Monday Morning Math: Happy New Year!

Good morning everyone! Happy 2023!    Today’s post is in honor of two new years.

The first is the new calendar year: 2023.  If you want to do something fun (which of course you do) then you can see if you can use exactly the digits 2, 0, 2 and 3  and different math configurations to write the numbers 1-100.  For example: .  Or  .  (There are examples all over the internet, so a quick search reveals many solutions should you wish.)

The other is the Lunar New Year, which began yesterday.  This is also known as the Spring Festival, and is observed around the world, including China, Indonesia, Japan, Malaysia, the Philippines, Singapore, South Korea, Taiwan, Thailand, and Vietnam: we are now in the Year of the Water Rabbit in many countries, and the Year of the Cat in Vietnam.

In honor of the New Year we’ll talk about the mathematician Jing Fang  京房.  He was born in China 2100 years ago (78 BCE, during the Han Dynasty).  He was a mathematician and, appropriately enough for this post, he described astronomy – the solar and lunar eclipses.

But these weren’t his only accomplishments.  He was also very good at making predictions using the Yijing, or I Ching. There are 64 hexagrams, each made up of 6 rows that have one long or two short marks in each row.  While this isn’t mathematics, it does lead to the math question to ponder: namely, can you explain why there are exactly 64 configurations?

Picture from Wikimedia showing a diagram that Gottfried Wilhelm Leibniz owned.

And his math-connection doesn’t end there either – he used mathematics to describe music theory, particularly that 53 fifths (technically “just fifths”, which may or may not be the same as a perfect fifth – but you can here one here) was almost exactly 31 octaves.  It took more than 1600 years for anyone (said anyone being Nicholas Mercator) to caculate the difference between the two more exactly than Jing Fang. 

Math Moons and Music – a good way to start the year.

Sources: Wikipedia and…just that, because the few other sources I found had the same information.

Happy Thanksgiving!

(This was originally published on November 27, 2008, but I ran across it in looking up Thanksgiving Math and thought it might be good to update the links and repost!)

While 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 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

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

(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).

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:

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.

Monday Morning Math: the witch of Agnesi

Happy Halloween!  For this day, it seems appropriate to talk about the Witch of Agnesi. Oooohhhhh!  Spooky! 

But this witch is a curve.  It’s described with the algebraic formula:   (or, more generally, as – in the previous equation I used ).

The original construction is described geometrically, starting with a circle of radius .  Here’s a short video of how it is constructed

So how did this curve get its name?   Maria Gaetana Agnesi was a wicked smart woman who was born in Milan, Italy, in 1718, about 15 years after this curve was first studied by Pierre de Fermat and Guido Grandi.  By the time she entered her teens she spoke 7 languages, and by the time she entered her twenties she was also accomplished at philosophy and mathematics, which she discussed with her father’s visitors, part of an intellectual salon.  After her mother died Maria Agnesi took over running the house and wrote a calculus book for some of her 20 younger siblings.  This book was published when she was 30, when the study of Calculus itself was only decades old.  It was over one thousand pages, and Agnesi was granted an honorary appointment at the University of Bologna.  Agnesi spent most of her adult life focused on theology and serving others, particularly people who were poor or sick.

In 1901, John Colson translated it into English, and here is where a significant mistranslation occurred.  Agnesi had called the above curve averisera (related to the word for “turning”), but the word for “witch” is similar, avversiera, and that’s what Colson used. 

(Public domain from Wikimedia)

Sources:

• Kat Eschner’s article and Evelyn Lamb’s article in Smithsonian Magazine
• Frank J. Swetz and Victor J. Katz’s article in Mathematical Treasures
• Elif Unlu’s biography of Maria Agnesi

Monday Morning Math: Happy Divali!

Good morning!  Today is Diwali: for those unfamiliar with the day, it is a celebration of the victory of light over darkness that originated as a Hindu religious festival and is now observed by people in many religious traditions. (More details can be found in, for example, the Times of India.)  Many years ago I ran across an article discussing mathematics and Diwali and saved the link.  Alas, when I looked for it now the original article appears to be gone, but in doing a new search I found a few more recent posts:

First grade teachers Michelle L and Michelle M talked about patterns and symmetry with their students, using designs on rangoli.  Rangoli are intricate patterns made on the floor out of flowers, rice, sand, or other materials:

CC Subhashish Panigrahi

CC by Rajeev Nair

CC by Dinesh Korgaokar

One thing I find interesting about this is that the symmetry is different in each case – it’s not always 8 or 12 for example.  

And for those of you craving something a little more Calculus-oriented, take a look at this video:  

Enjoy the week!

Monday Morning Math: Star Quilts

Happy Monday!  In honor of Indigenous Peoples’ Day our topic today is Star Quilts.  Here are some examples:

Here is the  Blue Day Star Quilt, probably of the A’aninin (Gros Ventre) in Montana, made in 1990-91.

Here’s the Fall Time Star (Bright Fall Day) Quilt by Marie Kinzel of the Lakota (Teton/Western Sioux) in South Dakota, made between 1968 and 1988. 

Created by Marie Kinzel (Marie Strong Heart Kinzel)

Below is the Earth and Sky Quilt created by Chantelle Blue Arm of Cheyenne River Lakota Sioux in South Dakota, made in 2014.  

Created by Chantelle Blue Arm

You can see many more examples at the National Museum of the American Indian.  I first read about them in the January 2021 issue of Crux Mathematicorum, published by the Canadian Mathematical Society. The article “Explorations in Indigenous Mathematics: No. 1” by Edward Doolittle begins:

The starblanket design is popular among the Indigenous peoples of the Plains region, particularly in quilted blanket designs, but also in other crafts. In the Plains Cree language, the word for star is atāhk and the word for blanket is akohp, so starblanket is atāhkakohp. Chief Ahtahkakoop, so named because “the stars blanketed the sky, more numerous and brighter than usual” the night he was born, was one of the first signatories of Treaty Six. Ahtahkakoop Cree Nation is named after the chief.

As explained in more depth in that article, these beautiful quilts lend themselves to math explorations. For example, and this will be familiar to anyone who had Problem Solving with me back in 2021:  How many tiles are there in a Star Quilt? (I added that the answer should have n in it, although each person needed to specify what n represented.) I remember enjoying reading these solutions, and I think people liked coming up with them, too.

Monday Morning Math: Presidents’ Day edition

Today is the day we observe Washington’s Birthday, popularly known as Presidents’ Day, and that makes it a good day to look at the Pythagorean Theorem.  Specifically, a proof of the Pythagorean Theorem.  Specifically, the proof created by James A. Garfield, the 20th president of the United States.

The Pythagorean Theorem, often written as , says that squares on the legs of a right triangle, added together, have the same area as a square on the hypotenuse,   Garfield’s proof, published 5 years before he assumed the presidency, used a trapezoid.  Here’s a photo from the New-England Journal of Education on April 1, 1876

(from Mathematical Treasures)

Garfield’s proof compares the area of the trapezoid (the height times the average of the parallel sides) with the area of the three triangles that make up the trapezoid.  Garfield wrote out the details for you above, but feel free to try it using the common a b and c notation.

Monday Morning Math: Valentine’s Day

Happy Valentine’s Day everyone!  I thought I’d give you a valentine.  Maybe a cardioid like this:

By AtomicShoelace – Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=39636709

Or, if you don’t want to have to turn your head sideways, here’s a heart made out of Sierpinski Triangles:

Here’s how to make it

And here are interlocking Mobius Valentines!

And here’s how to make these!

Have a happy day!

Happy Pi Day 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

and enjoy.  Happy Pi Day!

If it’s Pi Day, that means…

…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

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:

• 
• 
• 
• The x-value for which is as large as possible.
• 
• 
• 
• (from OEIS A001113)
• (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, …]
  

(Any other good ones?)

The Mystery of the Fibonacci Pumpkin

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.

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

See it down there?

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].

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!

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

Happy 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.

Look 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

So you want to make a pie for Pi Day, but you don’t want to decorate it with the traditional symbol .  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

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

or volume , 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)

or the slightly more complicated

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:

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

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

A Pop-Up Sierpinski Valentine Card

Last 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.

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.

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

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.

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

Repeat, making four new rounded cuts.

And then tucking in the folds.

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

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:

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

Hanukkah 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  , , and is equal to the greatest common divisor of  , , and .  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.

For example, if and , this says that the greatest common divisor of  , , and is equal to the greatest common divisor of  , , and .  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.

Visually, 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”).