Plenty of cell phones can access the internet, but the iPhone and the iPod Touch have already seemed to capture the attention of the phone and wi-fi public. What are the options for those who want to tease their brain with an iPhone or iPod touch?

I've already started work on a new section for the Mental Gym, called the iPhone Mental Gym. At this writing there are only two items in it, Albert Omoss' Sudoku, and my own original Knight's Tour. This version is a special adaption of the Knight's Tour at the Mental Gym, where you can also learn how to do the Knight's Tour.

If you don't have an iPhone, you can try them out online. If you want a fuller iPhone/iPod Touch experience, you can also try out TestiPhone. Just type http://www.testiphone.com/?url= and the web address in your browser's address bar, and try out your own online iPhone! For example, typing http://www.testiphone.com/?url=http://members.cox.net/astonishment/iphone/iknightquiz.html, and you'll get this as the result.

These two aren't the only way to challenge your brain, of course. Want to perform the 40 30s 4 15 feat? First, learn how to do it, and then use stephan.com's iFifteen.

How about a calculator for feats like the root extraction feat? Since numbers for many mathematical feats can become large quickly, I recommend Garth Minette's Belfry SciCalc (TestiPhone version).

I don't want to completely rob you of the joy of discovery, however. Start by checking out AppSafari and iPhone Widget List, and do a little exploring.

There's only one more day to go until the season premiere of Numb3rs!

In my previous entry, I briefly referred to Charlievision, where Charlie Eppes (David Krumholtz) visualizes a mathematical approach to a problem. Wouldn't it be nice to understand these mathematical concepts in more detail?

CBS thinks so, and has teamed up with Texas Instruments to create the We All Use Math Everyday site. This site features exercises for individuals, as well as students and teachers, that directly relate to each individual episode.

If you prefer the principles organized by name, instead of by episode, check out Redhawke's All The Math section. If you wanted to find out more about, say, Pi, you would scroll down to the P section, and click Pi. Once there, you get a brief description, and links that help you learn more about the concept.

Those previous two site I've actually mentioned before. There are two great references that have just become available recently that should be required reading for any Numb3rs fan.

Back in August, the new book Numbers behind Numb3rs: Solving Crime with Mathematics became available. NPR's Math Guy, Keith Devlin, got together with Gary Lorden, who is the principal math advisor for Numb3rs, and described math used in the first 3 seasons of the show. This is a great reference to help quell your curiosity about concepts you've seen on the show, but may not have fully understood.

WARNING: If you don't want to know anything more than the commercials are telling you about the show, stop reading now! The following links may contain spoilers and/or plot details.

The newest Numb3rs reference has shown up just this week! Wolfram Research best known for their MathWorld site and their Mathematica software, has just announced their The Math Behind Numb3rs site! This is the richest resource yet for Numb3rs mathematics!

Already they have an explanation of the math used in Trust Metric, the season premiere episode. In this episode, Charlie will be using set covering deployment, the illumination problem and the Mathematics of Friendship (you'll see!). Not only are these concepts discussed on the page itself and the MathWorld site, but you can learn more about them via free interactive Wolfram Demonstration downloads.

Probably the most fun part about this site is that you may be able to learn about the math before the episode airs. Keep this site as your secret source, and you can astound your friends and family with details about the principles during the show. They'll think you're a genius!

It's almost time for the 4th season of Numb3rs! If you're not familiar with the show already, check out my original Numb3rs post, as well as the whole Numb3rs category to get a better idea of the show.

The best way to get an idea of the show, of course, is to watch it. Season 1 (13 episodes) and Season 2 (24 episodes) are already on DVD. Season 3 (24 episodes) will be released this Tuesday, along with a Three Season DVD boxed set!

If you have an iPod, you can already purchase Season 1 (by episode only), Season 2 and Season 3 from the iTunes store. Update (Oct 1): The subscription to season 4 is now available, as well. Even if you're not sure about purchasing whole seasons yet, buying just the season 3 episodes The Mole and The Janus List, a total of only $3.98, is a great way to get ready for the season premiere.

Since it's been almost 4 full months since the season-ending cliffhanger aired, and was just re-aired last Friday, I have no qualms about discussing it. In the 4th episode of season 3, The Mole, Don's (Rob Morrow) FBI team is investigating the death of a Chinese interpreter. The murderer turns out to be Dwayne Carter, who saved agent Colby Granger's (Dylan Bruno) life in Afghanistan, when they both served in the Army's Special Forces.

This doesn't seem important until the season-ending cliffhanger. In The Janus List, a list revealed by a poisoned British secret agent reveals that Colby Granger is a double agent for the Chinese government. The last thing you see in that episode is both Carter and Granger being taken away to be tried for treason.

There are other questions that will hopefully be answered this season. Will Megan (Diane Farr) decide to stay on with the FBI after her DOJ assignment? Since Larry (Peter MacNicol) has a new outlook on life after being on the space shuttle, what will happen when he sees Megan for the first time? While we've seen "Charlievision", like in this scene, the The Janus List showed us the first instance of "Donvision", when he realizes that the bombs on the bridge look like a G major scale. Will Don start understanding Charlie (David Krumholtz) better in the new season?

The new episode will air on Friday, September 28th. In this clip, director Tony Scott talks about the making of the season premiere:

So, what can we look for in the new episode? I'll let the season 4 premiere trailer speak for itself:

The classic Day of the Week For Any Date feat isn't hard with practice, but there's no shame in wanting an easier approach.

To make it easier, I've just released the 2008 version of my Day For Any Date Calendar! Besides being an attractive and functional 2008 calendar, it comes with an instruction booklet. The booklet details how to use the calendar to determine the day of the week for any date without looking at the month in question!

Not only has the calendar been updated and re-designed for 2008, but I've also reduced the price, The 2006 and 2007 versions went for $31.95. The 2008 version has been reduced to only $24.95!

The calendar uses the same basic idea as Zeller's Congruence, except that the math has been greatly simplified for you, and ingeniously hidden within the calendar design itself.

For a more detailed look at the calendar itself, click here, and click on the months to see their respective pictures. You can also see this by going to the calendar page and clicking on the View Larger Products: [1] link.

Also, don't forget to take a look around the rest of the store. Remember, anything you buy there helps keep Grey Matters running!

Believe it or not, I haven't covered all the mathematical visualization sites on the web yet.

When you talk about hundreds and thousands of anything, it's somewhat easy to visualize. When you start talking about millions, billions and beyond, then it becomes harder to relate and understand. We'll start with an introduction to the concept by the Fat Boys:

How about visualizing time? How long was a million seconds, minutes or hours ago? How about a billion? How about a trillion? Over at How much is a Million? A Billion?, you can get an idea of just how long ago that was. Keep in mind that this site was posted in 2000, so at this writing, one million hours ago would be the year 1892, not 1885.

If you're curious about going all the way up to 1 nonillion (1,000,000,000,000,000,000,000,000,000,000) and beyond, this site uses paper folding to get the idea across. Just to keep you grounded, most pieces of paper can't be folded in half more than 7 times, although Britney Gallivan has folded a piece of paper in half 12 times.

Usually when you're talking about millions, billions and beyond, you're talking about money. Thanks to the MegaPenny Project, you can see how big various amounts up to 1 quintillion pennies appear. If cows are more your thing, you can see a similar visualization at the MegaMoo project.

Now that we can more easily visualize millions of dollars, how about actually having millions of dollars to spend? That would be nice, so why not try the lottery? Well, this lottery simulation (Java required) can show you why not very quickly. It simulate your average lottery, in which you pick 6 different numbers out of 50, but it goes farther than that. It assumes that you play $1 each week, and keeps track not only of your wins and losses, but what you would have if you invested that dollar in a savings account at 5% interest. After just a year or two, the difference is already shocking. Let it run for 20 years, and you'll be in awe of the amounts involved (but you can visualize them in pennies!).

Back in the original Visualizing Pi post, I mentioned various versions of the Powers of 10 concept. The most detailed version I found of this would have to be Nikon's Universcale page (Flash required). Like the previous versions, it zooms in and out, but there is more detailed descriptions as you cross the various size boundaries. You can stop and examine each item in detail, as well. There's not much instruction required to use this site. Just run your mouse over just about anything on the screen, drag to change speed and direction, and click anything that is highlighted.

For my millions and billions of readers (hey, I can dream, right?), this should be plenty to explore until next time. My next post should be up about 350,000 seconds after this one.

I'm not the only one who has been thinking about visualizing mathematical concepts!

At roughly the same time I was writing my previous post, Math Blog was posting thought-provoking mathematical videos. The videos that were chosen are very effective at making hard-to-imagine concepts easier to understand. Do you think you can understand 10 dimensions, or how to turn a sphere inside out with poking a hole in it?

The idea of making mathematical and scientific concepts easy to visualize isn't just a passing fancy, either. The National Science Foundation has hosted the Science and Engineering Visualization Challenge since 2003. Interesting past winners include Still Life: Five Glass Surfaces on a Tabletop (seen in the graphic above) and Flight Patterns.

Planetary Motion from Eudoxus to Copernicus, which only made honorable mention, is especially interesting. It's a visualization over time of how humanity pictured the universe. Planetary Motion reminds me a little of James Burke's documentaries (Connections and The Day The Universe Changed).

Speaking of TV shows, there aren't that many shows today, other than Numb3rs, that regularly feature mathematical visualizations. I remember a PBS show called Square One TV that used to teach an amazing variety of mathematical skits to hit the point home. As an example, how do you make the summation of positive and negative numbers visual? Here's Square One's approach:

You might think it would be hard to visualize would be Einstein's Theory of Relativity. If that's true, the people who created Al's Relativistic Adventures have proved that wrong. This is an interactive Flash demonstration that takes you step by step through the numerous principles involved in relativity, and even quizzes you to make sure you're following along.

It's a shame that math isn't taught to both the left brain and the right brain. As a final thought on this matter, I suggest reading If We Taught English the Way We Teach Math.

There are plenty of Pi posts here on Grey Matters. If you're really into it, you can even learn 400 digits of Pi!

Learning the digits are one thing, though. How do we move Pi from the left brain's logical side to the right brain's “big picture” side? In other words, how do we go from memorization to understanding?

Back in January, I posted John Reid's animation, which effectively depicts that Pi is the circumference of a circle divided by its diameter:

If you download the free Mathematica Player, you can download an interactive version for your system. The Mathematica version breaks up the pattern along the radius instead of the diameter, so you may want to compare both versions.

The great thing about this is that, if you took the time, you could actually carry this out as an experiment. It's there that you would run into the next problem. Sure, you might be able to determine Pi to 1 or maybe 2 digits this way, but the question will pop up as to how all the other digits are determined.

To answer this question, NOVA has provided an article on Archimedes' method for approximating Pi. In the article, a visual demonstration is provided. If you have Flash installed, you can try out the interactive version. If not, you can still check out the non-interactive version on the site, or download the interactive Mathematica player version.

Since we have all these digits, what do they mean in practical terms? To find out, let's try a fun thought experiment. We'll work with the biggest circle we can, and measure it to the smallest detail we can.

Let's say we're working with a circle that is approximately the size of the entire universe, and whose circumference we know. If we wanted to know the distance across this circle to the nearest proton, how many digits of Pi would you need to use?

To help visualize the scale we're talking about, watch the classic 1977 educational film, Powers of Ten:



Starting from the farthest point out, each power of 10 closer naturally requires another power of 10 (another decimal point) in Pi for accuracy to that level. To get down to the proton level, we have to journey 40 steps (from 10²⁴ meters down to 10^-15 meters). This means we would need “only” the first 40 digits of Pi for the circle in this thought experiment!

You can also find Java and Flash versions online. The Powers of Ten is such a visually powerful way to encapsulate the scale of the universe, it has been adapted numerous times for films and TV shows, such as Contact, Men In Black and The Simpsons.

If 40 digits of Pi are the most that will ever be needed, why do we keep on calculating digits? Now you're trying to understand humanity, which is much harder to understand than Pi.

Yes, it's time once again for the Carnival of Mathematics! The 16th Carnival of Mathematics is now up over at the Learning Computation blog.

This is one of the richest carnivals yet! It covers everything from teaching math in school to mathematicians themselves, and more than just a dash of computer science.

At the end, Kurt apologizes for being late, but really doesn't need to apologize. Not long ago, we had several early Carnival of Mathematics posts, so this is just a reversion to the mean.

Look for the next carnival over at MathNotations on September 21st!

Have you been practicing the Day of the Week For Any Date feat? Here's a quick test: On what day of the week does September 6th, 2008 fall?

If you took the time to work through that, then you should have come up with Saturday. Those of you who have performed this for friends, family or even total strangers, know that without any sort of proof to back it up, this feat falls flat.

My first solution to this problem was to provide two PDF files, as part of my Train Your Brain and Entertain CD-ROM, that constitute an 8400-year perpetual calendar. From the comments I've received, many suggest that this calendar is great in any place you might have other books, such as home or school. However, it's not handy to carry when you're just out and about.

That's why I developed a far more portable version! The new version is a Perpetual Calendar for your iPod, iPhone, PDA or cell-phone!

The calendar comes as a series of 176 .JPG images, with a resolution of 360 by 270. For your iPod or iPhone, you load them in as you would any other photos, via iTunes. If you have another cell phone or PDA in which you can display .JPG images, you can load these images as per your particular unit's instructions.

Once you have them loaded in, and accessible in numerical order, the perpetual calendar itself is easy to use. The first 8 pages are index pages, each of which list 50 years and a letter. Using our earlier September 6th, 2008 example, you would scroll to the 2000 - 2049 index to find the letter corresponding to 2008 (images below are at full size):

The next 168 images are images of individual months, each with a letter next to it. Since 2008 corresponds to J, and we're looking for September, we scroll through the images to find the one labeled September J:


This way, you can show that September 6, 2008, really does fall on a Saturday!

I have two important tips for using these images effectively. First, load the images in the numerical order in which they're named. Properly ordered, you should see the 8 index pages first, running in order from 1800 - 1849 to 2050 - 2099, followed by the calendar pages running from January A, then February A and so on up through December N. This way, you can find the page you need quickly and effectively.

Second, let the person who gave you the date see the entire process, so that they know that the calendar is accurate. If they want to see another date, such as today, for which they do know the day of the week, you can show them that date, as well.

Did I scare you with the title phrase? It's understandable. It brings up images of math class pressure during tests.

Could a math professor get something so simple as the Pythagorean Theorem wrong? In the math professor's class, where they are in control, probably not. However, if we replace the pressure of a test with the pressure of speaking on the radio, then it has actually happened. Peter Alfeld, of the Department of Mathematics at the University of Utah, admits to making a simple Pythagorean mistake on the radio.

The story may sound embarrassing, but it is nice to see people own up to their mistakes. By admitting to this mistake, however, Professor Alfeld does risk joining other notable people who had problems with the Pythagorean Theorem.

Probably the most famous instance of this is when the scarecrow from The Wizard of Oz got the Pythagorean Theorem wrong (or, more accurately, the script writers). This mistake was later parodied by no less than Homer Simpson, but at least someone was there to correct him:

Just so everyone is on the same page, the Pythagorean Theorem applies only to right triangles. It states that, given two legs of a right triangle, whose lengths are a and b, with a hypotenuse of length c, then a² + b² = c².

The mistakes the fictional characters above was applying the formula to isosceles triangles (although the formula does hold true for isosceles right triangles). Professor Alfeld's mistake was different. The ladder against a wall did form a right triangle, but he neglected to realize that the ladder itself should have been the hypotenuse.

When students are tested on the Pythagorean Theorem, the questions usually provide the length of two of the sides, one of which may or may not be the hypotenuse, and ask you to find the length of the remaining side. However, what if you were only given the length of one of the legs (not the hypotenuse), and asked to find the lengths of the other two sides?

At first, this would sound impossible. If you're given a length of one leg as 8 inches, wouldn't there be too many possibilities for the other sides? Common examples would include 6² (36) + 8² (64) = 10² (100) and 8² (64) + 15² (225) = 17² (289).

There is a surprisingly simple way, given just the length of one leg (again, not the hypotenuse) to generate all the Pythagorean triples that involve only whole numbers or those ending in .5. I'll leave you with this slideshow that explains it quite well: