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!
Published on Sunday, June 24, 2012 in
math,
videos
100 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.
Here'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!
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.
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.
• 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!
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?
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):
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.
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.
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.
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:
Today, 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?
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!
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.
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.
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!
Today is 6/28, making it Tau Day! Tau is a constant equal to 2 * Pi.
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