Quaternions

In honor of St. Patrick’s Day, it seems fitting to commemorate Sir William Rowan Hamilton and his discovery/invention of the quaternions, an extension of the complex numbers. This happened while he was walking with his wife near Dublin, his native city. He described the experience in an 1865 letter to his son, the Reverand Archibald H. Hamilton:

Turning Sequences into Blankets

Suppose you want to make a blanket (or placemat, or wall hanging,…) and you want it to be, you know, mathy. One way is to pick your favorite sequence of positive integers and use that sequence to create the blanket.

“Johnny”, or the interaction of odds and probability

I was recently listening to the album Johnny the Fox, recorded by the second most famous Irish rock band. The opening song on the album, “Johnny”, is a tale of a young man being pursued after having committed a crime. The climax of the song finds him in an alley, surrounded by police, and the song’s narrator describes his potential fate:

Five to one he gets away

That’s the odds I’m gonna give

Five to four they blast him away

Three to one he’s gonna live

But what exactly does that mean? Giving odds of 3 to 1 means that if someone were to bet that Johnny gets away, for each 1 dollar that was wagered, the house would pay winnings of 3 dollars. If I offer 3:1 odds, I must believe that for every 1 dollar winning bet, there will be a matching 3 dollars bet on losing outcomes. Thus 3 to 1 odds corresponds to the judgement that 1 out of every 4 dollars wagered will be on that particular outcome, a probability of 1/4.

In the case of the song “Johnny”, we find that the odds being offered (5:1, 5:4, and 3:1) correspond to the probabilities 1/6, 4/9, and 1/4, which add to (6+16+9)/36 = 31/36. Since these probabilities don’t add to 1, this isn’t a zero-sum game, and someone has the advantage. But who?

When it comes time to pay off the bets, the situation is quite clear. One of the potential outcomes has actually occurred, and the payoff odds (assuming a zero-sum game) will simply be the ratio of the total money wagered divided by the amount wagered on that outcome.

In the case of three possible outcomes, if we assume that the total wagers have been $a, $b, and $c on the three outcomes, and set $s = $a+$b+$c, we’d have odds of

s-a:a

s-b:b

s-c:c

with corresponding probabilities a/s, b/s, c/s, adding to 1.

Alternately, if the house takes a cut of the total wagers as profit (as happens at parimutuel betting facilities, such as race tracks, in the USA), the amount of money available to pay to gamblers isn’t $s, but rather $s-p, where $p is the profit from the race for the house. In that case, the odds offered would be

s-p-a:a

s-p-b:b

s-p-c:a

whose probabilities would be a/(s-p), b/(s-p), and c/(s-p), with sum s/(s-p), a probability that is slightly more than 100%.

In general, we’d expect the sum of the probabilities from the odds to add to more than 1, and it appears that the scenario described by lyricist Phil Lynott in “Johnny” isn’t one he will profit from. Good thing Phil was a musician and not a bookie.

In the first race on March 14 at Bay Meadows Race Track, the actual odds after the race were 21:10, 309:10, 3:5, 31:5, and 9:1; as probabilities, these become 10/31, 10/319, 5/8, 5/36, and 1/10, whose sum is 4335479 / 3560040 = 4335479 / (4335479 – 775439), so in practice they had set aside $755439 out of every $4,335,479 wagered (roughly 18%).

There are several notions of probability lurking here. The bookie setting odds is trying to anticipate how gamblers are going to wager. In theory, gamblers are trying to anticipate the actual likelihood of each outcome occuring, then comparing that to the odds offered, and maximizing their expected value. But the bookies know this, and act accordingly. And from there it gets complex.

Shades of: In theory, theory and practice are the same. In practice, they are quite different. (Paraphrasing Einstein, or Popper, or Yogi Berra, or any one of a number of 20th C philosophers….)

Happy Pi Day!

From Dinosaur Comics last year. (Click for a legible version.)

Lemon Pie Recipe

Yes, it’s March 13 and the countdown to Pi Day has begun! While people gather round and sing Pi Day Songs, you can make Aunt Mattie’s Lemon Pie. Aunt Mattie worked for my grandmother’s family, and was known for her pies: chocolate, butterscotch, and lemon. My grandmother once carefully measured and wrote down the ingredients that Aunt Mattie used, giving the recipe shown here. (Hmmm. It looks like this card could well be that original recipe. I am, perhaps, the messiest cook I know.)

Pi Day Sudoku

In honor of Pi Day, Brainfreeze Puzzles (“we turn coffee into puzzles”) created a Pi Day Sudoku on a 12×12 grid.

The rules are a little different from standard Sudoku, in part because the blocks are jigsaw pieces rather than 3×3, and in part because the first 12 digits of π are used instead of the standard 1-9. Each row, each column, and each colored block (“jigsaw region”) contains the first 12 digits of pi

3 1 4 1 5 9 2 6 5 3 5 8

in some order. In particular, there are two 1s, one 2, two 3s, one 4, three 5s, one 6, no 7s one 8, and one 9.

As a bonus, on Brainfreeze’s Pi Day site there are instructions for how to (possibly) win an autographed book by completing the puzzle. Woo hoo!
The contest is over: you can find the solution here.

The Newsletter and Studying Abroad

Our department‘s Winter Newsletter is here! Which, admittedly, is primarily interesting for people who have been to our department or at least stopped by and asked to borrow a calculator or snag some of the food that often is around, usually right within sight of the sign that says no food is allowed. Today it’s homemade raisin bread left over from a late afternoon Science/Math Division meeting.

Follow up to Song Charts

Remember TwoPi’s post on Song Charts — charts or graphs used to illustrate the title of a song (or sometimes its lyrics)? Wishing there were more? It’s your lucky day!

Mr. Fantasy created some and posted them here.

Samjshah posted about them, and pointed out the similarity to the blog indexed.

The flikr pool is growing rapidly — yeah, there are a few more I recognize!

And finally, in the Comments of the original post Ken Knowles pointed out a whole article in the Village Voice using math charts to analyze the song “This is Why I’m Hot” by Mims. It’s a fabulous read. And if, like me, you’re Hopelessly Out Of Touch with most kinds of music, here’s a 4-minute video to accompany the article:

Only have 2-minutes? Then look at this version instead.

Math Card Tricks: More Counting Cards

Looking for another card trick? This one only requires simple counting and moving cards around but it’s mostly done behind your back so it’s a little tricky (as it were).

Start with a Volunteer From The Audience. Tell The Volunteer to think of a number n from 1-15, and then, while your back is turned, to count down that many cards from the top of the face-down deck and memorize the n^th card. For example, if The Volunteer thinks of the number 3, then the card to be memorized is the 3^rd card from the top. The cards should remain in the same order — The Volunteer can look at the card, but doesn’t move anything around.

More on Daylight Savings Time

Back in November, on the 6th day of existence of this blog (aahh, it seems so long ago!), I wrote a post giving a brief history of Daylight Savings Time. There was a pop quiz at the end: “Do you know which Department controls time laws in the United States?” and I had every intention of answering, but, well, I forgot.

So for all of you who have been waiting four months for the answer, here it is! (Drumroll, please). It’s the Department of Transportation. From “Saving Time, Saving Energy”:

In 1918, the U.S. Congress made the U.S. rail zones official under federal law and gave the responsibility to make any changes to the Interstate Commerce Commission, the only federal transportation regulatory agency at the time. When Congress created the Department of Transportation in 1966, it transferred the responsibility for the time laws to the new department.

Speaking of Daylight Savings, you can find a world-wide overview of DST on this webexibits site. Interestingly, in the US, Canada, and Mexico most of the country observes DST but there is a portion of each country that doesn’t. Likewise, different portions of Antarctica have different rules: Rothera (a British base) does not use DST, but McMurdo and Amundsen-Scott South Pole Station (US bases) do.

For problem-solving fun, see Monday Math Madness!

There are still a few more days to submit solutions to the first ever Monday Math Madness! This biweekly contest is being jointly sponsored by Sol on Wild About Math! and Quan and Daniel of Blinkdagger. The first problem was posed last Monday, and since the winner (of a $10 Amazon gift certificate!) is chosen randomly from the submitted good solutions there’s still time for it to be you — solutions are due Sunday night.

For details and the premiere problem, visit Monday Math Madness is here!.

A Perfect Carnival!

The 28^th Carnival is here! This edition is hosted by Tyler and Foxy’s Scientific and Mathematical Adventure Land and, as they point out, it’s the last time that the Carnival will be a perfect edition until #496.

Tyler and Foxy’s Scientific and Mathematical Adventure Land covers Math, Computer Science, Other Sciences, Religion, Politics, and all sorts of other things. Their post on how many characters are in the n^th row of Pascal’s Triangle is something I’m sorely tempted to give to my own students to explore. The bloggers each have individual blog as well (Tyler at Powerup and Foxy at FoxMaths! 2.0), so there’s lots of neat stuff to look at in addition to the Carnival.

R.I.P. Gary Gygax

E. Gary Gygax, co-creator of Dungeons & Dragons, passed away on March 4 at the age of 69 at his home in Wisconsin. Gygax was at least partly responsible for making the Platonic solids cool, in particular the icosahedron (d20). Order of the Stick has a nice tribute to Gygax, as does Penny Arcade. We nerds of the world will miss him.

Update (3/7):  xkcd has a tribute up now.

Language and Algebra

A couple nights ago at dinner our 7-year old said that water was distasteful. Initially this surprised me, because the lad likes water, but it turns out that he meant that it has no taste. It’s full of no taste [(distaste)-full, as it were, with his interpretation of dis as “no”] as opposed to being the opposite of tasty. While this conversation brought to mind this recent xkcd comic, it also made me think about the associativity of language. English is not associative: “the happy teacher’s cat” could be “the (happy teacher)’s cat” or “the happy (teacher’s cat)”, depending on just who is happy.

And English is not commutative* (“Sylvia overcame Calculus” versus “Calculus overcame Sylvia”), although with all its declensions I think that Latin might well be.

What about the distributive law? Dr. Seuss’s “Green Eggs and Ham” is illustrated with both “green eggs” and “green ham”. But for a distributive law to really apply we’d really need two operations, and I’m not completely sure what the first one was. Nor am I sure how much farther I can take this line of thought or if I’ve already pushed it way past where it should go.

*Yoda gets a free pass on this rule.

Conformal Mappings – Now with Holes!

Professor Darren Crowdy, from Imperial College, London, has solved a 140-year-old problem in mathematics by discovering Schwarz-Christoffel mapping formulas for multiply connected regions of the complex plane. Schwarz-Christoffel mappings are conformal mappings (WP, MW) from the upper-half plane to a polygon. They are used extensively in potential theory, fluid dynamics, and aircraft design, but until recently could only be applied to simply connected regions (and some doubly connected regions).

According to Crowdy (from the Times article above):

This formula is an essential piece of mathematical kit which is used the world over… Now, with my additions to it, it can be used in far more complex scenarios. In industry, for example, this mapping tool was previously inadequate if a piece of metal or other material was not uniform – for instance, if it contained parts of a different material, or had holes. With my extensions to this formula, you can take account of these differences and map them on to a simple disk shape for analysis in the same way as you can with less complex shapes without any of the holes.

There is a longer discussion of the mathematics behind the Schwarz-Christoffel formula, including some of the history of its solution, at SIAM. For some great visualizations of conformal mappings, see the Virtual Math Museum, as well as the link above.