Pi vs. Tau vs. Eta

Published on Thursday, June 28, 2012 in , , , , ,

Michael Hartl's Tau-based unit circleToday is 6/28, making it Tau Day! Tau is a constant equal to 2 * Pi.

It sounds simple, and suggests that Tau shouldn't be a big deal. However, there is an entire movement to have Tau take Pi's place!

Let's start with good old familiar Pi (π). The following video, The Story of Pi, is a great refresher course on a very useful constant:

One of the most useful places for Pi is throughout the unit circle. If you need a refresher, here's a clear and simple Unit Circle tutorial.

With those concepts in mind, let's have Vi Hart explain why Tau (τ) is a better alternative to Pi.

There's plenty of support for Tau. Dr. Kevin Houston has another excellent intro to Tau, and Vi Hart even recently released a Tau Day song.

Michael Hartl, who began the Tau movement, offers a much deeper look at the effects of Tau in a 51-minute lecture, as part of his full Tau Day website. The Tau Manifesto itself is also available on the site for free.

Bob Palais classic article π is Wrong! (PDF link) is also considered to be a must-read for Tau enthusiasts.

Much of that was around for Tau Day 2011, however. This year, however, Tau faces an entirely new challenge!

The new challenger is Eta (η), which is ½π, effectively the inverse of Tau. David Butler's video shows the surprising uses of Eta, some of which rival Tau AND Pi!

That only leaves us π Day being on March 14th (3/14) and τ Day being on June 28 (6/28) is well and good, but when would we celebrate η Day? January 57th? Perhaps February 26th?

Why are we looking solely at the constant for change? How about changing the measuring unit? Since π is defined as a circle's circumference to its diameter, why not measure the circle in diameter lengths instead of radius lengths?

We could call these units “diameterians”, which would be equal to 2 radians. A complete circle would have π diameterians.

Notice that even from the seemingly simple shape of a circle, we can look at it from so many different ways. That's been part of the fun of putting up this post.

What do you think is the way to go? π? τ? η Diameterians? I'd love to hear your thoughts in the comments!


The Amazing Alan Turing

Published on Sunday, June 24, 2012 in ,

Jon Callas' picture of the Alan Turing statue at Bletchley Park100 years ago this weekend, on July 23, 1912, a remarkable man by the name of Alan Turing was born.

He made several amazing breakthroughs, many of which affect the way we live still today. His life story is not only astounding, but tragic, as well.

If you talk to people who are already familiar with Alan Turing's work, there are two topics that are guaranteed to come up almost immediately.

The first is his work in helping to decode the complex codes used by Germany in World War II. They used a machine called the Enigma. It encoded letters, much like a simple substitution cipher, but done in a way that the letters substituted for any given letter could change throughout the code.

Regular Grey Matters readers will be familiar with James Grime, who regular lectures on the history and use of the Enigma machine. In honor of the 100th anniversary of Alan Turing's birth, he has made the following video:

The other topic that comes to mind from Alan Turing's life is the Turing Test, which is basically a standardized way of determining whether a computer could truly be considered to have intelligence. Ian Watson's video, The Turing Test, explains the concept in more detail:

The common thread through all of his work was a fascination with the human mind. What was its nature? What were the processes and potential? That's why so much of his work was related to computers.

Even though the first electronic computers weren't developed until the 1940s, Alan Turing was already writing about programming machines step by step, and even storing those programs in the mid-1930s! In the “Giant Brains” episode of the documentary The Machine That Changed The World, from about the 42:40 mark onwards, you can learn more about his incredible contributions to computer science.

In his later works, he even focused on biology. If you go back to my Iteration, Feedback, and Change: Chaos Theory post and watch The Secret Life of Chaos video there, from the 3:21 mark to the 15:30 mark, you'll learn about how his work on ectogenesis, specifically dealing with the process of identical cells becoming different tissues, influenced much of modern chaos theory.

If you're interested in more details about his life, you can find 2 excellent feature-length videos about Alan Turing. First, there's The Strange Life and Death of Dr Turing (Part 1 is here and Part 2 is here), a 1992 Horizon documentary. The other is Breaking the Code: Biography of Alan Turing, a dramatized depiction of his life from 1996.

Looking through these resources, you'll find repeated mentions of his suicide by eating a cyanide-laced apple. His mother never accepted the suicide claim, and recent research is beginning to support her beliefs.

On the 100th anniversary of Alan Turing's death, Google created a custom doodle in his honor, based on the machine concept that had originally brought him notoriety among mathematicians. Even though it's no longer posted, you can still find it in the Google Doodle archives here.


Calendar History

Published on Thursday, June 21, 2012 in , ,

DafneCholet's Calendar* photoHere's a simple puzzle: In 2012, October 4 falls on a Thursday. On what day does October 15, 2012 fall?

Work it out: October 15 is 15 - 4, or 11 days later. Put another way, October 15 is a week and 4 days later. What day of the week is 4 days after Thursday? Thursday → Friday → Saturday → Sunday → Monday - So October 15, 2012 must be a Monday! If we check, sure enough, October 15, 2012 falls on a Monday.

Now try a similar puzzle: October 4, 1582 also fell on a Thursday. On which day of the week did October 15, 1582 fall? Using the same logic, October 15, 1582 should also fall on a Monday, right? Wrong!

October 15, 1582 was a Friday. How can that be?

October 15, 1582 was not only a Friday, but it was also the day immediately following October 4, 1582. How did that happen? The History channel gives us a brief answer:

October 4, 1582, in Italy Poland, Portugal, and Spain, was the last day they used the old Julian calendar (named for Julius Caesar). As part of the conversion, the calendar was moved forward 11 days, and the new calendar was dubbed the Gregorian calendar, since it was introduced by Pope Gregory XIII.

That explains how the change occurred, but why was the calendar changed in the first place? The answer takes us back to the Council of Trent, at which the Roman Catholic Church worked out how to deal with Martin Luther and his Protestant followers.

Starting at about 6 minutes into the following video, James Burke explains how the Council voted to liven up the churches, but then faced the problem of knowing the right occasion to honor for any given Sunday.

If you watch the following segment up to about the 3:30 mark, you'll learn about Nicholas Copernicus' proposal of a sun-centered system, and the Roman Catholic Church's surprising response.

The reason there are so many problems with finding a workable calendar in the first place is due to quite a few reasons. One of the biggest challenges is that we have to work out our calendar by observing phenomena in out space solely from our perspective on the Earth. Also, our calendar is trying to sync up events that don't really have anything to do with each other. In the following video, C. G. P. Grey does a wonderful job of explaining the challenges of working out the Gregorian calendar:

As I mentioned earlier, the only countries that actually made the change in 1582 were Italy Poland, Portugal, and Spain. Britain and its American colonies didn't make the change until 1752. Ancestry Magazine's article, Time to Take Note: The 1752 Calendar Change covers many of the unusual results the calendar change had, including double dating and the new Quaker dating.

Back in 1982, New Scientist magazine's Making a firm date article covered a larger history of the development of our modern calendar. The information about the problems faced by seasons not properly aligning with the calendar is easy to laugh at now, but it's easy to understand how it could be confusing.


Still More Quick Snippets

Published on Sunday, June 17, 2012 in , , , ,

Luc Viatour's plasma lamp pictureJune snippets are here! This time around, I'm going to challenge you to learn at least one new thing.

• Let's start with your memory, and a video titled Upgrade Your Memory With Nelson Dellis & Fusion-io. It's one of the better introductions I've found to the Link System, the most basic of all memory systems. You can learn more about the Link System via the other videos in this playlist:

• I've mentioned documentary host James Burke numerous times on this blog, such as my Annotated James Burke post, and several times in my Iteration, Feedback, and Change series of posts. The documentaries mentioned there were made available thanks to YouTube user JamesBurkeWeb.

The JamesBurkeWeb posts are unfortunately, divided up into 10-minute segments, which some viewers can find tricky to play. Just recently, however, YouTube user JamesBurkeConnection has posted the Connections documentaries as full-length videos!

• If you want to learn to square 2-digit numbers in your head, I've posted numerous resources already on the site. But there's nothing wrong with trying out new resources, as they can sometimes provide an element that the others were missing. The June 8th podcast of Math Dude Quick and Dirty Tips presented their approach, titled How to Square Two-Digit Numbers in Your Head. Give it a listen, and try it out!

• Maybe you're struggling with some much more complex math, such as the rules for derivatives. If so, two new posts from BetterExplained's calculus section may be of help. The first one is titled How To Understand Derivatives: The Product, Power & Chain Rules, and the other one is titled How To Understand Derivatives: The Quotient Rule, Exponents, and Logarithms. If you're already struggling with calculus, these posts are very helpful.

Some of these links can get into very detailed discussions, so try at least to learn about one of them, and save the others for later.


US Flag Day

Published on Thursday, June 14, 2012 in , , , , ,

Marlon E's photo of United States flagsIn the United States, June 14th honors the adoption of the United States Flag and the founding of the US Army.

There are numerous facts about the United States that are notoriously challenging to memorize. If you start now, can you memorize any or all of the US facts in this posts by the 4th of July?

Probably the shortest and quickest list of US facts to memorize is the 13 original US colonies. Over at memory joggers, they recently posted a wonderfully imaginative and vivid story that will quickly have you recalling all 13 colonies in the order they joined.

In my recent posts about Joshua Foer's TED lecture and playing card memorization, I talked about the Memory Palace technique. Besides short-term uses like playing cards, it can also be used for long-term memorization. MrDaley.com's post A Memory Palace – Or How to Memorize the U.S. Presidents shows you how to tackle this classic list with surprising ease.

One of the toughest USA-related memory feats, however, is recalling the names of all 50 states. The video below was part of now-defunct promotion for funding of a book teaching mnemonics for US state names.

Unfortunately, it doesn't teach all 50 states, but perhaps you'll take inspiration from this approach, and develop your own mnemonics for the remaining states (Don't forget Alaska and Hawaii):

For even more United States-related memory challenges, check out my Memorize United States of America Facts post over in the Mental Gym!


Fun with Wolfram|Alpha

Published on Sunday, June 10, 2012 in , , , ,

Wolfram|Alpha logoWolfram|Alpha is an amazing site, and I've already talked about it several times on Grey Matters.

In this post, we're going to take a closer look at Wolfram|Alpha, and see just how useful it is.

If you're new to Wolfram|Alpha, go to the link, and try clicking the Random button (lower right) one or more times. Once you've found a search that interests you, such as annual deaths from auto accidents in the Czech Republic, click the equals sign, and watch what happens.

If you've never used Wolfram|Alpha before, you might be expecting search results, but that's not what you get. Instead, Wolfram|Alpha analyzes your request and tries to give you the answer! For example, annual deaths from auto accidents in the Czech Republic results, we not only learn that there were an estimated 1,063 auto-accident-related deaths in 2006, but how many deaths per day is the average, and even a graph showing that such deaths declined significantly from 2003 to 2006.

Wolfram|Alpha's biggest strength is its ability to understand a request and give you the results directly. Initially, Wolfram|Alpha got many bad reviews because people thought of it as a search engine, but the reviews improved dramatically, once people started understanding its true nature. Its own example pages are a good place to start learning about its capabilities.

Sometimes, Wolfram|Alpha's ability to understand plain language seems almost magical, but you need to remember that it's a computer program. Instead of being frustrated that it doesn't seem to give you the results you want, try and meet it halfway by learning how to make your input clearer. The article 7 easy ways to get more from Wolfram Alpha is a great place to start.

You can also find regular uses for it in posts such as 10 Search Terms To Put Wolfram Alpha To Good Use Everyday and 10 tips to harness the hidden potential of Wolfram Alpha. There are plenty of other similar articles on the web, as well.

As with anything you start learning, you'll learn more quickly if you also have fun with it. Sure, you can have Wolfram|Alpha work through formulas for you, but try changing each number in the formula, and see what effect it has on the end result. This way, you'll begin to better understand the formula (Hat tip to 25 Cutting-Edge Wolfram Alpha Tips for Serious Students for this tip). As I point out in my post Memorization VERSUS Understanding?!?, the more you understand, the less you have to memorize.

Speaking of having fun with Wolfram|Alpha, check out the WolframAlphaSecrets site. You'll find a good selection of Wolfram|Alpha's lesser-known abilities here, from the impressive flights overhead (Wolfram|Alpha link - click any flight name for even more details) to the humorous batman equation (Wolfram|Alpha link - you appreciation for this will, of course, depend on your favorite Batman logo).

Once you've got the basics down, you're learning from having fun, and possibly even using it to find needed answers, it's time to learn how to make it more accessible. There's already an amazing array of Wolfram|Alpha apps, plugins and addons just for that purpose. The apps even add handy features such as a complete history of your queries, and the ability to store favorite queries.

Probably the most jaw-dropping way of accessing Wolfram|Alpha is via Apple's Siri. By opening Siri and proceeding your request with the word Wolfram (tip: pronounce it Wolf-rum, not Wolf-ram), you can enter your queries verbally, almost like computer interactions on Star Trek! TAUW features a great article on 10 cool things you can do with Wolfram Alpha and Siri, and even posted a video of Siri and Wolfram in action (YouTube link):

The main limitation with Siri is that it can only be used on the iPhone 4S, at this writing. However, it is reported that Siri may be coming to the third-generation iPad as part of the upgrade to iOS 6.

Are you longing for a futuristic interface to Wolfram|Alpha, but don't have an iPhone 4S? How about handwriting your entry to Wolfram|Alpha? Yep, it's already possible, thanks to the Web Equation interface at VisionObjects.com. The following video shows how it is used, and will work with many touchscreen devices, as it doesn't require any special browser plug-ins:

The arrows at the top left of the Web Equation interface are undo and redo functions, very handy if you make mistakes.

The video creator mentions LaTeX, but doesn't explain much about it. LaTeX is a web standard developed to display math formulas more precisely on the internet. For example, instead of typing “the limit of sin(x) over x, as x approaches 0” and hoping your reader can see how that would usually be written, you can display it just as you would see it on paper, or in Wolfram|Alpha for that matter:

You can use the LaTeX code generated by the Web Equation interface in your a LaTeX editor, such as codecogs.com's online equation editor.

Have you found any uses for Wolfram|Alpha you'd like to share? I'd love to hear about them in the comments!


Free Mathematical Magic

Published on Thursday, June 07, 2012 in , , , , , ,

Ianmacm's number pictureToday, we're going to have a little fun with math.

If you don't believe that math can be fun, then you haven't spent nearly enough time at this blog. To prove it, we'll need to add a little magic.

Don't worry, I'll start off slowly. Let's take a look at some mathematical magic tricks from Magic Roadshow Online Magazine #132 (June 2012 issue). The first one is called Calendar Cards:

With their card on top of the face-down deck, ask the spectator how many weeks are in a year? When they answer '52'... deal five cards face down on the table and then two cards face down on top of the five. Pick up all seven cards as one and drop them back on top of the deck.

Next, ask how many months are in a year? Most folks will reply '12', and you then deal twelve cards face down on the table. As before, pick up all twelve and put them back on top of the deck as one.

"How many days are in a week?" When they answer '7', repeat the process of dealing seven cards face down and then replacing them on top of the deck.

"And lastly, how many parts do we divide the day into..?" The answer is '2' - AM and PM. Deal two cards face down and replace them on the deck as before.

At this point, the spectators card is back on top of the deck. You can now reveal it any way you wish...
Why does this work? Start with the first step, in which 7 cards (5 + 2) are dealt. What happens to the cards? Card 1 becomes card 7, card 2 becomes card 6, and so on. In simpler terms, card x becomes card 8 - x. What does 8 have to do with anything? It's simply one more than the number of cards that were dealt (7, in this case).

At this point, card 1 has become card 7. Next, 12 cards are dealt, so each card x moves to a position equal to 13 - x (Again, 13 being 1 more than the number of cards that were dealt). A quick look at this 13 - x chart reveals that card 7 is now card 6.

Next, 7 cards are dealt again, so we go back to the 8 - x table we used before, and see that the 6th card is now the 2nd card. Dealing 2 cards for AM & PM moves card x to 3 - x (do you see why?), which moves card 2 to being card 1.

This is a fun principle to play around with. For example, what happens to the cards when you spell your first name and then your last name? If they both have the same number of letters, this will just bring the cards back to their original order, but what happens if your first name is longer than your first? Or shorter? How can you get the original top card back to the top?

There are several other good mathematical magic routines on that same page. There's Double Reveal, which takes advantage of the formula ((2x + 5) * 5) + y. Comedy Reveal displays some surprising qualities of multiplying by 33, then multiplying by 3367.

The No Hands Trick: Calculated Digits is another good one with which to experiment. When you have the spectator perform what is effectively the formula ((2x + 2) * 5) - 7, you show what seems to be an amazing prediction. Looking at the table of possible answers, it's much simpler than it first appears. What would you do if you wanted your prediction to be a 4 instead of 3? How about a 7? Play with the formula and see if you can work it out.

One of the best things about the above tricks is that they're all a great way to get students involved in a deeper understanding of algebra. These tricks can help people understand that you can start from what seems to be complex, and find (and use!) the important underlying patterns.

Mathematical tricks don't only have to rely on algebra. For a wider variety of mathematical principles applied to magic tricks, check out Peter McOwan's and Matt Parker's new FREE ebook (PDF), The Manual of Mathematical Magic. There's no sign-ups or any other requirements, you simply go to their site, click the PDF link, and save it to your computer after it loads. The routines are divided by the type of math employed, so you can easily find an appropriate trick for any student.

If you have any favorite mathematical magic routines of your own, I'd love to hear about them in the comments!


Memorize Playing Cards

Published on Sunday, June 03, 2012 in , , , , ,

Shreyas Patil's photo of playing cardsIn last Sunday's post, Joshua Foer spoke about the memory palace technique.

It is an amazingly powerful technique, but not just for memorizing speeches. Today, we'll look at a fun side of the technique - memorizing playing cards!

The first important part of memorizing playing cards is, surprisingly, to forget the playing card side of it for now. First, you're going to build the memory palace in which to house your memories.

I've put together a series of videos that can help you better understand the memory palace technique itself. The videos in the playlist range from short and simple to detailed multi-part videos, so you can choose the type of instruction that's best for you.

Out of all the videos in the playlist, only this clip from the TV show The Mentalist brings up how to use it with playing cards.

Once you have a place for your memories, you need to make the playing cards themselves more vivid. You do this by relating each card to a person, either a person you know directly, or a famous person. Celebrities work well for this, as they're often thought of as icons in some particular field, and the better their iconic status in your mind, the quicker and more effective their image will come to mind.

Before you begin putting these two mental constructs together, make sure you have them solidly in your mind. You want to be able to walk through your memory palace easily, without hesitation. When a card is named, you should instantly be able to picture the associated person, and when a person is named, you should immediately be able to name the associated card.

Ron White recently posted an excellent article about using the memory palace technique to memorize playing cards over at The Art of Manliness' blog. This is a great article and walks you through the entire process in a way that's simple to understand.

Videojug also has an instructional video on memorizing playing cards that is clear and very helpful.

Even if you never enter a competition, and just use this to amaze your friends or win the occasional bar bet, the process of actually learning to memorize playing cards is still valuable. It helps expand your sense of what is possible, and you experience an incredible sense of accomplishment the first time you achieve it!