Articles by Ian Luke Kane

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Basic Calculators and Overkill

September 29, 2008 in Editorial | 2 comments

I really enjoyed reading this post about the physicality of a calculator over at Social Mathematics. The idea of having a simple calculator available to you in your workplace or home office setting is intriguing to me. I must admit that I do enjoy having a basic little calculator at my disposal. I know that I can open up a spreadsheet on my computer to take care of some maths, or that I can simply use my brain, but there’s something about using a calculator for basic arithmetic that is appealing to me. I tend to use the calculator as a sort of extension to my short term memory.

Of course, it’s quite possible to take this idea to the extreme. Many moons ago at a previous job, there was an individual who worked there that was not the sharpest crayon in the box. We’ll call this individual “Slowbe”, and we will assume for this story that Slowbe was a man.

Slowbe was well known for his occasional mental lapses, and I remember hearing a story about him sitting in a meeting with several others, going over various numbers in a spreadsheet. This was a spreadsheet that Slowbe had created himself. As the group was looking at the numbers, Slowbe decided that he wanted to add up several numbers in the spreadsheet. So he left the conference room, walked around 300 meters round trip to his desk, and returned with a pocket calculator.

Now, I’m not sure if you, the reader, are familar with Excel. But when you have a spreadsheet open, and there are several numbers within this spreadsheet which you’d like to add up, doing this requires about 0.4 seconds worth of work. In essence, Excel can act quite easily as a really stinkin’ powerful calculator. Everyone in the meeting (most of whom were IT people), sort of looked at one another with “What just happened?” kind of faces.

This, my friends, is overkill.

Picture by draggin of a Little Professor Calculator created by Texas Instruments in 1976.

iPhone Math Application: Mandelbrot

September 12, 2008 in Software | 2 comments

For those of you interested in the iPhone, a developer named alpheccar has released an application for the device named Mandelbrot, which is a fractal generation tool of the famous Mandelbrot set. It’s a fantastic little free application that allows the user to zoom in on various pieces of the fractal using the typical touch screen commands of the iPhone. Users are also able to discover Julia sets within the fractal, and to switch back and forth between the Mandelbrot and Julia sets.

Many color palettes are included, and the most recent version (1.3.3) allows for palette animation, which creates a great visual effect. The animation is relatively fast, and can be either automatically or manually set. By selecting manual animation, users have the option of setting a slide bar that controls both the speed and accuracy of animation. It’s also possible to save animations to the photo book on the phone.

It’s a great for education and visual entertainment. I really appreciate that this developer took the time to develop such an application and then to distribute it freely. That’s a great gesture. For those of you with an iPhone, a direct link to the application can be found here.

FYI, the photo from this post was taken within the Mandelbrot application and later exported to my computer. Good stuff. Good job, alpheccar!

Funniest Pie Chart EVER

September 12, 2008 in Entertainment | 11 comments

I mean, how can you argue with this?

UPDATE: Thanks to ll for finding attribution for this image. It comes from GraphJam, and the pie chart is here. The author is Jamie Schimley. I’m looking forward to browsing through more hilarity at this link.

ORIGINAL: Thanks to AdArena.net for this one. You should check them out. They’ve got some great stuff posted. I’m not sure where this picture originates, but it certainly made me laugh out loud. If anyone has further attribution on this photo, let me know. It’s truly hilarious.

Nineteen Squared is Three Hundred Sixty One

July 29, 2008 in Mental Math | 7 comments

There are a few math stories that I tell rather often, and this is my favorite of the bunch. Other people seem to enjoy it as well. Given that I’ve never committed the story to writing, I thought that it was time to do so.

During my undergraduate experience, we had an optional two week winter session that allowed for individuals to take simple introductory courses in order to meet various graduation requirements. One particular winter, I signed up to take a psychology 101 course, and the second day of the class we learned about behaviorism and conditioning. I started thinking about various ways that I had conditioned myself, and I realized that there was a single recurring thought that oftentimes pass through my mind with no discernible pattern or regularity. The thought started when I was around 17 years of age, and at the time of this story I think I was 22. It didn’t matter if I was brushing my teeth, out with friends, driving my car, etc. It didn’t matter if I was daydreaming or having a serious conversation. The thought was this:

“Nineteen squared is three hundred sixty one.”

Now, I have no recollection of learning this fact. I never memorized it, and I cannot think of any practical reason that this thought would stick so strongly in my brain somewhere (I learned later that a full size Go board was 19×19, alas). But the thought came to me nonetheless.

During my senior year of undergraduate work, I had a number theory class, where the professor wouldn’t allow us to use calculators for arithmetic problems we would work through. One day, in class, we were working on a rather extensive arithmetic calculation when the professor suddenly hesitated at the blackboard. He turned to us, and asked, “Does anyone know what nineteen squared is?”

Well, I instantaneously answered, “Three hundred sixty one.” Not even one second later. Everyone in the class sort of turned and looked at me with expressions that said something like, “Who in the world keeps track of the square of nineteen?” or “How did he calculate that so fast?”

It was as if the whole of my life had led up to that one moment. If was as if the math gods had prepared me for this solitary moment, where the haunting thought that appeared for no reason whatsoever would ACTUALLY be useful.

Needless to say, when class ended, I was slightly afraid to leave the room. Given that I had obviously fulfilled my purpose in life, I felt that an anvil would imminently fall from the sky and put an end to my misery. But to my great benefit, no anvil fell. And ever since that day, the recurring thought has left my mind. Well, except when I tell this story.

(Photo by rexhammock)

Carnival of Mathematics #37

July 24, 2008 in Logic/Math | 16 comments

Carnival of Mathematics

Welcome to the 37th edition of the Carnival of Mathematics!

In preparation for this edition, I actually managed to secure an exclusive interview with the number 37, and have included a small portion of our conversation below:

Logic Nest (LN): So 37, what have you been up to lately?

37: Oh, not much. I’ve always had a fairly good life given that I’m not only a prime number, but a lucky, irregular, AND unique prime. It’s summertime where I live, so mostly I hang out by the pool with my good friends 16, 21, and 28. We’re in a band together called the Padovan Sequence.

LN: Wow. That’s very interesting. I’ve heard that some people think you’re unlucky though. What do you say about that?

37: That’s totally a fabrication. Just because I’m the number 666 divided by its digits added together [37=666/(6+6+6)] doesn’t mean a thing.

LN: Understandable. I can see the confusion. I’ve heard that there’s a website out there that’s all about you, is that true?

37: Yes, and I must admit that I’m a bit embarrassed about it. Just because I pop up in all sorts of scientific, cultural, and historical situations doesn’t mean that I should have a fansite. I mean, come on now, people…

And it went on like that for a while…

Speaking of prime numbers, let’s kick the carnival off with this article submitted by Jeffrey Shallit from Recursivity about a Rutgers graduate student named Eric Rowland who has proved a new prime-generating formula that’s quite simple. There are some great comments on this post that include various programming implementations of the formula.

Over at Walking Randomly, Mike Croucher has posted his second Integral of the Week involving an exponential function and the square root of pi. The twist on this problem is that he gives you the evaluation and asks you to prove it. In addition, he’s asking readers to exclude the common evaluation method of converting the integral to polar coordinates. He’s taking solutions via the comments on the site. There are already a few proposed solutions, but take some time to think it over before jumping straight to the comments!

“A” presents an editorial on Being Bad at Math posted at It’s the Thought that Counts. This post is about the popular idea that it’s acceptable to confess a total lack of math ability, even though equivalent statements about difficulty in something like one’s native tongue would be seen as embarrassing. This post explores a cultural brushing off of mathematics, and how this idea should no longer be tolerated in the twenty-first century.

Another great lesson in math and culture comes from Barry Leiba, who points out a personal pet peeve of mine in his article That’s a mean median posted at Staring At Empty Pages, namely that people often incorrectly equate “median” with “average”, even at the New York Times. This one should get the blood of you stats people out there boiling!

Given the impending American presidential election, Barry Wright, III presents an educational post entitled Plurality Winner, Condorcet Loser? at fashionablemathematician - mathematics. The contents of the article explores various ideas that Barry is exploring from Donald Saari’s Basic Geometry of Voting, which is a text he is using “both for research purposes and to prepare to TA a class on the mathematics involved in Democracy, voting systems, and the like”. By definition, “a Condorcet winner is one which is ranked higher than every other alternative in a majority of decisions” while a “plurality winner is an alternative which receives more first-place votes than any other alternatives”. As the title implies, there is an interesting case when one can be both a plurality winner and a Condorcet loser. Good stuff.

In The Universe of Discourse : Period Three and Chaos posted at The Universe of Discourse, Mark Dominus gives us some information about Möbius functions, which are of major importance in complex analysis, where they correspond to certain transformations of the Riemann sphere. In particular, he looks at Möbius functions with real coefficients. In this post he talks about functions with a periodic point of order 3 (where f(f(f(x))) = x for some x) in connection to the Sharkovskii’s theorem. Both of these concepts are explained more fully at the link above.

Denise presents Math History on the Internet posted at Let’s play math!. She presents links for some WONDERFUL historical resources available on the web. As she says, “the story of mathematics is the story of interesting people. What a shame it is that our children see only the dry remains of these people’s passion. By learning math history, our students will see how men and women wrestled with concepts, made mistakes, argued with each other, and gradually developed the knowledge we today take for granted.” There’s some really great stuff available at this link for anyone interested in picking up some mathematics history!

In his post Playing with Permutations at Reasonable Deviations, Rod Carvalho proposes a 2-player game. The goal is to find out whether a necessary condition is also sufficient. This game blends Combinatorics with Algebra, and even Algebraic Geometry. It’s an interesting game to consider and builds on a few other posts that Rod has written since January 2008.

Ron Cook from The Endeavour gives us an explanation of Random Inequalities in this three part series. Random inequalities are often used in Bayesian clinical trial methods, and should interest all the stats people who are reading. The first part introduces the reader to the concept of random inequalities, the second part shows how they are analytically evaluated, and the third shows how they are numerically evaluated when analytical evaluation is not possible.

Lastly, Ξ_Heather wants us to think about Burnt Pancakes and Godzilla at her article 4, 6, 8, 10, 12, 14,?.What comes next? posted at 360. As she explains, “the Burnt Pancake problem involves pancakes of different sizes, each with one burnt side, piled up on top of one another.” It’s great content explained in an entertaining manner. FYI, Godzilla evidently wears a chef’s hat when cooking pancakes.

Here are a few more submissions that have come in since I initially published last night:

Alvaro Fernandez presents Top 10 Brain Training Future Trends posted at Sharp Brains.This article discusses the concept of “brain training”, or how we keep our brains fit. This is particularly interesting given that mathematics is commonly perceived as a game for the young, as evidenced by this XKCD comic. Take care of your brains, people!

Are you aware that there is an Encyclopedia of Triangle Centers? David Eppstein is, and he describes another kind of triangle center, different from the ones at the Encyclopedia, here.

Catsynth asks the question, “What do you get when you mix a cat and a Fourier Transform?” in this post. Education and entertainment ensue! The lesson to be learned is simple: be careful of what mathematical transforms you perform on your pets. Obviously.

That’s all for this edition! If you’d like to post any additional articles to this edition of the Carnival, please contact me. I’ll be taking submissions through Sunday evening. Otherwise, stay tuned for the next edition which will be hosted by CatSynth.

The Pirahã People and Numeracy

July 16, 2008 in Linguistics, Logic/Math | No comments

I came across this article today on Science Daily that talks about the Pirahã, which, according to Wikipedia, are “an indigenous hunter-gatherer tribe of Amazon natives, who mainly live on the banks of the Maici River in Brazil”. The Science Daily article introduced me to the fact that this tribe has no concept of precise numbers. While they do use indefinite numerical terms such as “some” and “more”, this group does not seem to have any representation for concepts such as “one” or “two”. As MIT professor Edward Gibson states, “here is a group that does not count. They could learn, but it’s not useful in their culture, so they’ve never picked it up.” Absolutely fascinating. You should certainly check out the two links above, especially the portion in the Science Daily article that describes some of the experiments carried out by Gibson and his MIT team that have further illuminated this portion of the Pirahã culture.

This article intrigued me so much that I dug a bit deeper, and found that Daniel L. Everett, the Chair of Languages, Literatures, and Cultures from Illinois State University, has spent a good portion of his career working with the Pirahã people. He has collaborated in the past with Gibson on various projects in the past. Some info can be found here. There’s a great New Yorker story that was published in April 2007 on Dan here that’s certainly worth a look. Here’s a teaser from this article:

The Pirahã, Everett wrote, have no numbers, no fixed color terms, no perfect tense, no deep memory, no tradition of art or drawing, and no words for “all,” “each,” “every,” “most,” or “few”—terms of quantification believed by some linguists to be among the common building blocks of human cognition.

It’s a very long article, but it paints a beautiful picture of linguistics, cognition, faith, and personal relationships. It’s packed full of great questions. There’s a LOT that’s in these writings I’ve linked to that I haven’t even brought up (including the idea of recursion in linguistics), so I urge you all to read more! There are also some great links for further reading in the Wikipedia article linked to above, including several scholarly papers.

Why Tape Tears

July 16, 2008 in Logic/Math | No comments

A friend let me know quite a while ago about this story presented on NPR’s site entitled “Mathematicians Explain Tape’s Tendency to Tear”. It’s an explanation of a recent Pedro Reis article in the journal Nature Materials describing the annoying tendancy of tape to narrow while unpeeling it from the roll. As the article explains, Reis’ work “could help engineers test thin films for strength and reliability” The audio of the story is also available on the NPR site.

I love this story because I can imagine Pedro first thinking about this problem while unpeeling a roll of tape. I don’t know if the inspiration actually came this way, but its a great mental image that conveys the idea that some of the most interesting problems to solve are right under our noses.

Here’s the abstract of the paper from Dr. Reis’s website:

Thin adhesive ﬁlms have become increasingly important in applications involving packaging, coating or for advertising. Once a ﬁlm is adhered to a substrate, ﬂaps can be detached by tearing and peeling, but they narrow and collapse in pointy shapes. Similar geometries are observed when peeling ultrathin ﬁlms grown or deposited on a solid substrate, or skinning the natural protective cover of a ripe fruit. In this work, we have shown that the detached ﬂaps have perfect triangular shapes with a well-deﬁned vertex angle; this is a signature of the conversion of bending energy into surface energy of fracture and adhesion. In particular, this triangular shape of the tear encodes the mechanical parameters related to these three forms of energy and could form the basis of a quantitative assay for the mechanical characterization of thin adhesive ﬁlms, nanoﬁlms deposited on substrates or fruit skin.

LaTeX Render

July 10, 2008 in LaTeX, Software | 1 comment

I spent some time today reinstalling the LaTex Render plugin for Wordpress. I’ll now be able to put some wonderful looking $\LaTeX$ graphics into the blog. I’d like to thank Steve for spending time both writing and debugging the code. So for the time being here is your mathematical moment of zen, for which I also need to thank Steve:

$\pi = \sum _{k=0} ^{\infty} \dfrac{1}{16^k} \left[\dfrac{4}{8k+1}-\dfrac{2}{8k+4}-\dfrac{1}{8k+5}-\dfrac{1}{8k+6}\right]$

Feel free to also check out the LaTeX in Wordpress post that I wrote a while ago. It should still be mostly up to date. It’s also possible to leave comments using LaTeX syntax as long as they are wrapped in [ tex ] [ / tex ] tags.

Stuff White People Like: Statistics

June 26, 2008 in Statistics | No comments

If you haven’t yet been introduced to the Stuff White People Like blog, you’re in for a treat. There is a new article on statistics that you can read here. Basically, the blog is comprised of witty, yet strangely accurate descriptions of…well…things that white people like. It’s truly hilarious. For instance, take the first line of this article, “White people hate math. If you want to befriend white people, mention “that weird Asian calculus teacher who drew perfect circles” and how much you hated his class…” Awesome.

LOLCats + Math = Hilariously Cute

June 17, 2008 in Entertainment | No comments

I mean, how can you resist this? Hilarious, cute, and math oriented. I hope this brings a smile to your face today!

more cat pictures

Another caption given for this picture in the comments is “Delta Kitteh knows the difference”. Hehehe.

Purity of Mathematics: XKCD

June 13, 2008 in Entertainment | 1 comment

I’m sorry, but this XKCD comic was so wonderful that I simply had to post it. If you don’t read this comic religiously, shame on you. I had to shrink the photo down a bit to make it fit, so feel free to click on the comic to head over to the XKCD page.

The Abacus and Mental Math

April 30, 2008 in Mental Math | No comments

I must confess that I’ve never learned to use an abacus (or a slide rule, for that matter). I came across the following video, and thought that it would act as another great view into the wonderful world of mental math. It’s quite tremendous what the human brain is capable of. Check it out:

Midnight Due Dates

March 23, 2008 in Editorial | 1 comment

I’ve always hated when professors have midnight due dates for assignments. This is because there doesn’t seem to be a grand understanding of what a midnight due date actually means. For instance, consider that I’m taking two courses, and that each of them has an assignment due at midnight on April 1. For Professor X, midnight on April 1 actually corresponds to 00:00:01 on April 1, i.e. the very first moment after midnight on April 1. For Professor Z, midnight on April 1 corresponds to 23:59:59 on April 1, i.e. the very last moment before midnight on April 2. This has bothered me so much that I’ve done a small amount of research, and have come to realize that Professor X is actually correct.

From the official Greenwich Mean Time website, “Every day starts precisely at midnight and A.M. starts immediately after that point in time e.g. 00:00:01 A.M.” This may seem to be obvious, but there’s a remarkable amount of confusion over the issue. Professors who understand this call for due dates of 11:59:59 PM on a particular day. The airline industry always rounds similarly so as to not confuse customers. So if you ever have an assignment due at midnight, it may be in your best interest to check with your professor about what she or he really means!

Here are a few links other than the official one above that will give you more information about noon and midnight:

Look Around You - Maths (Hilarious!)

January 10, 2008 in Entertainment | 2 comments

I hadn’t heard of the “Look Around You” BBC television series before yesterday, but I was nearly on the floor laughing by the end of the following video segment about “Maths”. Read up on the series at their Wikipedia entry. I’m sure the other episodes are equally as hilarious. In my opinion, there are few things better than British humor mixed with math. Enjoy the video!

Point Nine Repeating Equals One

January 9, 2008 in Logic/Math | 1 comment

Let’s be honest…there are certain subjects that a math-ish kind of blog must mention at some point. One of these obligatory topics happens to be the “0.9999999… = 1″ proof. It’s one of those facts that delights the mathematically inclined. It’s sort of like the joke that Grandpa always tells when the family gets together: you know it’s coming, and you know how much pleasure he gets out of relaying the joke, but for goodness sake, this is the 99th time you’ve heard the punchline. At any rate, there is a set of about 15 math facts that people love to talk about simply because they’re all totaling mind-blowing or sound totally nonsensical. I tend to think that the “0.9999999… = 1″ proof belongs in the latter category.

The previous digression leads me to mention the Things of Interest blog, and their absolutely fantastic post on various forms of the “0.9999999… = 1″ proof. You can find that post here. In case one proof doesn’t do it for you, this site offers several, each of which occurs at a various level of mathematical rigor. There will definitely be a proof for you here that you’ll understand.

Mathemagic (Arthur Benjamin at TED)

January 7, 2008 in Mental Math | 2 comments

This is perhaps one of the most amazing videos I’ve every watched on the Internet. I was literally left speechless. I can accomplish certain small feats of mental math, but this is absolutely unbelievable. Arthur Benjamin shows us some inspiring abilities of the human mind. I’m sure he has spent a fair amount of time learning his methods, and that to him his abilities are perfectly normal (in some sense!), but it’s great to watch someone with this talent. I highly recommend watching the video in its entirety.

Do Humans Differ from Machines?

August 15, 2007 in Machine Learning | 1 comment

Slashdot is linking today to an article that considers the implications of the 10 year anniversary of the defeat of chess grandmaster Gary Kasparov by IBM’s Deep Blue computer. The article (here), written by philosopher Daniel Dennett, considers the possible differences, or lack of differences, between humans and machines. I’ve linked to other pieces considered by Daniel Dennett on this blog, and I consider him to be an articulate and fair judge over matters of this type. It is highly worth your time to read this piece and to think it over a bit.

Predicting War

June 14, 2007 in Statistics | 2 comments

A few days ago on Slashdot there was an article about a statistical model that claims to be able to accurately predict the result of a war nearly 4 out of 5 times. Here’s a snippet from the University of Georgia’s press release on Dr. Patricia L. Sullivan’s study: “‘If you know some key variables – like the major objective, the nature of the target, whether there’s going to be another strong state that will intervene on the side of the target and whether you’ll have an ally – you can get a sense of your probability of victory,’ said Sullivan, whose study appears in the June issue of the Journal of Conflict Resolution.” Very interesting. Statistics is a beautiful, and very misunderstood, field. When I hear about claims like this my ears definitely perk up. In general, studies like this propose that particular variables (such as a poor military strategy) are predictive of other events (such as a military victory). There’s obviously a cause/effect chain reflected in this type of idea. And believe it or not, there is a LOT of study in the area of cause/effect relationships. People like Peter Spirtes at Carnegie Mellon University spend a lot of time studying these causal relationships.

So while that claim that a statistical model can predict the outcome of wars should be taken with a grain of salt, everyone should consider the fantastic amount of research (and quality science) that is going into these types of causal models.

Primer on Paradoxes

June 13, 2007 in Logic/Math | 2 comments

I came across this wonderful introduction to several famous paradoxes quite a while ago, but haven’t taken the time to inform you all about it. Daniel Haggard presents a non-technical explanation of five age-old paradoxes that have both delighted and confused humanity. It’s a very accessible read and I recommend it for everyone interested in the strange logical conundrums that surround us. I particularly enjoy his section on Newcomb’s Paradox, which boggles my brain every time I think about it. Honestly, I’m glad that paradoxes exist. I mean, nothing I say is true, right?

The Math of Sudoku

June 11, 2007 in Games | No comments

There’s a wonderful article over at ars technica about the mathematics of sudoku. Agnes M. Herzberg and M. Ram Murty recently wrote an article in which they explain some of the mathematical underpinnings of the popular puzzles. They explore several fundamental questions such as “does a given sudoku have a unique solution?” The link above provides a great introduction to the formal paper, and also includes a link to the PDF version of the paper. Check it out. One of the great tidbits from the article includes the stat that there are over 5.5 BILLION unique sudoku puzzles. Yikes. That’s keep even the most committed addicts busy for a while!

Failing the Turing Test

May 27, 2007 in Machine Learning | 1 comment

I can’t remember exactly how I came across this hilarious article, but I highly suggest checking it out (warning: it contains what some might deem slightly offensive language). It’s all about a guy named Jason who keeps being mistaken for a robot in his instant messaging conversations. It’s highly amusing. For those of you who don’t know, Alan Turing proposed a test in 1950 to gauge whether or not a computer can think. Here’s briefly (and incompletely) how the test goes:

Recruit two humans, one to participate in the test and the other to judge it.
Recruit a computer whose inventor claims it can think.
Put the human judge in a room that contains only a device capable of receiving and sending text messages.
Have the human judge type questions into this device which she would like to ask the human participant and “thinking” computer.
One question at a time, the “thinking” computer and human participant answer these questions by writing out text answers and transmitting them back to the device in the room with the judge.
If the judge cannot determine through the answers to these questions who is the human participant and who is the “thinking” computer, the computer wins and passes the test. In other words, since the computer tricked the judge, it can be said to think.

This is a simplistic version of the test, but it’s definitely the gist of it. In Turing’s paper he guessed that by the year 2000 a computer would have been built that was able to pass the test. He was wrong. Even now, in 2007, a computer hasn’t been built that has consistently passed this test. Interesting, huh? The full text of Turing’s paper which details the Turing test can be read here. The article is called “Computing Machinery and Intelligence”. It’s an accessible paper for anyone with these types of interests. And for more information regarding the Turing test, check out its Wikipedia article here.

Sound Math and the Bible

May 17, 2007 in Logic/Math, Theology | 1 comment

I’m usually quite interested in the attempts of individuals to apply math to the Bible, and the Church Hopping blog has a fun little article about some people that are actually using interesting mathematical principles on the text of the Bible. Check it out.

UPDATE: Daniel from the Logos Blog has contacted me about a great post over on that site. As he said, “Thought you might be interested in today’s Logos Blog post looking at The Top 50 People in the Bible and using the IBM Many Eyes visualization. Cool thing is that anyone can play around with the charts and data…” Check it out here.

$25,000 Turing Machine Prize

May 16, 2007 in Logic/Math | No comments

Wolfram, the makers of the software Mathematica, are offering a $25,000 prize to the first person who can prove whether the above 2, 3 Turing machine is universal. From the website (here):

“A universal Turing machine is powerful enough to emulate any standard computer.

The question is: how simple can the rules for a universal Turing machine be?

Since the 1960s it has been known that there is a universal 7,4 machine. In A New Kind of Science, Stephen Wolfram found a universal 2,5 machine, and suggested that the particular 2,3 machine that is the subject of this prize might be universal.

The prize is for determining whether or not the 2,3 machine is in fact universal.”

What a great idea! I’m really curious to see how long it take for someone to claim the prize. If you’re interested in understanding more about what a Turing machine is, please check out the above links.

TED Talk Suggestions/The Problem of Evil

April 28, 2007 in Theology | No comments

I don’t know how many of you are familiar with the TED conference, but if you don’t know anything about it I suggest checking out their website here. The yearly conference focuses on technology, entertainment, and design, and it defines its mission as “spreading ideas”. You can watch most of the conference talks on this website. There are some serious gems that you should watch. I suggest the following:

Also, I just watched the following presentation made by the Reverend Tom Honey. As the brief synopsis of the talk indicates, “It’s a classic problem in theology: How can the existence of evil be reconciled with a God who is supposed to be all-loving, all-knowing and all-powerful? Many Christian thinkers have attempted answers to this question. In the days following the thousands of personal tragedies recorded during the South Asian tsunami of 2004, Tom Honey pondered those answers and found them wanting. Instead, he penned his own, personal, and sometimes dramatic response to the tsunami. This is a courageous talk for a Church of England vicar to have given. It concludes that certain traditional concepts of God just won’t do … and calls for believers and nonbelievers alike to dig deeper in their quest for truth.”

I was extremely impressed with his thoughts. None of them were particularly new to me, but this talk does provide a good theological redux of the issues involved in the current debate over the problem of evil. It’s an extremely personal and honest appraisal of thought, and I enjoyed listening to it.

Math that Doesn’t Stick

April 27, 2007 in Editorial | No comments

Glue! Memory is a strange thing. I was thinking about this the other night in the context of a few simple math facts that have somehow always alluded my memory. For instance, whenever I have to mentally compute either 7+5 or 8+5 I really have to think about it. I’m not quite sure why. My conjecture is that I was absent from primary school on that particular day, and simply never recovered. I had a similar experience with the lower-case cursive letter “k”.

Does anyone else out there have a similar experience? I’m sure that this phenomenon is fairly universal. Science is teaching us some amazing and unexpected stuff about how the mind works, and so I suppose that it’s not surprising that sometimes “easy” facts escape us.

Windows XP Login Logout Loop (Virus)

April 17, 2007 in Software, Technology | 25 comments

Don't let the virus win! I’m writing this post with the hope that it will be helpful to people who face the same computer predicament that I did a few days ago. Here’s a little bit of background information: Last Tuesday I met John Chol Daau, who is from Sudan. He grew up as one of the Lost Boys of Sudan, forced to leave his home and wander hundreds of miles through Africa to survive. If you don’t know much about this particular humanitarian issue, I suggest spending a small amount of time reading up on it. Anyway, John told me that his PC was experiencing a debilitating virus, and asked if I would look at it. I said that I would. After spending quite a bit of time reading through various website forums, here’s a short description of the problem and its solution:

Problem: The PC (which runs Windows XP with SP2) starts normally. The Windows splash screen appears correctly and then the login prompt correctly loads. You can then enter your user name and password like normal, but as soon as you try to login you are IMMEDIATELY logged back out again. The desktop doesn’t even load. It moves immediately back to the login window where you can then enter your user name and password again. No matter how many times you try to login you always experience this immediate logout. Even if you try to login to the computer in safe mode you still experience the same problem. This problem is documented on Microsoft’s website here.

Solution: I’m sure this behavior can be caused by many different problems, but the most common cause is a virus. If you’re familiar with the Windows registry, this virus changes a few registry key values that makes it impossible to login to your computer. If you’re not familiar with the registry, don’t panic. I’ll post links to a few articles that very clearly explain how to fix this problem. Basically, the virus makes two very simple changes to your computer that render it useless. In order to fix the problem, you have to change these two things back to the way they were while your computer was working.

Easy Fix: The “easy” solution to this problem can be found here. In order to use this fix you have to have your Windows XP install CD. This is the CD that contains the files necessary to install the operating system on your computer. You probably have this disk stashed in a drawer somewhere. You should note that there’s a difference between the Windows XP install CD and the recovery CD that may have shipped with your computer. It’s actually possible that when you bought your computer that it didn’t actually come with a Windows XP install CD. Sometimes computer manufacturers will only ship you a recovery disk, which is altogether different. You need your Windows XP install CD so that you can run an application called the Recovery Console. The link above should provide documentation on how to use the Recovery Console. Unfortunately, this fix didn’t work for John’s computer, but it may work for yours.

Slightly Harder Fix: This fix is the one that ended up working to fix John’s computer. A detailed explanation of this fix can be found here. It requires you to have access to another Windows PC with a CD burner (even if it’s a friend’s computer). You have to download a program called BartPE, which is one of the greatest recovery tools that exists. For this particular problem, BartPE will enable you to quickly change the two settings that the virus messed up. You may need a Windows XP install CD for this method as well. But it may be possible for the program to find what it needs from your friend’s computer without having to have access to this disk.

If you have any questions, please feel free to contact me. The above links should give you the tutorials you need to fix the problem. And if you use a PC you should use a virus protection program! If you don’t, you’re asking for trouble! Good luck!

CERN Goes Boom

April 9, 2007 in Logic/Math, Technology | No comments

Here’s yet another reason why you should make sure to learn basic math. I suppose that this is one way to lose your job…

Clever Hans Summary

March 22, 2007 in Mental Math | 1 comment

A random horse. Is it possible to teach a horse how to do math? Around the year 1900 it looked like the answer was a resounding yes. My friend Trevor suggested that I write something up about Clever Hans, who was a horse capable of performing mathematical feats on par with a young human teenager. Hans could add, subtract, multiply, divide, work with fractions, differentiate musical tones, and understand the German language. At least, Hans could apparently do all of these things. Several other pages have described the Clever Hans case in great detail, so I’ll refer you to them. Please check out one or more of the following links to learn about an interesting phenomenon that has less to do with math and more to do with psychology:

E8 Structure Solved

March 20, 2007 in Logic/Math, Technology | No comments

There was an announcement yesterday that a collaboration of mathematicians from the United States and Europe have mapped the structure of E8, which is a 248-dimensional Lie group. It’s actually even more rich than that, but I think the concept of a Lie group is intense enough for one post. What interests me most about this particular problem is that there was some SERIOUS computer horsepower that went into the solution. As the Yahoo! news story (link) indicates, “While the human genome, which contains all the genetic information of a cell, is less than a gigabyte in size, the result of the E8 calculation, which contains all the information about E8, is 60 gigabytes in size.” Yikes. Amongst other practical applications this result will provide some good information for physicists who study string theory. The reason for this is that structure of E8 is both symmetrical and extremely complex. Please check out the American Institute of Mathematics page on the E8 project here for more information. There’s a lot of great information on their site. So what does the structure of E8 look like? Here’s the picture:

Guinness Beer and Random Sampling?!

March 18, 2007 in Statistics | 1 comment

As if we didn’t already have enough to thank Guinness for, I learned something very interesting today about a connection between this fine beer and statistics. I’m reading the book “Randomness” by Deborah Bennett (Amazon link), which is an introductory text concerning the basics of probability theory and statistics. It’s an accessible read, and contains several nuggets of interesting historical information. I suggest checking it out if you’re interested in this sort of thing. Here’s what Dr. Bennett has to say concerning the Guinness connection to statistics:

“The best-known early demonstration of a random sampling experiment was performed by William Sealy Gosset, a research chemist working for the Arthur Guinness Son and Company Ltd. in Dublin. Gosset was studying the relationship between the quality of Guinness beer and various factors in the beer’s production. The brewery was continually experimenting with soil conditions and grain variety that might produce improvements in crop yield, and Gosset was intent on bringing all the benefits of statistics to the brewery’s agricultural experiments.”

In short, Gosset discovered the t-distribution while working on this problem for Guinness. If you’ve ever taken an introductory course in statistics you’ll probably remember working with t-tests. They’re a way of correctly analyzing small sample sizes, where “small” usually means samples less than size 30. There are some other fun facets of Gosset’s work on this problem, including the fact that he published his findings in papers under the pseudonym of Student. Read Gosset’s Wikipedia page here for other general pieces of information.

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Proof of the Week: Introduction

March 15, 2007 in Logic/Math | No comments

I’ve decided to try to do a weekly feature called “Proof of the Week,” where I’ll explain a mathematical proof that I find particularly illuminating or intriguing. Part of the reason that I write so many math posts on this blog is that I feel that much of the beauty of math is an acquired taste. So my desire is to help serve as a “waiter” who introduces people to some of the fascinating tidbits of the subject. I know a lot of people who run (or roll their eyes) when they hear the word “math.” It brings back terrifying memories of grade school multiplication tests and what not. I don’t blame you. My fourth grade math teacher used to slam a book shut at the end of every minute long mad-dash times test. It scared the bejeus out of me every time. Even so, I still love math.

Most of the proofs I’ll be talking about from week to week won’t be overly intense. I’m sure that many of them will require some general knowledge background, but nothing too academic. My hope is that by explaining some interesting results that you too might see a little bit more of the grandeur contained in this subject. I remember when I took my first proof-based math class during my sophomore year of college. I knew that a lot of rigorous math had to do with proofs, but it wasn’t until my 20th year of life on this planet that I learned what they were really all about. And here’s one of the many revelations I came to rather quickly:

Math is nowhere near as objective as I thought it was growing up. In other words, I always thought that there was a unique answer to every problem. Because of this, I think that many people regard math as some sort of rigid 60 year old person wearing starched clothing who eats the exact same three meals a day and whose house is painted a single shade of grey. To use another image, many people view math problems as some sort of assembly line. You insert a problem at the beginning of the line, perform a bunch of robotic methods, and the answer plops out at the end of the line. If this is your view of math, no wonder you think it’s boring! There’s no art in these images. There’s no movement or color in these pictures.

Math is nowhere near as simple as an assembly line. At least not at its heart. But since most of us grow up learning rote methods to solve problems many of us find the subject to be too tedious or mundane. And I don’t blame you for thinking that. What I WOULD like for you to consider is that you’ve been misled. Like any other academic discipline, math is a growing organism. Hopefully in these “Proofs of the Week” I’ll be able to illuminate some of the beauty that is contained in math. The first of the series will be up in a day or two. Stay tuned!

Happy Pi Day!

March 14, 2007 in Entertainment | No comments

Since today is March 14 (3.14), I’d like to wish everyone a happy pi day. Read more about the wonderful number of pi here and here. I’m sure that some of you have seen the following visual representation of pi before, but this animated gif should give everyone a refresher on exactly what pi is. This informative animation was created by John Reid. [Note: I have this animation set so that it will only loop 6 times in total. If you'd like to see it again please refresh or restart your browser.]

Let’s Go Pens!

March 13, 2007 in Editorial | No comments

As a lifetime resident of Pittsburgh, Pennsylvania, I’d like to take this moment to express my happiness at learning that the Pittsburgh Penguins hockey organization will be staying in the city. Thank you to Mario Lemieux and the countless others who put forth both effort and patience in making this deal a reality. If you have no idea what I’m talking about, read a little bit about the outcome of this long ordeal here. I grew up in a family that followed hockey closely, and I loved watching Lemieux and company win two Stanley Cup championships for the city of Pittsburgh in the early 90s. I’m glad that I’ll be able to see more hockey in this city! Let’s go Pens!

Mandelbrot Fractal Video

March 10, 2007 in Logic/Math | 1 comment

Fractals are beautiful things. If you don’t know what a fractal is, you should read this for a general overview. The most famous fractal (and one of the most mathematically simple) is the Mandelbrot Set, which is named after its discoverer Benoît Mandelbrot. For awhile I’ve wanted to include some sort of video of the Mandelbrot Set “in action”. The following video shows what happens when you “zoom in” on a portion of this fractal. It’s quite interesting. Suffice is it to say that if I ever fall into a bottomless pit, I hope that bottomless pit is like falling into the Mandelbrot Set. At least that way there would be good stuff to look at. There are several other videos out there on the web that show other perspectives of zooming into this particular fractal, so if you like what you see here head over to YouTube or what not and search for some more! The math rock song in the video was written by Jonathan Coulton. If you listen to the lyrics they actually explain a little bit about how to graph this particular fractal. Check out his website here. [Warning: For those of you with sensitive ears, the song that accompanies the video has a few curse words scattered throughout!]

Turing Machine Muffins!

March 9, 2007 in Logic/Math | No comments

Voila! Turing machine muffins! What a delicious idea. If I had used this method while learning about these universal machines I probably would’ve been much happier. What’s a Turing Machine, you ask? Read about them here and here. Check out other pictures of muffin madness here and here. Thanks to Boing Boing for this info!
Mmm...Turing Muffins

Is America Too Damn Religious?

March 8, 2007 in Editorial, Theology | No comments

NPR. Yesterday I listened to a fantastic podcast from the NPR program Intelligence Squared U.S.. From the website, “Intelligence Squared U.S. brings Oxford-style debating to America - one motion, one moderator, three panelists for the motion and against.” The specific program I listened to examines the question, “Is America Too Damn Religious?”, which is a particularly fascinating question to me. The panelists present a scope of different opinions on this issue, and most of the comments are well thought out. Everyone is generally respectful, which is a trait I find important in this type of programming. The reflections are at times theological, political, and practical, which was an interesting mixture to listen to. The entire program can be found here. It’s possible to download a free MP3 version of the program on this page. For those of you with iTunes, you can also find an abbreviated version of the debate by looking for the Intelligence Squared U.S. podcast (which is also free). The entire debate runs about 1.5 hours (the abbreviated podcast is about 1 hour), but I think it’s worth it. For those familiar with the issues surrounding this debate I wouldn’t expect to find too much new information, but what I thought was interesting was the particular representation of viewpoints associated with the panelists.

Using PrintScreen in Boot Camp: MacBook Pro

February 24, 2007 in Mac | 51 comments

I just spent about 20 minutes trying to figure out how to take screen shots in Windows XP on my MacBook Pro (using Boot Camp). I thought I would write up the simplest answer of how to do this, in the sense that you don’t have to install anything or remap your keyboard. Here’s all you have to do:

Under the Start menu, select All Programs, then Accessories, then Accessibility, and then On-Screen Keyboard. When you do this the Windows XP on-screen keyboard will appear on your screen. This on-screen utility includes the PrintScreen button (labelled psc in the same position as where the F13 key should be located. That’s it.

If you want to take a screen shot of only the active window (using the Alt+psc command), make sure that the on-screen keyboard isn’t in the area of the active window, otherwise it will be included in the screen shot.

Anyway, I searched around for far too long to find that little piece of information, so hopefully someone in the same predicament will easily be able to find this post (and the answer)! Since the Apple external keyboards actually include the F13 key (while the MacBook and MacBook Pro built-in keyboards do not), you can use PrintScreen slightly easier with the plugged-in keyboard.

UPDATE: As pointed out by Stuart in the comments below, if you’re using Boot Camp v. 1.3, Fn + F11 now maps to print screen. You can print only the active window with Option + Fn + F11. Thanks for the tip!

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Flatland…The Movie?!

February 3, 2007 in Entertainment, Logic/Math | 1 comment

A bit of geometry. Yes, it seems that Edwin A. Abbott’s wonderfully original novel about the travels of the square named A. Square through one, two, and three dimensional space will soon be brought to video. The website for Flatland: The Movie can be found here. The trailer for the movie is available on the website or on YouTube here. Here’s the synopsis of the movie:

Flatland: The Movie is an animated film inspired by Edwin A. Abbott’s classic novel, Flatland. Set in a world of only two dimensions inhabited by sentient geometrical shapes, the story follows Arthur Square and his ever-curious granddaughter Hex. When a mysterious visitor arrives from Spaceland, Arthur and Hex must come to terms with the truth of the third dimension, risking dire consequences from the evil Circles that have ruled Flatland for a thousand years.

Well, it sounds like there has definitely been some license given to modify the orginal plot of the novel. But I have to say that the plot modifications were immediately forgiven once I found out that Martin Sheen was going to do the voice of A. Square. Who can argue with that? Also, Tony Hale, of the late TV show Arrested Development, will be playing the King of Pointland. At any rate, as the website explains, “The movie will be part of an educational DVD, which will include the original text from Abbott’s book.” Also, it looks like it will be coming out in spring 2007, which isn’t too far away! If you’re dying to get a copy you can sign up on the website for priority access to the DVD. While you’re anxiously awaiting its release, I suggest reading Abbott’s original work. It’s a really quick read and is imaginative and original.

Poincaré Conjecture (The Proof is in the Method)

February 2, 2007 in Logic/Math | No comments

The wondrous sphere. Though I’m a little bit late on this, Science Magazine recently published a great article on the scientific breakthroughs of 2006. Topping the list was the proof of the Poincare Conjecture, which I’ve posted about several times on this blog. You can read their synopsis of the breakthrough proof here. It turns out that from the media’s perspective the drama behind the proof is almost greater than the mathematical result. Basically there was a lot of name calling among some members of the mathematical community concerning who made certain contributions toward the eventual proof. Sad. Apart from the soap opera, the author explains the Poincaré Conjecture in a very accessible way, which should be understandable by anyone who’s interested in reading it. This proof will be a huge deal for mathematics over the coming decades, and should help mathematicians better understand topics such as the “Navier-Stokes equation [of fluid dynamics] and the Einstein equation [of general relativity].”

Perhaps the most interesting thing to note is that the article focuses not only on the result of the problem (the proof itself), but also the methods used to solve the problem. This is an hugely understated part of the mathematical process. I’m of the opinion that when the general populace thinks about math that they are fixated on two things: the problem and the answer. What people tend to overlook is the process of problem solving. In math, there are not always clear-cut methods that explain how to get from point A to point B. A lot of thought is sometimes necessary to figure out how to traverse the path. The Poincaré Conjecture is a monumental achievement not only because of the end result, but also because of the original steps the solvers of the problem (especially Grigori Perelman) took to get there. These steps will be used in other problems; they are not exclusively tied to this one specific problem. Once again, congratulations to Perelman and the other mathematicians who had a hand in making this historic achievement!

Pigs and Vegetarianism

February 1, 2007 in Editorial | 4 comments

The pigs are driving! If you haven’t read this article written by Jeff Tietz for Rolling Stone magazine, I highly suggest reading it. The piece presents the pork industry through the business of Smithfield Foods, which is the largest pork producer in America. The article suggests that one of every four pigs in America is slaughtered by this company. There’s a lot of familiar ground covered for those who know about industrial food production, such as the living conditions of the pigs, unsanitary excrement levels, and animal antibiotic consumption. The article paints a bleak picture, but manages to do so while giving the reader a fair amount of readable statistics. There’s also a historical portion that tells of the story of Joseph Luter III, the chairman of Smithfield Foods. It’s interesting to read the economic implications of the growth of such a large company. The keyword of the entire article is pollution. This specific company (and others like it) exposes nearby people and land to a copious amount of pollution. One statistic relayed is that Smithfield’s largest processing plant “dumps more toxic waste into the nation’s water each year than all but three other industrial facilities in America.” Yikes.

Part of what I like about this writing is its focus on the impact on humanity. Oftentimes these types of stories are sad stories only from the perspective of the animal (which is still true), but doesn’t measure the ways that businesses like Smithfield Foods are adversely influencing human lives. Tietz focuses on the ways that pollution generated on the farms sickens people and keeps them from leading normal lives. Fish who used to live in the areas waterways are now dead, ending the employment of countless local fishermen. Workers in the hog plants die while becoming overwhelmed with the toxic fumes they breathe. People living in the area have contaminated drinking and bath water. The list goes on.

I’ve never been the type of vegetarian who gets angry with omnivores. But articles like this one give me reason to keep up my chosen eating habits. Our current methods of food production are not sustainable. When millions of gallons of pig shit per year are dumped into our rivers, that is NOT sustainable. When pigs are pumped full of drugs that breed antibiotic resistant germs that make people sick, that is NOT sustainable. And on and on. Okay, my rant is over. Read the article, it’s full of information that people should know.

Brief Macbook Pro Review

January 30, 2007 in Mac | No comments

I am the proud owner of a 15″ Macbook Pro. It has 2GB of RAM. My iBook only had 256MB of RAM. Think about it. I must say that I’m extremely pleased so far with the purchase. Even Dreamweaver and MS Office are running really well under Rosetta. I really can’t complain at all about the performance. I’ve also installed Boot Camp, so I’m also running Windows XP. I’ve decided not to worry about either Vista or Office 2007 until the summer. By that point a lot of people and businesses will have moved over. So far I have to say that Windows is running quite well. Once I do some heavier duty stuff I’ll report back on what I’ve found. I thought about using Parallels but decided against it for the time being. I’ll wait until Leopard to see if Apple will enable users to switch between operating systems without having to reboot. If that’s the case I have no reason to buy Parallels (other than to easily play with Linux). Otherwise I’ll have to reconsider. As far as my other software needs are concerned, TeXShop was already available as a universal binary, and TeX installed really easily on the machine. Time will tell. If anyone has any particular questions about what I think about the computer let me know. The self-portrait in this post was taken with the built-in iSight camera.

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Pi Music Experiment

December 26, 2006 in Art, Logic/Math | 1 comment

Music to my Ears. I love when people intentionally mix together mathematics and art, and one of the best examples of this merger that I’ve seen for awhile can be found here. As the site itself says, “this experiment attempts to convert the first 10,000 digits of pi into a musical sequence.” You have the ability to choose several preset music scales, or can choose 10 notes either manually or randomly. It takes a few minutes to play through the sequence, and the sounds are quite transfixing. Even though this meshing of pi and music is somewhat artificial, the result is wonderful. It’s worth checking out.

“Your Age by Eating Out” Explained

December 18, 2006 in Logic/Math | 3 comments

My friends Tim and Megan over at The Franktuary recently sent me a math puzzle and asked me why it worked. I thought it would be a fun little exercise to explain on the site. So here’s the puzzle:

Giant Food! 1. First pick the number of times a week that you would like to go out to eat (more than once but less than 10).
2. Multiply this number by two.
3. Add five.
4. Multiply by fifty.
5. If you have already had your birthday this year add 1757. If you haven’t, add 1756.
6. Subtract the four digit year you were born.

You should have a three digit number.
The first digit of this was your original number (i.e., how many times you want to go out to eat in a week).
The next two numbers are your age.
It’s said that this is the only year (2007) that this will work.

Here’s why this works:

First, you should notice that this puzzle has nothing to do with the number of times that you’d like to eat out every week. You could ask a person to randomly pick a number between 1 and 9 (including 1 and 9) and the puzzle would work out the same way.

If we go through step by step here’s why this works:

1. Let’s call the number that you initially pick x. It’s important to the puzzle that this number be between 1 and 9, including 1 and 9. You cannot pick 10. After we go through the explanation of the problem you should quiz yourself and ask why you cannot pick 10.

2. Multiply your number by 2. Now we have 2x.

3. Add 5 to this number. Now we have 2x + 5.

4. Multiply this number by 50. Now we have (50)(2x + 5). If we multiply this out we have 100x + 250.

5. Let’s assume that you’ve already had your birthday this year. According to the puzzle we should next add 1757 to our number. So we have (100x + 250) + 1757. Again, if we simplify we get 100x + 2007. Notice that the second half of this equation (after the plus sign) is the four digit year. Once we’re done with the puzzle you should ask yourself why we only add 1756 if we haven’t had our birthday yet this year.

6. At this point we’re supposed to subtract our four digit year of birth from our number. Since I was born in 1981 I’ll subtract this number. So our number is now (100x + 2007) - 1981. Simplifying, we get 100x + 26. This final equation is split into two parts. The second part (after the plus sign) will be your age. I am 26 years old. This is correct. The first part of the equation will be a multiple of 100, and will always be a three digit number. In other words, if you consider the three individual digits of this number, it will always be digit x followed by two digits of 0. So the number represented by the first part of this equation will basically be the number x00, if that makes sense. If you add 26 (or whatever your age is) to x00, you’ll always get x26.

The first digit will always be the number of times you’d like to eat out every week and the last two digits will be your age. And there’s the answer!

Here’s another two quiz questions for you:

1. Will this puzzle work for people of ALL ages, or just for specific ages?
2. How could you modify this puzzle so that it works in 2008 (or any other year for that matter)?

Well, hopefully that explains the puzzle. Let me know if you have any questions!

Google and Port Authority in Pittsburgh?

December 6, 2006 in Editorial, Technology | No comments

One of the major gripes I’ve heard people make about the Port Authority transit system in Pittsburgh is that it’s impossible to plan trips. The best method is generally to talk with people who have been using public transportation for longer than you have. Port Authority set up a route search engine through its website about a year ago, but it was downright terrible. I know this system can still be accessed, but I’ve been disillusioned enough by it to forget about its existence. Well, it looks like Google has taken a step in the right direction and created Google Transit, which is a product currently available in 8 U.S. cities, including Pittsburgh. It’s exactly what it sounds like it would be: a way to search public transportation lines in the city of Pittsburgh. I haven’t thoroughly tested it yet, but the interface is basically an extension of Google Maps. Good stuff. I suppose that having Google in Pittsburgh will lead to this type of perk. I’m sure there are some kinks to be worked out, but I’m EXTREMELY happy about this development. I generally think that public transportation is a great way to travel, and one of the hindrances has definitely been route planning, at least for this city. This type of online aid should help with this p