Archive for the ‘Holiday-related Math’ Category

Happy Pi Day 2015!

March 10, 2015

Pi Day is coming up, and this year is a special one:  instead of just 3.14 on March 14 we get 3.1415.  Woo hoo!  So grab a Sudoku

(from Brainfreeze Puzzles:  Digits 1-9 in each row, column, and square, plus digits 31415926 in each block of pink)

and grab a beverage of your choice

GodzillaPiBeer2015

and enjoy.  Happy Pi Day!

If it’s Pi Day, that means…

March 14, 2011

Brainfreeze Puzzles must have a new Pi Day Puzzle up — woo hoo!!!!!

The Rules are to fill in this pie-shaped circle so that the numbers 1 through 12 appear:

  • exactly once in each double-wedge of the same color,
  • exactly once in each pair of opposite wedges, and
  • exactly once in each ring around the center.

As in previous years, they are having a contest for correct entries (information on this website and this pdf file), so no hints or solutions are to be posted in the comments until the contest closes on June 1.  [If/when they print a solution, we’ll post a link to it.]

If you missed Brainfreeze’s earlier puzzles, here are the ones from:

2010

2009

2008

Things that equal e

February 4, 2010

e-day is coming up on Sunday, and I’ve already started making goodies to share on Friday (not wanting to fall into the trap of burning everything again).  Instead of writing “e” on the top, I’m thinking of putting in one of the following:

  • \displaystyle\lim_{n \to \infty} \left( 1 + \frac{1}{n} \right) ^n
  • \displaystyle\lim_{n \to \infty} \frac{n}{\sqrt[n]{n!}}
  • \displaystyle 1+1+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+...
  • The x-value for which \displaystyle x^\frac{1}{x} is as large as possible.
  • \cosh{1}+\sinh{1}
  • \cos{i}-i\sin{i}
  • \displaystyle \frac{\sinh{\pi}}{\pi}+\frac{2\sinh{\pi}}{\pi}\cdot\sum_{n=1}^{\infty}\frac{(-1)^n}{1+n^2}\left(\cos{n}-n\sin{n} \right)
  • (from OEIS A001113) \displaystyle \left(\frac{16}{31}\cdot\left(\sum_{n=1}^{\infty}\left(\frac{n^3+n+2}{2^{n+1}n!} \right)+1\right)\right)^2
  • (also from OEIS A003417) The number represented by the cool looking continued fraction [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12, …]
    1+1+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{4+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{6+\cfrac{1}{1+...}}}}}}}}}

(Any other good ones?)

The Mystery of the Fibonacci Pumpkin

October 31, 2009

We have a mystery on our hands.  Greater than the mystery of what Frankenstein has to do with polygons.  Greater even than the mystery of how to come up with a good Math Halloween Costume.    The mystery is:  Do pumpkins have some strange connection with the number 3?

It all started when we were getting ready to carve Jack O’Lanterns.

Go Godzilla Go!

And we noticed this odd thing at the bottom of the pumpkin:

Halloween 011 small

See it down there?

Weird Alien marking

What the heck is that?  The hole in the center goes all the way through to the outside, but the perfect 120° angle markers are only on the inside of the pumpkin.  Did this come from a metal spike or something, and it’s just artificial?  Or do pumpkins somehow naturally divide into three parts, like a banana.  Hey kids — you can try this at home!  Take a banana, break it in half, and then stick your finger down the middle.  It will naturally split into 3 pieces lengthwise.  I learned this in college [in the dining halls, not a classroom].

Banana thirds

Apparently, I just now discovered via The Sneeze,  this happens to bananas because the ones sold in the supermarket are triploid organisms, which means that they have 3 sets of each chromosome instead of 2.  That’s a little weird, no?  A little unnatural?  Actually, unnatural is exactly what it is:  triploid organisms occur (exclusively?  This part I’m not sure on) when a biploid organism is crossed with a tetraploid, giving the average of 2 and 4 sets of chromosomes.  This is bad for the resulting triploid banana, which is sterile, but good for the person who eats it, because the sterile banana contains no seeds.  [No, not even those black spots, which according to this official sounding page are “the remains of aborted ovules that did not mature into seeds”  and EEEWWWW who else is suddenly grossed out by bananas?]

But pumpkins don’t seem to be triploid organisms.  At least, they aren’t on the list of examples I found on Wikipedia, though watermelons are.  So the mystery remains:  is this figure naturally occurring, or artificial?

Happy Halloween!

Happy Halloween!

The Fibonacci connection:  I’ve seen the fact that bananas split into 3 pieces used as an example of how Fibonacci numbers appear in nature.  That our bananas might be a hybrid of those that have 2 and 4 sets of chromosomes, though, and that presumably split into 2 or 4 pieces seems to discount the whole Fibonacci relationship for bananas since 4 isn’t a Fibonacci number.

The Actual Dresden Codex

May 5, 2009

dresden_codex_p09Happy Cinco de Mayo!    A year ago we celebrated the day here by talking about some Cinco de May math, branching out into Spanish Colonial Mathematics [including a really cool multiplication technique called The Method of the Cup].

But that was a pretty broad geographical region, so today it’s back to Mexico, with Guatemala, Belize, and maybe a little bit of Honduras and El Salvador thrown in.  (Aside:  I often make students learn where countries are when I teach The History of Mathematics, and a few times I’ve had them memorize all the countries of Central America for the first test.   One year I had the great idea that I’d test them on a map with no political boundaries, reasoning that there is some value in being able to identify countries by other landmarks.   Although I’d warned them about this, it turned out to be a truly awful test to grade — some people started off wrong, so while the countries were right in relation to each other they were all either too far north or too far south, and figuring out partial credit was a bear. I’ve never given a test like that again.)

So anyway, here’s the region we’re talking about:

Yep, it’s the Mayas!  I’ve seen a lot of math books that talk about Mayan mathematics, because it’s pretty straightforward (until it isn’t).

A dot means 1:  •
A line means 5:  ______
And a shell means zero.

The Mayans used a modified Base 20 system, so that the group of numbers at the bottom counted the 1s, the next group up counted the 20s, the next group up counted the 360s [and yes, it should really be 400 because twenty 20s is 400, but for some reason they used 360 here.  I’ve seen speculation as to why — it might have something to do with the fact that there are just over 360 days in a year — but no one knows for sure], the next group up counted the 7200 [because that is twenty 360s], the next group up counted the 144000 [because that is twenty 7200s], etc.

But the point of this post is that because the Internet is such an amazing place, you can see images of the actual original sources that use this!  Sadly, there aren’t that many:  most of the writings were destroyed, and only about four codices remain.  Codices are folded books, like this one (from the St. Andrew’s web site).

This is the Dresden Codex, and it’s about a thousand years old.  It was made of Amatl paper [fig bark covered with a lime paste], and although the picture above is in black and white it was actually done in color.  Indeed, when it came to numbers the groups (1, 20s, 360s, etc) were written in alternating black and red so that it was easy to tell where one group ended and other began.

dresden_codex_annotated1Look at that number in the lower left-hand corner.   There are 7 (1s), then 1 (20), 4 (360s), 6 (7,200s), 3 (144,000s), 11 (2,880,000), 3 (57,600,000s), and 6 (1,152,000,000s), giving a sum of over 7 billion.  It’s a pretty big number, and only uses 2 symbols [since the zero wasn’t  necessary in this case].

When I started teaching about Mayan numbers, there weren’t many pictures of the Dresden Codex available so I made do with a photocopy of the one of the pages.  But the Internet is an amazing place, and now you can actually see the entire thing in color in not one but two places!

The first place I found it is this site (archaeoastronomie).  It has the best pictures; my favorite for looking at numbers is page 51 and the pages around it [the zooming is really impressive, although it doesn’t necessarily zoom to the page you want].

And more recently there’s the Foundation for Advancemet of Mesoamerican Studies, Inc. They also provide downloadable .pdf files here [on the order of 100 Mb, but the download page offers the option of downloading in sections].  Hooray for the Internet!

Things that equal Pi

March 13, 2009

piSo you want to make a pie for Pi Day, but you don’t want to decorate it with the traditional symbol \pi.  What other expressions could you use that are equivalent?

You could go with the elegant:  a picture of a circle and the ratio of the circumfirence to the diameter

\frac{C}{d}

In a similar vein, you could move up a dimension to area

\frac{A}{r^2}

or volume \left(\frac{3V}{4r^3}\right), although in this case you’d have to draw a sphere and I can tell you right now that I’d lose points for clarity.

If geometry isn’t your thing, you could decorate your confection with an infinite sum, perhaps the Madhava-Gregory-Leibniz series (discovered by Madhava of Sangamagram, India about 600 years ago, and then rediscovered by James Gregory of Scotland and Gottfried Wilhelm Leibniz of Germany 200 years later)

\frac{4}{1}-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}+\cdots

or the slightly more complicated

\sqrt{\frac{6}{1}+\frac{6}{4}+\frac{6}{9}+\frac{6}{16}+\cdots}

found by Leonard Euler of Switzerland in 1735.  Or even the Bailey-Borwein-Plouffe formula (which is, face it, kind of fun to say) that was discovered only 14 years ago(!) by Simon Plouffe of Quebec, Canada:

\displaystyle\sum_{k=0}^{\infty}\frac{1}{16^k}\left(\frac{4}{8k+1}-\frac{2}{8k+4}-\frac{1}{8k+5}-\frac{1}{8k+6}\right)

Incidentally, Simon Plouffe and Neil Sloane are the authors of the Encyclopedia of Integer Sequences, which gave rise to the online version.

But back to \pi.  Do you prefer products?  Then maybe you’d want to turn to Wallis’s product, discovered by John Wallis of England in 1655:

2\cdot\left(\frac{2}{1}\cdot\frac{2}{3}\cdot\frac{4}{3}\cdot\frac{4}{5}\cdot\frac{6}{5}\cdot\frac{6}{7}\cdot\frac{8}{7}\cdot\frac{8}{9}\cdot\cdots\right)

We’ll end on a more radical note:  the Viète formula, which was named after François Viète of France, but actually found by Euler.

2\cdot\frac{2}{\sqrt{2}}\cdot\frac{2}{\sqrt{2+\sqrt{2}}}\cdot\frac{2}{\sqrt{2+\sqrt{2+\sqrt{2}}}}\cdot\cdots

A Pop-Up Sierpinski Valentine Card

February 12, 2009

sier-12Last year we made interlocking Möbius Valentines.  This year we’re going staying three-dimensional, and making a pop-up card that forms a Valentine Fractal!

Start by taking a sheet of paper and folding it in half.

sier-1

Next, you want to make a cut along the fold.  Measure halfway along the fold, and then make a cut of half the width.  Since you’re going to want to form a heart, make  it a bit rounded like the top of the heart, and then fold it over to make a crease.

sier-2

sier-4

Once you’ve creased it, refold that bottom part so that it sticks into the middle of the card, like this:

sier-5

Now iterate!  Along the original fold, measure halfway down each part and cut ¼ of the width (instead of ½), still forming the top of a heart.  As before, fold those pieces down to create creases.

sier-6

And then refold along those creases to tuck the folds inside:

sier-8

Repeat, making four new rounded cuts.

sier-9

And then tucking in the folds.

sier-10

And again!  Cut, fold over, and tuck in.  And here’s the result!

sier-12

Happy Valentine’s Day Everyone!

You can also make a plain old traditional Sierpinski Triangle with this method.  Just make straight cuts instead of rounded ones, and it ends up looking something like this:

sier-15

I first saw this design in the book Fractal Cuts by Diego Uribe, published in 1994, although according to this article from Mathematics Teacher (which includes a template) it might be older than that.   In those instructions you do all the cutting and folding first (without tucking the pieces into the center) and then push out the pop-up part all at once at the end, but I found that to be really frustrating; tucking in as you go along is a lot easier.  As far as I know the Valentine variation is my own doing, but it’s certainly possible that someone else has done it before.

Star of David Theorem

December 21, 2008

star-of-davidHanukkah starts today at sundown, so in honor of the holiday here is the Star of David Theorem.  In simplest terms, this theorem says that the greatest common divisor of  {{n-1} \choose k}, {n \choose {k-1}}, and {{n+1} \choose {k+1}} is equal to the greatest common divisor of  {{n-1} \choose {k-1}}, {n \choose {k+1}}, and {{n+1} \choose k}.  To see why it’s called the Star of David, look at the following visual.  the greatest common divisor of the blue corners and the greatest common divisor of the purple corners are equal.  Together, the two triangles form the Star of David.

star-of-david

For example, if n=4 and k=2, this says that the greatest common divisor of  {3 \choose 2}, {4 \choose 1}, and {5 \choose 3} is equal to the greatest common divisor of  {3 \choose 1}, {4\choose 3}, and {5 \choose 2}.  As it turns out, this is one of the less interesting examples because both sides simplify to gcd(3,4,10), which is 1.  So let’s look for another example.

Where there are binomial coefficients, Pascal’s triangle can’t be too far behind.  Sadly, when the above Theorem is placed visually into Pascal’s triangle, it ends up looking kind of turned and squooshed.

pascal-star-1Visually, the top star illustrates the previous not-so-interesting example of how gcd(3,4,10) is equal to gcd(3,4,10).  But the lower star illustrates that that gcd(36,210,165) is equal to gcd(84,45,330), and this is a little less obvious.  In this second case, but greatest common divisors are equal to 3.

According to Wolfram’s Mathworld, the Star of David Theorem was first stated by H. W. Gould in 1972, and there were several generalization in the years immediately following.  Apprently the association with Pascal’s triangle wasn’t noticed until 6 years ago, however, by B. Butterworth in this article (which is originally about using Pascal’s triangle to illustrate the song “The Twelve Days of Christmas”).

Homemade Holiday Gift #1 and 1.5

December 20, 2008

Godzilla wearing a butterflySo there are a lot of cool math things out on the web, that’s clear with a quick google.  But suppose you want to try something homemade?  One option is to make cookie ornaments.  But not with real cookies, because there’s pretty much a 100% chance that the cookies would start sporting bite marks, and then they’d disapper entirely leaving only little crumb covered ornament hooks, and that would just be sad.

No, it’s better to make ornaments out of something a little less tasty.  My favorite is Penzey’s Cinnamon Applesauce Holiday Ornaments:  instructions formally written out in a .pdf file here.  Essentially, you take a jar of applesauce, drain it overnight by putting it on a dishtowel or cheesecloth over a bowl (supposedly this is unnecessary, but enough water drains out that it seems to be quite a useful step).  Then you add a bunch of cinnamon, which is pretty cheap if you buy it at a dollar store.  Stir in enough for it to become a stiff dough (it’s edible, but about as tasty as pure cinnamon),  roll it out thin, cut out shapes, and poke out a hole for yarn or ribbon  Then you cook them for 175° [assuming your oven goes that low] for 6-8 hours.  That’s right, 6-8 hours:  you pretty much need to play on being home all day.  On the bright side, your house will smell GOOD!], decorate them, and add something to hang them from like yarn.

You can use your favorite cookie cutter (perhaps in the shape of pi).   In this case I used a butterfly to represent the butterfly effect from chaos theory [an idea gathered from the Halloween Costume suggestion here].

butterfly-ornament

That symbol is gamma because I thought that γ deserved to step out of π’s shadow for the holidays.

Godzilla wearing a butterfly

So that’s Homemade Holiday Gift #1.  For 1.5, I was going to show you how to make Gem Magnets, but it turns out that with their glass tops, photographing them requires a lot more skill than I have.  Or maybe a better camera.  They look something like this:

magnets

See what I mean about the photography?  But they’re still pretty cool to make:  you get magnets, then cut out circles either from pictures or from plain paper that you can draw on (perhaps your favorite math symbol, which is the faint tie in to mathematics with this post).  Then you glue one of those clear stones on top, and lo and behold you have a homemade present!  Woo hoo!  (And if you like directions that are a little more precise, you could try here.  If words like “silicone” scare you, just use an all purpose adhesive that sticks to metal and glass).

Happy Holidays!

Happy Thanksgiving!

November 27, 2008

thanksgivingWhile you’re baking your turkey and pondering why the temperature continues to rise even after the bird comes out of the oven or thinking about how Game Theory showed that tryptophan affects trust and cooperation, you might also be wondering, “What kind of math did the folk at the first Thanksgiving study?”

The answer is…I don’t know. So I looked around at what textbooks would have been available. I couldn’t find any math texts of the Wampanoags from around 1621, and while I was enticed by a reference to “our earliest native American arithmetic, the Greenwood book of 1729” from David Smith’s History of Mathematics Vol.2, in looking further it seems that the phrase “native American” was used only in contrast to having been originally published in England, Spain, or other non-American country. So as far as the Wampanoags are concerned, I’ll admit to remaining in the dark about what specific kinds of math they would have been learning (formally or informally).

Then I looked into the Pilgrims. As they hailed from England, it seems likely that they’d be using a an English text. And for this they had few choices: there was a 1537 book entitled An Introduction for to : Lerne to Reckon with the Pen and with the Counters after the True Cast of Arsmetyke or Awgrym [that last word is Algorithm], which was a translation of a book in Latin by Luca Pacioli. Wikipedia mentions that another textbook was published in 1539, but I couldn’t find reference to the title or author (though I suppose I could add one myself).

A more likely candidate would be Arithmetick, or, The ground of arts by Robert Recorde, first published in 1543 and listed in several places (e.g. this site) as “The first really popular arithmetic in English”. The title is short for The grou[n]d of artes: teachyng the worke and practise of arithmetike, moch necessary for all states of men. After a more easyer [et] exacter sorte, then any lyke hath hytherto ben set forth: with dyuers newe additions. Here’s a woodcut from the 1543 version

robert_recorde-1543

and here’s the title page from the 1658 edition, with its fancy modern spelling:

recorde-title(Heh heh — I just ran that page through Adobe’s Optical Character Recognition, and it pretty much didn’t recognize a thing.)

I couldn’t track down any copies of what exactly was in the book, but this article about medicine in English texts quotes quite a bit from Recorde’s book. For example:

In 1552, Record amplified his arithmetical text with, among other material, a “second part touchyng fractions”. This new section included additional chapters on various rules of proportion, including “The Rule of Alligation”, so-called “for that by it there are divers parcels of sundry pieces, and sundry quantities alligate, bounde, or myxed togyther”.

The rule of alligation, we are told, “hath great use in composition of medicines, and also in myxtures of mettalles, and some use it hath in myxtures of wynes, but I wyshe it were lesse used therein than it is now a daies”. Despite Record’s regret about the adulteration of wine, the first problem the Master uses to exemplify his discussion of alligation in fact deals with mixing wines; the second involves a merchant mixing spices; and the remainder involve the mixing of metals. None of the examples involve medicine, although the merchant’s spices are once called “drugges”.

A third part was added in 1582 by John Mellis. Here’s a page (from Newcastle University) showing one of the pictures in the margin (possibly from that year, or possibly a later edition; I couldn’t tell for certain).

Records Arithmetic

Robert Recorde is actually better known for a later book: The Whetstone of Witte, which was published in 1557 and so is another candidate for An Original Thanksgiving Math Text.  It was called because, according to the poem on the front, “Here, if you list your wittes to whette, Moche sharpenesse therby shall you gette,” and appears (using Google books) to have been well-known enough to be referenced by Shakespeare in Act 1 Scene 2 of As You Like It (written around 1600) when Celia says, “…for always the dulness of the fool is the whetstone of the wits. How now, Witte! whither wander you?”

The Whetstone is best known for its introduction of the equal sign =, which Recorde explains below:

whetstone-equal-1

Which reads as

Nowbeit, for easy alteration of equations.  I will pro(?provide?) a few examples, because the extraction of their roots, may the more aptly be wrought.  And to avoid the tedious repetition of these words “is equal to”, I will set as I do often in work use a pair of parallels or Gemowe lines of one length, thus: ======, because no two things can be more equal….

So there you have it: math that some of the first Thanksgiving folk could possibly have studied.

thanksgiving

Some WWI tidbits for Veterans Day

November 11, 2008

uncle_sam_pointing_fingerI was just reading the article “Dr. Veblen Takes a Uniform:  Mathematics in the First World War” by David Alan Grier (from the American Mathematical Monthly, Dec 2001).  The full article is about the Captain Oswald Veblen and the math folk who worked with him at a military facility in Aberdeen, Maryland, but there are plenty of additional tidbits  For example:

Over 150 mathematicians served in the First World War. Many took conventional military roles but half found ways to employ their mathematics. They worked as surveyors, assisted cartographers, and taught trigonometry to officer candidates.  (p. 923)

A lot of the work involved making ballistic tables, which was not such a simple task:

These mathematicians did part of their computing at the actual test ranges, where they served as observers, data collectors, and range officers. On the water range, a large range that ended in the Chesapeake Bay, the mathematicians were stationed on towers along the shore. After they observed the splash of a shell hitting the water, they would compute the range of the shot and telephone their result to the firing station. At the firing station, a second mathematician would adjust the range for changes in temperature, humidity, and wind. Once the series of shots was completed, a mathematician at a central office would compute ballistics coefficients and create range tables.  (p. 927)

Which sounds about as exciting as alphabetizing names for the phone book.  By hand.  In the mud.  Nonetheless, it had to be done and so it was.  Another set of computations, not as cold as the above,  involved solving differential equations numerically to computer trajectories.  The main staff started doing this, but then they got bored and made the enlisted men take over.  In Washington DC, however, there weren’t enough men to do they job and so the Diff Equ was handed off to women:

By early summer, [Forest] Moulton hired eight women. All had graduated from prominent universities during the prior two years. All had been mathematics majors. They came from University of Chicago, Brown University, Cornell University, Northwestern University, Columbia University and the George Washington University. For these women, the war was an opportunity to play a role, perhaps only briefly, on the public stage.  (p. 929)

Rosie the Riveter, meet Connie the Computer!  Actually, the only women mentioned by name is Elizabeth Webb Wilson, who turned down nine other job offers before taking on this job.  And apparently in their spare time, at least some of the women involved also fought for suffrage.

All in all, it was a good read.  Happy Veterans Day!

Frankenstein, Great Expectations, and Polygon

October 29, 2008

What’s the connection? Mary Shelley (born Mary Wollstonecraft Godwin) wrote Frankenstein; or, The Modern Promethius when she was a teenager, in 1818. The original Dr. Frankenstein’s monster didn’t look like the guy to the left: in the 3rd edition of the book (published in 1831) he looked like this:

So what does this have to do with Polygon? Well, Mary Shelley was born in The Polygon! The Polygon was here:

Some sites indicated that The Polygon was the name of the actual house, but after surfing the net when I really should have been grading doing some research I’m pretty sure that The Polygon was that immediate neighborhood, not one single building.

For example, in a book (Memoirs of the Author of a Vindication of the Rights of Woman) that her dad (William Godwin) wrote about her mom (Mary Wollstonecraft, who died 11 days after Mary was born), The Poygon is mentioned twice:

It is perhaps scarcely necessary to mention, that, influenced by the ideas I had long entertained upon the subject of cohabitation, I engaged an apartment, about twenty doors from our house in the Polygon, Somers Town, which I designed for the purpose of my study and literary occupations. Trifles however will be interesting to some readers, when they relate to the last period of the life of such a person as Mary. I will add therefore, that we were both of us of opinion, that it was possible for two persons to be too uniformly in each other’s society. Influenced by that opinion, it was my practice to repair to the apartment I have mentioned as soon as I rose, and frequently not to make my appearance in the Polygon, till the hour of dinner.

In digging around some more, I discovered someone else who lived in The Polygon: Charles Dickens! He wasn’t born there, but moved to 17 The Polygon, Somers Town in 1827 (more than a decade after Mary had left) at the tender age of seven when his family was evicted from their previous abode. [He only lived there about a year before moving.]

Finally, The Keeper of All that is Good and True says, “In 1784, the first housing was built at the “Polygon”, now the site of a council block of flats called “Oakshot Court”.” So I’m convinced that The Polygon is that neighborhood, maybe the plaza (which would likely be in the shape of a polygon). And the word Polygon is mathy, and Frankenstein is a pretty Halloweeny book, and Charles Dickens has some scary stuff it it (not monster-scary, but those debtors’ prisons don’t sound like much fun), so it all seems to fit the season.

The Double Möbius Star of David

September 29, 2008

Rosh Hashana begins today at sundown. It’s often referred to as the Jewish New Year, since it marks the start of the year 5769, but unlike the January celebration for the year 2008 it’s celebrated over two days. This is because Rosh Hashana always starts on the first day of the lunar month of Tishri, and traditionally months were officially declared by the Sanhedrin when a new moon was sighted. Messengers were sent out to share the date of the new month, but the most distant communities didn’t get the news in time and so to be safe they celebrated the start both 29 and 30 days after the previous month’s start. This custom continued even after the exact start of the month could be predicted mathematically.

As a math activity for these two days, you can make a Star of David out of interlocking Möbius strips (where “interlocking” is actually more like weaving). The half-twist in the Möbius strips means that the pieces will lie flat.

(I actually ran across this pattern on this jewelry site, which sells double Möbius stars in gold. Something to add to the collection of Möbius clothing!)

TwoPi observed (when looking over a draft of this post) that since in general it is possible to form interlocking links by cutting a twisted loop in half lengthwise, it might be possible to create this design by starting with a single loop and then cutting it. I tried, but wasn’t able to come up with anything: a loop has to have an odd number of half-twists to lie flat (like the Möbius strip itself), but if it has an odd number of half-twists then it will be a single twice-as-long loop when cut.

On the other hand, loops with an even number of half-twists form two interlocking loops when cut lengthwise, but the resulting loops will also have an even number of half-twists and so look a little bulgy. The closest I came was starting with a loop with two half-twists, but the resulting design just didn’t look as nice as the picture above.

Pirate Money

September 18, 2008

Ahoy Matey! September 19 is International Talk Like a Pirate Day! Time to break out the parrots and rum and enjoy a pirate comic or two.

In keeping with the pirate theme, I sprinkled doubloons, gold coins, and pieces of eight on a test I just gave (the words, not the actual coins). And that got me thinking — just what are these pieces of money?

A piece of eight is a silver Spanish coin, also known as the Spanish dollar, which was used starting around 1500 and which was the basis for the US dollar. Its name comes from the fact that it was worth eight reales (so it might make more sense to call it an eight-piece coin). Indeed, for those with sharp scissors, the coin was sometimes cut into eight slices, each worth one reale, that were referred to as bits. Thus two bits was worth a quarter dollar, which is a useful fact to know if you should need a shave and a haircut. For that matter, there’s all sorts of other “bit” terminology: a dime, being just short of a bit, was called a “short bit” and a “long bit” was 15 cents.

Spanish Piece of Eight from 1739 (King Philipe)

Spanish Piece of Eight from 1739 (King Philipe)

But what about doubloons? A doubloon is a gold coin, worth twice as much as some other Spanish coin. According to Wikipedia, a doubloon is worth two ducats, and therefore worth four pieces of eight or 32 reals. But wiseGEEK says the doubloon is worth sixteen pieces of eight, and therefore worth 128 reals. Who to trust, who to trust. I like the name wiseGEEK, so I’ll go with their story (plus they admit that lots of different coins have been called doubloon). Suffice it to say that doubloon=gold and they’re good to have aplenty in your treasure chest.

Pirate flag by oren neu dag, Creative Commons License.

Which direction is Mecca?

September 1, 2008

The month of Ramadan begins today.  This is an opportunity to refrain from food and drink (during daylight hours) and to focus on God. During this and all other months, Muslims pray five times a day towards Mecca. But this brings about an interesting mathematical question: what direction does Mecca lie in?

Perhaps surprisingly, there are two different answers for calculating the Qibla (the direction for prayer). One method is the “constant compass” distance. To visualize this, imagine a world map that uses the Mercator projection. Mark your location and Mecca, and draw a line between them. From Rochester, NY that direction is about 103° clockwise from North, or roughly Southeast (but more East than South).

Another method is to use the shortest distance between two cities. The easiest way to visualize that is to get a globe, put one finger on your location and one on Mecca, and then to turn the globe so that both fingers are lying on an imaginary “equator”. What you’re really doing is finding a great circle — a circle that cuts the globe in half — that passes through both Mecca and your town. From Nazareth College, this turns out to be about 50°-55° clockwise from North, or roughly Northeast (but more East than North). The north part is somewhat surprising since Mecca is south of Rochester, and I thought I’d simply made a mistake the first few times I calculated the distance using spherical trigonometry while teaching History of Mathematics.

There is not complete agreement about which way is the more appropriate, and some of the online Qibla calculators (such as this one) allow you to use either method.  It reminds me a bit of the Millennium, with people celebrating on both January 1, 2000 and January 1, 2001, although the question here isn’t which is technically correct but which of the two mathematical models for direction gives the appropriate direction for religious purposes.