement for it).

As expected, the service is extremely good at mathematical calculations. In this case the only limit is the timeout imposed on each query to exclude those that require far too many resources to complete. An example of a calculation that is executed is “integrate e^-x^2”.

For the record, this service is currently under a fair bit of stress because of the initial curiosity of many worldwide, therefore the calculation of the integral above has shown an error message a couple of times (a tribute to HAL 9000):

All sorts of disciplines are represented by the examples on the site, with a particular focus on scientific and engineering ones. The results are elegantly presented and well organized, often illustrated and shown in table form, particularly when the user is asking for a comparison between different data sets. There are also fun tributes, including those to Monty Python and Douglas Adams. But how well does this system work when we step outside of the predefined examples provided by Stephen Wolfram’s team? The results range from exceptionally good to downright disappointing. Let’s see a few examples.

Let’s compare IBM, Google and Sun. As you can see, the results are definitely excellent. The comparison is almost exclusively numeric in nature, but the answer provided by WA jives with what I was hoping for. The same is true if you look for statistical information on a single large company like Telecom Italia.

Since this is a “computational engine” we can even try to perform a few calculations starting with the data that we found to be available in the comparison tables for the companies above. For example, “employees IBM/Google” will show us the ratio over the last few years between the number of IBM and Google employees (IBM currently hires almost 20 times as many people). We can calculate the revenue for each IBM employee by running “(market cap / employees IBM)”, but the engine fails to compare this parameter between companies: “(market cap / employees IBM) / (market cap / employees Google)”, despite the fact that the data for such a calculation is all there in the knowledge base (and as you can see there was already a revenue/employee row in the initial comparison table).

Moving on to something else, we can try to compare two cities like Toronto and Milan, and obtain very useful details. It’s also possible to calculate only certain attributes by running, for example, “population Toronto/Milan” or “distance Toronto Milan”. In the first case the data was updated to 2004, and is perhaps a little outdated now despite the fact that we are talking about demographic data. But the real problem arises when we try to compare the regions of Ontario and Lombardy. We’d expect to see nice geographical, demographical and economical comparisons between the two, yet instead we get nothing. Wolfram|Alpha will report information about little towns like Unionville (NC), but totally ignores a region like Lombardy or a Canadian province like Ontario (with a population of 13 million people). It would seem that this is a big hole in the knowledge of this service. Sure it’s fairly easy to fix, but it’s a symptomatic weakness nevertheless.

If we consider for a moment searches about famous people, we’ll find an excess of zealousness in trying to be concise and objective. In fact, when searching for Barack Obama, the results are limited to his place and date of birth, and stating that he is a head of state. We may be spoiled by Wikipedia, but a photo or a few more details at least were to be expected. For example, at a bare minimum, it could be indicated that we are reading the page of the 44th President of the United States of America. But Wolfram|Alpha reserves this type of treatment for all famous people, including Stephen Wolfram himself.

In the comparison between “Paul Erdos vs Euler” one would expect a nice parallel between these two great, prolific mathematicians (for example, you might expect comparisons on the number of publications, total number of pages, the most significant discoveries, and so on). Instead, the results are limited to a comparison of births and deaths. This is quite disappointing no matter how you look at it. On a side note however, it must be said that WA is quite good at interpreting misspelled names (e.g., Paul Erddsos).

The Natural Language Processing (NLP) capabilities of Wolfram|Alpha are good enough to use the service without encountering any major frustrations, but it doesn’t appear to be a particular revolution or advancement in the field of artificial intelligence either. It is also clear that we are not dealing with a Google-killer nor a Wikipedia-killer, but rather with an innovative new tool that can be used in addition to the existing ones. That said, elsewhere I mentioned that I personally think that this is a step forward for humanity. It may be a small step, but I stand behind that bold statement.

We are at the forefront of a service that will be useful for research and to anyone with a need for correct data as quickly as possible. The knowledge base will have to grow, some data will have to be updated, and the engine will need to permit more freedom in the kind of calculations that are allowed to be performed. There can be little doubt thought that we are witnesses to the birth of something ambitious that has the potential to accelerate the advancement of our civilization.

An Italian translation of this article is available on Stacktrace.it.

It was “buzz worthy” for sure, and all the attention quickly attracted the interest of numerous scientists from many disciplines. As soon as the mathematicians, and particularly computer scientists, managed to get through its 1000+ pages, the first negative reviews began to pour in. Though, in all fairness, a few scientists had a little too much fun with this book and managed to showcase their comedic abilities by writing some of the most hilarious reviews known to humankind.

In this controversial best-seller, Stephen Wolfram primarily dissects the subject of cellular automata and its relevance to other scientific disciplines, in a systematic manner. It’s a book that covers a lot of ground and is arguably a remarkable piece of writing. Yet, the scientific community greeted the book with a fair dose of criticism.

So what went wrong? The main problem with A New Kind of Science is that it set very high expectations due to its author, title, and the numerous reminders of how important this material is, throughout the book.

The main accusations ranged from the book being called a display of Wolfram’s ego, to having very little “new” content, all the way to the more severe claims of not crediting other people’s work. For example, the idea of the universe as a cellular automaton was first presented by Konrad Zuse, so Wolfram’s “new” idea of a discrete, computable universe was anything but groundbreaking. On top of that, the most remarkable technical achievement revealed in this book was arguably the proof that the rule 110 cellular automaton is Turing complete. While this was conjectured by Wolfram, it was actually proven by his assistant Matthew Cook, who was refrained from publishing his results elsewhere by Wolfram’s lawyers.

It’s important to understand that, while perhaps not accepted as the breakthrough that Wolfram had hoped for, this book – and the methods for studying computational systems illustrated within it – is far from gibberish. Wolfram’s ambitious project failed in the eyes of the community due to the extremely high expectations that were set for this book. When you claim to have something radically new, you must be able to back that claim up in a convincing enough manner or else you’re bound to end up with egg on your face.

To be fair to Wolfram (for the few who are not familiar with his work) NKS is a controversial project, but he was already famous for having created the excellent program Mathematica (whose 7th version was recently released), one of the world’s most complete and advanced mathematical software.

Now Wolfram is at it again. According to his recent announcement, he is about to unleash something called WolframAlpha to the world, which combines both his work with Mathematica and NKS. In Wolfram’s own words:

I had two crucial ingredients: Mathematica and NKS. With Mathematica, I had a symbolic language to represent anything—as well as the algorithmic power to do any kind of computation. And with NKS, I had a paradigm for understanding how all sorts of complexity could arise from simple rules.

The project has been kept on the down-low for the past few years, while some of the brightest mathematicians and engineers employed by Wolfram Research, Inc. worked on it. It’s currently in private beta, but will go live in May of this year. From an initial glance, it would seem to be just another search engine a la Google.com. But is it? Not quite. It’s labeled as a “computational knowledge engine”, whose aim is to compute answers from the human knowledge available on the web. Whereas on Google you can search for strings and the results will be a series of relevant links, WolframAlpha will supposedly be able to parse and “understand” a query that’s inputted in English, and compute an answer based on the extensive knowledge stored in its system (assuming that a univocal answer exists). Conceptually speaking, it’s leaps and bounds more complex to get right than Google, which simply looks for matching strings and orders the results based on the popularity of the given keywords (For more information about the mathematics behind Google, read this book).

According to Nova Spivack, who had a chance to try out WolframAlpha, the service is able to compute factual answers to questions such as “What is the location of Timbuktu?”, “How many protons are in a hydrogen atom?,” “What was the average rainfall in Boston last year?,” “What is the 307th digit of Pi?,” “where is the ISS?” or “When was GOOG worth more than $300?”. This project has the potential to change the world as we know it, just like Google did. Several years ago Altavista was fine for most people’s search needs – or so we thought. It took Google to show us how much better off we could be search-wise, how much we needed Google, and ultimately how inadequate Altavista was. Unlike the case of Google and Altavista though, WolframAlpha would not replace Google, since the two services cover complimentary needs. Having access to a service that’s able to compute answers out of the chaos of the factual information that’s available to man would be a major breakthrough for humanity and computer science. And if an API (Application Programming Interface) were to become available, other developers would be able to tap into that with their applications.

Bold claims, high expectations. You understand why, two months away from experiencing something so potentially revolutionary, there is a lot of hype surrounding this project – but also major skepticism. For many this is A New Kind of Science all over again, especially since natural language processing and “computing knowledge” are extremely ambitious challenges in a realm where many have failed before. Pulling this one off would be a major accomplishment (that would dwarf Wolfram’s past achievements, including Mathematica), and, at long last, it would be the hard earned, practical validation of some of the methods and philosophies expressed in NKS by Wolfram.

I fully expect people to find bugs and have many simple questions, for which we will see bizarre answers. We’ll read blog posts about the whole thing and perhaps have a good laugh. But what interests me the most is whether, as Google did in the past, this new engine will be able to be practical and useful on an everyday level. Bugs are fair play and expected, but what we’re looking for here is a spark of true innovation thanks to the mathematical modelling of human knowledge.

I suspect that this engine will either have us in awe like Mathematica did, or leave us with mixed feelings – if not downright disappointment, like A New Kind of Science did for many. I can’t help but hope for the former, as I wait for my chance to try it out.

I’m hip-deep in the teaching of Mathematics 152, a discrete mathematics course titled “The Mathematics of Symmetry” designed by Paul Bamberg and taught at Harvard University. The course is seminar-style: the students take turns presenting the material in 5-to-15-minute assigned topics during class. The design puts an emphasis on learning to communicate mathematics, and so as I took over the course this semester I considered what I might do to further this goal. I wanted to encourage class participation, discussion and a sense of community, as well as tie the mathematics of the course to the wider experience of the students. So I tried an experiment: an assigned community math blog. The blog is open to and in fact aimed at the layperson public, but also serves as a community discussion board for the students. The 23 students in the class have been assigned 4 posting dates each, spread throughout the semester, which means the blog is updated at least once and sometimes twice a day.

I was inspired by my mother and father, who both assign “journals” to their students in psychology, english, classics and philosophy. The journals, updated regularly by students, are a sort of private diary of reactions to the course. They serve to draw connections with the sudents’ world outside the classroom, and encourage reflection on the material. Blogging, it occurred to me, is a sort of public journaling, and provide some of what my parents sought from the course journals, but this time in the form of a community project.

We’re now approaching the halfway mark for the semester, and I’ve been incredibly impressed with the students’ posts. They range from amusing to historical to musical to magical—even social commentary. There’s been no shortage of topic ideas, although I had hoped there would be more discussion via comments. I hope you’ll take a look at the blog and post some responses, so the students see that they are really reaching an audience out there on the internet: reaching beyond the course itself.

Site: The Math 152 Weblog.

Sure you can Google and with a bit of luck and skill find a helpful page. You may even decide to ask your question in a forum or in a proper newsgroup. However none of these resources are particularly interactive. There are many advantages which derive from a direct “realtime” conversation with other mathematically inclined people who are ready to provide assistance. There is an excellent Math chat that you can join if you want to ask for advice or get help on any mathematical subject. Below you’ll find the details needed to access it from your IRC client:

Server: irc.freenode.net
Channel: #math

If you are new to IRC and don’t have a client, you can use this web based interface to access it. In general though I do advise that you get a good client and familiarize yourself with the system. These are a few free clients that I recommend:

Windows: XChat 2
Mac OS X: Colloquy
Linux: XChat

Be friendly and respectful, don’t expect to be spoon-fed, and you’ll find this an invaluable resource. Occasionally you may even spot me there.

Rejecta Mathematica is a new, open access, online journal that publishes only papers that have been rejected from peer-reviewed journals (or conferences with comparable review standards) in the mathematical sciences. We are currently seeking submissions for our inaugural issue.

Considering that there will be very little editorial control (no peer reviews) of the published articles, it will be interesting to see how it’ll evolve. Will it be full of under-appreciated pearls of genius or will it become the domain of math cranks? I can’t help but hope that it will be a mix of both, if nothing else, for the entertainment value. We’ll see where it goes… and let us know if you intend to submit any papers to them. By the way, if you are shot down by Rejecta Mathematica too, chances are that your paper sucks. But if you really believe that there is merit to it, consider Math-Blog as your Rejecta Rejecta Mathematica.