Archive for Math

Happy π Day from Nine Kinds of Pie

π Day is upon us once again!  Here are 3.14 pieces of Pi (as it were).

1. The π sculpture in Seattle, Washington.

Pi sculpture, Seattle, Washington

Photo by Niall Kennedy.

2. π to 1000 decimal places

3.1415926535897932384626433832795028841971693993751058209749445923078

16406286208998628034825342117067982148086513282306647093844609550582

23172535940812848111745028410270193852110555964462294895493038196442

88109756659334461284756482337867831652712019091456485669234603486104

54326648213393607260249141273724587006606315588174881520920962829254

09171536436789259036001133053054882046652138414695194151160943305727

03657595919530921861173819326117931051185480744623799627495673518857

52724891227938183011949129833673362440656643086021394946395224737190

70217986094370277053921717629317675238467481846766940513200056812714

52635608277857713427577896091736371787214684409012249534301465495853

71050792279689258923542019956112129021960864034418159813629774771309

96051870721134999999837297804995105973173281609631859502445945534690

83026425223082533446850352619311881710100031378387528865875332083814

20617177669147303598253490428755468731159562863882353787593751957781

8577805321712268066130019278766111959092164201989

This website lists π out to its one millionth digit. So does this one.

3. My childhood obsession with this number.

I have never cared a great deal about the ratio of a circle’s circumference to its diameter.  But, when I was a child, I did memorize π out to its tenth decimal point. I also wrote it out on a piece of paper out to its hundredth decimal point — an item I discovered a couple of years ago when cleaning out my childhood archive (then stored in boxes in my mother’s basement). Why write it out to the hundredth digit?  Perhaps I planned to memorize the number all the way out to that decimal?

Most likely, its novelty captivated me.  I loved paging through the Guinness Book of World Records, and even bought a new edition of the book each year.  There, you could learn about Chang and Eng Bunker, or the longest word (that isn’t a technical term or proper noun): floccinaucinihilipilification, which (in case you were wondering) is the act of estimating something worthless.  π is not worthless, but my attraction to it derives more from its oddity than its utility.

The sarcastic among you might observe that an affection for arcane information is a fairly good description of my profession — college professor. And you wouldn’t be entirely wrong. But you would be belittling the fact that knowledge is worth pursuing for its own sake. Knowledge has no obligation to be useful. Or, to put this another way, do not be discouraged if its purpose may initially elude you.  You never know when knowledge might come in handy, or what ideas it might generate.

0.14.  Two Previous Posts on π

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Nine More Kinds of Pi: Happy Pi Day 2015!

Since this blog takes its name from an artist who wrote about pie and painted π (see last year’s post), I try to offer a little tribute to this beloved irrational number each Pi Day. Today, at 9:26:53 am and pm (twice!), the date will spell out the first ten digits of the number. Well, it will if you write the date in the American style: 3/14/15, 9:26:53.

Last year, I managed to offer nine kinds of π / pie. And this year,… here’s another nine!

1. Andrew Huang’s “Pi Mnemonic Song” (2013)

I’ve featured π music on this blog before (see here and here), but only just discovered this piece, thanks to Paul DeGeorge, who shared it on Facebook. Yes, there’s a minor glitch late in the video. As Huang acknowledges on the song’s YouTube page, “there was an editing error at 1:18 with a couple of the numbers, but the song does all truly line up with Pi.”

2. “Joy of Bubbles,” π art from Cristian Vasile (2014)

Cristian Vasile, "Joy of Bubbles"

Vasile describes the piece as the “First few thousands digits of pi displayed over radial paths. Small size dots.” In an article from last year’s Pi Day, the Guardian shares art from Vasile and others.

3. π in superballs (2015)

Pi in superballs (Photo by Philip Nel)

Yes, these are my superballs, on my bathroom floor. Why do you ask? Also, technically, they’re not superballs (a brand name) — they’re just bouncy rubber balls. But they do spell π out to the 11th digit.

4. Sandra Boynton’s Pig & Pi (2015)

Sandra Boynton, Pi

Every day, Boynton posts something funny via Facebook and Twitter — here’s what she has for today. (I hope the pig finds some pie soon….)

5. Stephen Doyle’s global Pi (2015)

Stephen Doyle's global Pi

Doyle’s photographic portrait cleverly places the number within a circle, half of a globe — presumably to signal global Pi Day. This piece of art accompanied Manil Suri’s “Don’t Expect Math to Make Sense” in the New York Times, 14 Mar. 2015.

6. Technische Universität Berlin’s Pi mosaic

Technische Universität Berlin's Pi mosaic

Located outside the Maths Department at the Technical University of Berlin.

7. Tim Habersack’s π in color (click for larger image)

Tim Habersack's Pi in color

Click on the above to see the larger image (it takes Pi out to 786,432 digits). As Habersack explains, he “Wrote a php script which reads in the characters of pi one at a time. I assigned a single color to each character, 0 = white, 1 = teal, 2 = blue, etc. I create an image, and starting at position 0,0, I draw in my pi calculation one pixel at a time.”

8. Michael Albert’s Pi collage

Michael Albert's Pi collage

From PBS’s “6 Ways to Celebrate Pi Day.”

9. The Barbury Castle Crop Circle (allegedly)

Barbury Castle crop circle

Located in Wiltshire, England, this crop circle is alleged to be a graphic representation of the first ten digits of Pi. Full explanation here, though I’m a bit skeptical of it, myself.  Whether or not it spells out Pi, it is an impressive crop circle.

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Nine Kinds of Pie

Happy Pi Day!  In recognition of 3.14 (today) and this blog’s Pi pie avatar (logo?), here are Nine Kinds of Pie (loosely defined).

Pi1. The ratio of a circle’s circumference to its diameter: 3.1415926535.  When I was a kid, I memorized the number out to its tenth decimal point. On a long strip of paper, I also wrote the number out to about 100 decimal points. Perhaps I thought that learning this irrational number would grant me some greater insight. Or, possibly, I was intrigued by the fact that this simple ratio would be represented by such an unwieldy and unending number. I’m not sure. But I still have a fondness for Pi (and pie!). For the truly obsessed, here is Pi out to 100,000 decimal points. Here’s Pi Day’s “Learn About Pi” page, the Joy of Pi’s “Pi Facts,” and Wikipedia’s essay.

Pi: one hundred digits


2. Pi was also important to Crockett Johnson. In his later years, he worked on the mathematical conundrum of squaring the circle — a problem that also intrigued mathematician and children’s author Lewis Carroll, a century earlier. Johnson even published his own original theorem on the subject.

Crockett Johnson, from the Mathematical Gazette (1970)
Crockett Johnson, algebraic proof from the Mathematical Gazette (1970)

He moved towards this answer, visually. He literally worked out the problem via his paintings, creating many variations on the idea, and ultimately arriving at Squared Circle (1968).

Crockett Johnson, Squared Circle (1968)

Then, to get the algebraic notation correct, he corresponded with mathematicians, who helped him express his idea in the less visual medium of the formula. It was published in the Mathematical Gazette in 1970.


3. As readers of children’s literature know, this blog takes its name from a scene in Johnson’s Harold and the Purple Crayon (1955), in which the title character “laid out a nice simple picnic lunch.”

Crockett Johnson, Harold and the Purple Crayon (1955): "There was nothing but pie."

I love the way Johnson’s tone both embraces Harold’s matter-of-fact tone and registers amusement at this claim. On the one hand, the third person narrator (via a literary technique known as free indirect discourse) tells us what Harold is thinking: our narrator is so closely aligned with Harold’s point of view that one could easily swap the pronouns and Harold’s name for “I.” These are Harold’s thoughts. On the other hand, they’re not entirely Harold’s thoughts. Johnson’s deadpan delivery of these lines also underscores the mild absurdity of having nine favorite kinds of pie. That is, there’s also an awareness here that Harold lacks — specifically, that “nine kinds of pie that Harold liked best” is  funny.


4. Or is it? This somewhat baffling pie chart lists Americans’ 10 favorite types of pie.

Pie chart of Americans' favorite types of pie

It’s somewhat baffling because the percentages don’t add up to 100 — which is the point of using a pie chart. The circle represents 100%, and then each slice some lesser percentage. But this chart doesn’t. As the chart’s caption explains, the total “adds up to more than 100 per cent because people were asked to rank their three favorite types of pie.” And that still doesn’t make the above chart any more illuminating — though it is pretty to look at.


5. Pies are among those foods that come in both sweet and savory varieties. When I think of pie, my thoughts drift to the sweet (apple, peach, pecan, blueberry), but there are are also savory pies (meat pie, chicken pot pie, potato pie, pizza pie). The Oxford English Dictionary, which finds the oldest use of the word “pie” (then spelled “pye”) in 1304, offers the following as its  first definition:

A baked dish of fruit, meat, fish, or vegetables, covered with pastry (or a similar substance) and freq. also having a base and sides of pastry. Also (chiefly N. Amer.): a baked open pastry case filled with fruit; a tart or flan.


6. Care for some Amblongus pie? Since this is (often) a children’s literature blog, here’s a recipe from Edward Lear’s “Nonsense Cookery” (which appears in his Nonsense Songs, Stories, Botany, and Alphabets, 1871).

TO MAKE AN AMBLONGUS PIE

Take 4 pounds (say 4 1/2 pounds) of fresh Amblongusses, and put them in a small pipkin.

Cover them with water and boil them for 8 hours incessantly, after which add 2 pints of new milk, and proceed to boil for 4 hours more.

When you have ascertained that the Amblongusses are quite soft, take them out and place them in a wide pan, taking care to shake them well previously.

Grate some nutmeg over the surface, and cover them carefully with powdered gingerbread, curry-powder, and a sufficient quantity of Cayenne pepper.

Remove the pan into the next room, and place it on the floor. Bring it back again, and let it simmer for three-quarters of an hour. Shake the pan violently till all the Amblongusses have become a pale purple colour.

Then, having prepared a paste, insert the whole carefully, adding at the same time a small pigeon, 2 slices of beef, 4 cauliflowers, and any number of oysters.

Watch patiently till the crust begins to rise, and add a pinch of salt from time to time.

Serve up in a clean dish, and throw the whole out of the window as fast as possible.

For any readers unfamiliar with Lear’s nonsense works,… you’re in for a treat even tastier than amblongus pie and gosky patties. Go read ’em!


7. John Cage and Lois Long’s Mud Book (written 1950s, published 1983) offers a recipe for mud pie — and yes, this is John Cage, the composer.

John Cage and Lois Long, Mud Book (1983)

 

John Cage and Lois Long, Mud Book (1983)

 

John Cage and Lois Long, Mud Book (1983)

 

John Cage and Lois Long, Mud Book (1983)

As Lane Smith and Bob Shea write in their post on Mud Book (and my source for these images), “Instructions any child can follow with ingredients easy to obtain. Notably, dirt, rocks, water, dirt and more dirt. But remember, mud pies are to make and to look at. Not to eat.”


8. Michael John Blake’s “What pi sounds like” is my favorite musical interpretation of the number. (It’s also literally a “musical number.” Get it?)

Many others have composed music inspired by Pi. Lars Erikson — composer of the Pi Symphony — even sued Michael John Blake (composer of the above piece), alleging plagiarism. Erikson lost.

If you like it (and if you don’t), you can  buy Michael John Blake’s “What pi sounds like” via iTunes.


9. One could make a long list of pie-themed music, too. The earliest one that comes to mind is “Song of Sixpence” (18th century): “Sing a song of sixpence, / pocket full of rye. / Four and twenty blackbirds / baked in a pie.” There’s A. A. Milne’s “Cottleston Pie,” performed here by Rowlf the Dog on the first season of The Muppet Show (1976).

According to an informal and completely unscientific survey of the “pie” songs in my iTunes, pies usually function metaphorically in music. Yes, there are “The Worst Pies in London” (from Sondheim’s Sweeney Todd) and Zoe Lewis’s “Pies for the Public” (from Sheep), but you’re more likely to encounter Don McLean’s “American Pie” (1971), Jay & the Techniques’ “Apple, Peaches, Pumpkin Pie” (1967), the Beatles’ “Honey Pie” (from The Beatles [White Album], 1968), David Wilcox’s “Wildberry Pie” (1991), Death Lurks’ “Happiness Pie” (from The Kids in the Hall soundtrack, 1996), or Louis Jordan and Ella Fitzgerald’s “Patootie Pie” (1946).


And what better way to express one’s appreciation for Pi and pie than by baking a Pi pie?

Pi pie

If you want to make one, the Nerdista has a recipe for her Pi pie (pictured below).

Nerdista's Pi Pineapple Pie

Happy Pi Day! Let’s have some pie. And Pi, of course.

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Free Pi!

No, Pi cannot be copyrighted, despite what one composer claims.  I had wondered why Michael John Blake’s beautiful YouTube video of “What Pi Sounds Like” had been taken down.  I’d linked to it in my “Happy π Day from Crockett Johnson” post, and then it… disappeared.  Blake explains why below:

Vi Hart has a truly excellent response to Lars Erikson, the composer who filed the claim against Michael Blake.  Check it out:

As Hart notes, Erikson has also written a melody based on Pi — but it’s a different melody.  I am not a legal expert, but I don’t think that Erikson’s claim has any standing: If this were a case of one melody sounding like another, then Mr. Erikson would have precedent.  See for instance, the case of George Harrison’s “My Sweet Lord,” in which Mr. Harrison’s piece was ruled to have borrowed from the Chiffons’ “He’s So Fine,” and Mr. Harrison was ordered to pay royalties to the song’s composer.  (I’ve complied a page of such borrowings — most of which have not resulted in lawsuits — on a blog post inspired by allegations made against Lady Gaga’s “Born This Way”.)

Compare the two Pi songs.  Here is the first movement of Lars Erikson’s Pi Symphony:

NewScientist has re-posted Michael Blake’s original video for “What Pi Sounds Like”:

You can also buy Michael Blake’s “What Pi Sounds Like” on iTunes.  When I listen to these two works, side by side, I find it a bit of a stretch to claim that Blake has somehow plagiarized Erikson’s work.  Yes, they both draw inspiration from 3.1415926535…, but sharing a common influence does not allow us to conclude that one work “stole” from the other.  Honestly, the main conclusion I draw from all of this is that Lars Erikson has the heart of a lawyer, and that Michael John Blake has the heart of an artist.

What do you think?

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Happy π Day from Crockett Johnson

Nine kinds of pie (from Crockett Johnson's Harold and the Purple Crayon)This isn’t the only kind of “pie” that Crockett Johnson was interested in.  In addition to “all nine kinds of pie that Harold likes best,” Johnson also drew inspiration from π (3.14159265…) — the ratio of a circle’s circumference to its diameter.

In early 1968, he began the project of “squaring the circle,” constructing a square with the same area as a circle — but using only a straight-edge and a compass to do so.  This is impossible, but he was either unaware of this fact or undeterred by it.  Not a trained mathematician, Johnson worked his way towards the answer visually.  He painted solutions, testing different theories on his canvas (which was actually mortarboard — canvas intimidated him).

By the middle of the year, he had arrived at a solution:

Crockett Johnson, Squared Circle (1968)

He painted two versions of this.  The other can be found at the Smithsonian’s on-line exhibit, Mathematical Paintings of Crockett Johnson.  I recommend you visit its site for a more complete explanation of the mathematics behind the painting.

Johnson next wrote up an algebraic formula, and sought the opinion of professional mathematicians.  Finally, in 1970, the Mathematical Gazette published his first mathematical theorem, “A geometrical look at √π”:

Crockett Johnson, from the Mathematical Gazette (1970)
Crockett Johnson, algebraic proof from the Mathematical Gazette (1970)

Johnson would publish his second original contribution to the field of mathematics in 1975, just prior to his death.

Learn more:

Other examples of Crockett Johnson’s work (from this blog):

 

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Back-to-School Special, Part I: Children’s Literature & Asymptotes

In my decade of teaching Children’s Literature at the university level, I’ve learned a lot.  But I never feel that I’ve learned quite enough to teach the grad class Critical Approaches to Children’s Literature.  I’m grateful that I’m teaching it now and not ten years ago, but it’s one of those courses that makes me conscious of the deficits in my knowledge.  And, on the whole, I see this process as a good thing — because it means that I’m moving closer to mastery of the subject… which, of course, is all one can do.  If the x-axis represents mastery, I’m moving along a curve that approaches but never actually intersects with the x-axis.  I get ever closer, but never arrive.

That curve, by the way, is called an asymptote.  It looks like this:

Horizontal Asymptotes

Above is a graph of y =1/x, taken from this website.  The line approaches zero (which, in my analogy, represents mastery of the field), but never reaches it.

Norton Critical Edition of Alice in WonderlandSo, the syllabus for Critical Approaches to Children’s Literature represents one stop along that journey.  What’s on it?  General themes include: didacticism, pleasure, nonsense, audience, genre, diversity. Theoretical approaches include: formalist, psychoanalytic, feminist, queer theory, cultural studies, and others.  As you’ll see (if you follow the link), we’ll be reading fiction by Helen Bannerman, J. M. Barrie, Francesca Lia Block, Anthony Browne, Lewis Carroll, Jean-Marie Leprince de Beaumont, Maria Edgeworth, Neil Gaiman, Jacob and Wilhelm Grimm, Heinrich Hoffmann, Ann Jonas, Guus Kuijer, David Macaulay, L.M. Montgomery, Walter Dean Myers, Marilyn Nelson, Charles Perrault, Jon Scieszka and Lane Smith, Brian Selznick, Dr. Seuss, Mary Martha Sherwood, Shaun Tan, Chris Van Allsburg, and Mary Wollstonecraft, among others.  A Wreath for Emmett TillAnd we’ll gain critical perspective from Robin Bernstein, Michel Foucault, Sigmund Freud, Felicity A. Hughes, Anne Scott MacCleod, Michelle Martin, W.J.T. Mitchell, William Moebius, Mitzi Myers, Perry Nodelman, Walter J. Ong, Lissa Paul, Jacqueline Rose, Jan Susina, and many others.

Yes, there are many other texts and theorists that could be included.  And I’m sure that I will change the syllabus again next time I teach it.  Indeed, I’d like to use Keywords for Children’s Literature, which I co-edited with Lissa Paul (due out from NYU P in May of this year).  In case you’re curious, whenever I use a book of my own, I donate any royalties I receive to an appropriate charity.  When I used my Dr. Seuss: American Icon (in a Seuss class), that worked out to about $1 per book.  So, it’s not much, but I don’t think it’s ethical to profit off of my students in that way.  Anyway, I’m sure this syllabus could be better — and not just because I now note a few formatting errors on the Schedule of Assignments.  (I’ll fix those before class on Wednesday.)  But I also think the syllabus will do the job, as I — and my students — travel along that curve, always approaching, never arriving, but learning a lot along the way.

UPDATE: 18 Jan. 2011, 10:45 am. Looking back at what I wrote (late last night, with minimal editing), there’s a major omission that I need to correct: Naomi Wood.  My Critical Approaches to Children’s Literature syllabus and course pack borrow heavily from hers.  Yes, my class has my own “stamp” on it — and that’s even more true of this year’s iteration of the syllabus.  (The initial syllabus, from Spring 2009, even more closely followed hers.)  But the general course plan is very much hers.  I’m fortunate to have helpful colleagues who share their knowledge, and I want to make sure that Naomi gets due credit here.  So, Naomi: I doff my hat to you!  And, yes, it is a red-and-white-striped topper.  How ever did you know?

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