Monday Morning Math: Origami

May 2, 2022 by

Good morning!  This is the first Monday in May and the trees and flowers are in bloom.  It’s also finals week at Naz, which means it’s the last Monday Morning Math of the school year.  I hope you’ve had fun – we’ll pick it back up in September, probably starting the Monday after Labor Day!  Same bat time, same bat channel.

For this final MMM we’re using a reader suggestion – Origami (thanks Phyllis!)  Origami is a pretty neat subject [one that I think we’ll come back to next year too] because you can use it to make things like a flapping bird:

or a modular pinwheel:

Want something more geometrical?  Here’s a whole page that shows you how to make all the Platonic Solids (symmetric three dimensional polyhedron made out of a single regular polygon) AND all the Archimedean Solids (same as Platonic, but using more than one shape) AND some stars.

Want something more practical?  How about a giant space telescope made out of origami: 

Too big? What about medical implants?

Or do you just want to see some pictures of origami?

Here’s a fish:

Here’s some more fish, each made from a dollar bill:

Here’s a lilac spider, in honor of the upcoming Lilac Festival:

And, finally, here’s a pretty figure in case you didn’t want to end with a spider:

Have a good summer/winter everyone [depending of course on which hemisphere you’re in]!  My own plans include making even more origami – I got this book by Thomas Hull but haven’t had a chance to make things from it yet.  I hope your own plans are equally fun!

Monday Morning Math: Illusions

April 25, 2022 by

Good morning!  Today’s post is about illusions – specifically, the kind of illusions you can verify with MATH!  

I kind of think calling them “Illusions” might be giving some answers away, but even suspecting the answers it can take some convincing, possibly by holding up paper to your phone or computer.

Which center circle is bigger?

Would lines m and n meet if they were extended?

Which segment is longer, AB or CD?

Will line m, when extended, meet point A, point B, point C, or none of these points?

We’ve posted about illusions before (e.g. here and here) and if you want to see more you can go to the contest for the Illusions of the Year! They include photos and videos of the best ones, with some hints of how they work:

Enjoy!

Monday Morning Math: Omar Khayyam

April 18, 2022 by

Good morning!  Our mathematician today is Omar Khayyam.

Omar/Umar Khayyam was born in Nishapur, Persia, (modern day Iran) in 1048. Not much is known about his mother, but his father was a doctor who hired tutors to teach Omar.   Omar Khayyam is known for his mathematics, including writing down the laws of algebra that we know today.  He was able to make progress toward finding a general formula for ax^3+bx^2+cx+d=0 similar to the quadratic formula:  Greek mathematicians had come up with solutions to the quadratic formula that used a straightedge and compass, but Khayyam conjectured that it was not possible to solve the cubic equation with just those tools, and so developed other means of finding the solutions geometrically, using a parabola.  (It would be more accurate to say solutions to cubic equations: although we write it as a single equation, at that time the quadratic and cubic equations were written as several different cases depending on whether the coefficients were positive or negative.)  It was 500 years before anyone found a more general solution than his.

Omar Khayyam was one of the earliest people to describe the Arithmetic triangle (which is sometimes called Pascal’s triangle, although this was 500 years before Blaise Pascal).  He also contributed to the fields of non-Euclidean geometry and number theory.

In addition to mathematics, Khayyam wrote about astronomy, geography, and music.  He is largely remembered for his poetry, especially the rubaiyat (aka  Rubā‘iyyāt, or quatrains)

The Moving Finger writes, and, having writ,
Moves on: nor all thy Piety nor Wit
Shall lure it back to cancel half a Line,
Nor all thy Tears wash out a Word of it.

(Translated into English by Edward Fitzgerald)

Khayamm passed away in Nishapur in 1131, and a mausoleum stands over his tomb

Sources:

Monday Morning Math: Dorothy Lewis Bernstein

April 11, 2022 by

Happy birthday Dorothy!  Today is the 108th birthday of the first woman to be president of the Mathematical Association of America.

Dorothy Lewis Bernstein was born on April 11, 1916, in Chicago, Illinois.  Her mother, Tillie Loyev (changed to Lewis upon immigration to the US), was born in Ukraine and her father, Jacob Louis Bernstein, in Russia.  Her parents married in Milwaukee (where her mother’s family lived) in 1912 and then moved to Chicago where her father was a dairy farmer.  Dorothy was the oldest of six children and grew up in Chicago and Wisconsin.  

Dorothy earned her BA and MA in mathematics from the University of Wisconsin Madison [the alma mater of yours truly.  Go Badgers!] and her PhD from Brown University.  She worked at several universities, including Berkeley where work was being done to support the army during World War II.  She then spent 14 years just down the street from Naz at the University of Rochester(!!!).

While at Rochester she was asked by C. B. Tompkins, who was working at Engineering Research Associates on a contract from the Office of Naval Research, to undertake a study of the current state of knowledge of existence theorems in partial differential equations. As she explained in her 1978 AWM talk, “some of the proofs could be used as basis for the computational solutions of non-linear problems that were just being tackled by high-speed digital computers.” Her 1950 book with Princeton University Press was the result of this undertaking. 

Supplementary Material for Pioneering Women in American Mathematics: The Pre-1940 PhDs

In 1959 she began working at Goucher College in Maryland, where she stayed 21 years.  Dr. Bernstein was a prominent researcher, and was active in several organizations both before and after her retirement in 1979.  She was the vice president for the Mathematician Association of America in 1972-73 and president (the first woman elected to that position) 1978-80.  She passed away in 1988 at the age of 73.

Sources:

Monday Morning Math: Angles in a Triangle

April 4, 2022 by

Good morning!  I’m teaching Geometry this semester, so triangles are on my mind.  And here are some facts about triangles that, even though I know them, still blow me away.

I was taught that the angles in a triangle add up to 180 degrees.  This is true, at least in Euclidean Geometry, which is geometry done on a flat surface like a plane or piece of paper..(That link was to an illustration using folded paper but there are many proofs too.)

But what if you’re not on a flat surface?  What if you’re on a sphere?  Then things get weird.  “Lines” on a sphere are defined by the shortest path between two points (formally called geodesics), and it turns out that if you draw a line between two points and keep going, it will cut the sphere in half, like the equator of a globe, or a line of longitude.  And if you put three of those together you get a triangle…but the angles don’t add to 180 degrees.  In fact, you can have have a triangle with not one but two right angles, as shown below:

You can even have a triangle with three right angles!  Or three obtuse angles!  And overall, there isn’t a fixed amount that the angles add up to: it can be anything from just over 180 degrees all the way up to just under 540 degrees (which would be a triangle with three really big angles, covering a significant amount of the sphere).   So weird.

That’s not the only thing that’s weird.  If, instead of being on a sphere (which bows out), you are on a hyperbolic paraboloid, which bows in and looks like a saddle, then everything is opposite and the angles of a triangle are smaller than they would be on a flat surface.  You can still have triangles with one right angle or one obtuse angle, but the other two angles will be a bit smaller, and some triangles will just have three very small angles.  As in spherical geometry, the angles of a triangle don’t add up to any one fixed amount, but can be any positive number less than 180 degrees. 

If you like exploring, the (free) program Geogebra [that mimics drawing with a straightedge and compass]  has tools that let you draw in spherical geometry and hyperbolic geometry. Enjoy! 

Monday Morning Math:1+2+3+…

March 28, 2022 by

Good morning!  It’s snowing today, it was sunny a few weeks ago, and who knows what will happen next.  In this spirit of surprise, today we’ll look at 1+2+3+….   Any idea what it is if you keep adding?  You might think you’d approach infinity, but actually….

….well, actually that makes sense.  But then this would be mighty short, so instead we’ll prove that the sum is -1/12.  This is called the Ramanujan Summation after the mathematician Srinivasa Ramanujan who was born in 1887 and who passed away in 1920.  

So, let’s get proving!  We’ll do this in parts:

Step 1:  Prove that 1-1+1-1+1-1+-… adds to 1/2.

We’ll call the sum of this sequence A, and do some fancy algebra:
Since A=1-1+1-1…. then if we subtract 1 (the first 1 on the right) we get:
A-1=-1+1-1+1…., which is the negative of what we started with.  That means  (A-1)=-A, so (2A-1)=0, and that means A=1/2.  All done!

This all assumes that we can treat infinite sums the same way as finite sums.  YMMV.

Step 2:  Prove that 1-2+3-4+5-6+…. adds to 1/4.

We’ll call the sum of this sequence B, and keep going with the fancy algebra.
Since B=1-2+3-4+5-6+….  let’s look at A-B
A-B=(1-1+1-1+1-1+…)-(1-2+3-4+5-6+…)   
Let’s reorder, putting the first terms together, the second terms, etc.
This is (1-1)+(-1+2)+(1-3)+(-1+4)+(1-5)+(-1+6)+…, which simplifies to
0+1-2+3-4+5.  And that’s just B!

So A-B=B, which means A=2B, so B is half of A, and therefore 1/4.

Step 3: .Prove that 1+2+3+4+5+6+… adds to -1/12.

We’ll call this sequence C.
Since C=1+2+3+4+5+6+…, let’s look at B-C
B-C=(1-2+3-4+5-6+…)-(1+2+3+4+5+6+)
Like we did in Step 2, we’ll reorder, putting the first terms together, the second terms, etc.
This is (1-1)+(-2-2)+(3-3)+(-4-4)+(5-5)+(-6-6)+…, which simplifies to:
0-4+0-8+0-12+…., which is -4-8-12-…
You can factor out a -4, and get -4(1+2+3+…), and that’s -4C!

So B-C=-4C, giving B=-3C, so C is (-1/3) of B, or (-1/3) of (1/4) and that, my friends, is -1/12!

What do you think?  If you think it makes sense, you’re in luck – there are some deep results in physics that use these ideas (although they are proved using something called the Riemann zeta function).   On the other hand, if you think there was some mathematical sleight of hand, well, you’re right also.  Treating infinite series like they are finite makes sense until it doesn’t, like adding up a bunch of positive integers and getting -1/12.

This subject was inspired by a reference in The Art of Logic in an Illogical World by Eugenia Cheng, and is also on a Numberphile video.  I used a post on Cantor’s Paradise for the notation, and Scientific American for additional background.

Monday Morning Math: The L’Hôpitals

March 21, 2022 by

Good morning! The math tidbit for today is a two-for-one special: the L’Hôpitals, who lived in France in the 1600s.  But we’ll start with the phrase that my brain jumps to when I see L’Hôpital, which is L’Hôpital’s Rule (also called L’Hospital’s Rule).  It’s about limits, so it shows up in Calculus.  Essentially, if you have a limit as x\to a of a fraction where both the numerator and denominator individually are approaching 0 or where both the numerator and denominator individually are approaching \pm\infty, then:

\lim_{x\to a} \frac{f(x)}{g(x)}=\lim_{x\to a} \frac{f'(x)}{g'(x)}

provided that second limit actually exists.  So, for example,

\lim_{x\to 0} \frac{\sin(x)}{x}=\lim_{x\to 0} \frac{\cos(x)}{1}=1.

Huzzah!

This rule was named after Guillaume François Antoine de l’Hôpital, where that last name was spelled different ways even in his lifetime, even by him.  He learned calculus by correspondence with Johann Bernoulli, who was busy figuring out calculus himself at that time, since the subject was only a few decades old.  In 1696 l’Hôpital published what is considered to be the first Calculus book: Analyse des infiniment petits pour l’intelligence des lignes courbes.  He thanked several people in the introduction, including Johann Bernoulli, but it wasn’t clear at that time how much of the book was really after l’Hôpital’s own work (none?) and how much was based on Bernoulli’s notes (all?).  Apparently Bernoulli was fine with l’Hôpital publishing the book, possibly because of the money l’Hôpital paid him, possibly because he was happy just to have these still-new ideas disseminated. L’Hôpital died in 1704 when he was about 43 years old.

A lot of that information  comes from the MacTutor biography, which also states, “L’Hôpital married Marie-Charlotte de Romilley de La Chesnelaye; they had one son and three daughters.”   The English Wikipedia page adds that his wife was “also a mathematician and a member of the nobility, and inheritor of large estates in Brittany” with a link to a page for her – in French – from a French biography that indicates that she worked in Geometry and Algebra and lived from 1671-1737.  The only other site online that mentions her is this dictionary, also in French, which indicates that she helped with the printing of the aforementioned Calculus book and impressed another math professor, Monsieur de la Montre, with her knowledge of Euclid.  So was she too involved in the creation of that first Calculus book but not mentioned?  It sounds like it, though the extent of her involvement is unclear.  

And we’ll leave on that uncertain note.  Any new information would of course be welcome.

References:

Monday Morning Math: Pi Day

March 14, 2022 by

Happy Pi Day everyone!  It’s Spring Break here, but it’s Pi Day, so worthy of celebration!  Here’s 7 math jokes about pi, because π ≈ 22/7

#7
What did pi say to its sweetheart?
You look radian today.

#6
Why did pi fail its driving test?
Because it didn’t know when to stop.

#5
What do you get when you divide the circumference of the moon by its diameter?
Pi in the sky

#4
How many bakers does it take to make a pie?
3.14

#3
What is the most mathematical kind of snake?
A pi-thon

#2
What do you need for dessert seafood?
Octo-pi

#1

Monday Morning Math: Maryam Mirzakhani

March 7, 2022 by

It’s Women’s History Month, and we’ll celebrate with the first (and so far only) woman to receive the Fields Medal in Mathematics – an award given every four years in recognition of outstanding mathematics. 

Maryam Mirzakhani was born in Tehran, Iran, in 1977.  She loved reading novels and writing as a child, and became interested in mathematics when her brother would come home from school and tell her what he learned – her earliest memory of math specifically was when he told her the story of a young Carl Friedrich Gauss being asked to add the numbers from 1 to 100 and surprising his teacher by seeing a pattern that allowed him to do the computation in moments.

Maryam Mirzakhani went on to participate in (and win) the Math Olympiads as a teen; she then went to Sharif University of Technology in Iran and then Harvard University in the United States, where she earned her doctorate.  

In 2014 Dr. Mirzakhani was awarded the Fields Medal. Her research is sometimes described as a complicated version of billiards, where a ball bounces along the edges of a table.  “You want to see the trajectory of the ball,” Dr. Mirzakhani explained in a video produced by the Simons Foundation and the International Mathematical Union. “Would it cover all your billiard table? Can you find closed billiards paths? And interestingly enough, this is an open question in general.” You can see a short (under 3 minute) video of Dr. Mirzakhani explaining her work here:

Mirzakahni passed away in 2017 from breast cancer.  She was 40 years old.

The 2022 Fields Medals will be given out at the  International Congress of Mathematicians this summer.  The conference was originally scheduled to be held in St. Petersburg, Russia, but due to the Russian invasion of Ukraine the organizers have announced that the conference will be held virtually, without any contributions from the Russian Government, and will be free for everyone who would like to attend.  

Sources:

Monday Morning Math: Voronoi Diagrams and Georg Voronoy

February 28, 2022 by

Good morning!  Today’s math is Voronoi Diagrams, but I’ll lead into it with some Geometry.  Suppose you have two points A and B, and wonder which points on the plane are closer to A and which are closer to B.  Maybe A and B are schools, and this is for figuring out districts, or maybe they are pizza places and you’re wondering where to order from.  It turns out that the border between the two regions is on the perpendicular bisector of A and B.

But what if there are three points? It’s a little more complicated, but the boundaries are still made up from the perpendicular bisectors of the different pairs of points.  And this idea continues even when there are more points, as this picture shows.

Diagrams like these are called Voronoi Diagrams named after Ukrainian mathematician Georgy Feodosevich Voronoy (also written as Georgii Voronoi).

Wikimedia Commons

He was born in the village of Zhuravka in the north central part of Ukraine on April 28, 1868, and while he was still in the equivalent of high school he solved and published the result of a problem in algebra.  He then went to the University of St. Petersburg in Russia, first as an undergraduate but eventually as a doctoral student under Andrey Markov (himself well known because of something called Markov Chains, which are ways of calculating probabilities).   Both his Master’s thesis and his doctoral thesis were awarded the Bunyakovsky prize for outstanding work in mathematics by the St Petersburg Academy of Sciences.

Voronoy became a professor at the University of Warsaw in Poland, where he continued to do research and also supervise students: one of his students was Wacław Sierpiński, for whom Sierpinski triangles are named.   When he was only 40 years old he developed severe gallstones, and passed away on November 20, 1908.

Sources:

Monday Morning Math: Presidents’ Day edition

February 21, 2022 by

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 a^2+b^2=c^2, 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

February 14, 2022 by

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:

image.png

Here’s how to make it

And here are interlocking Mobius Valentines!

image.png

And here’s how to make these!

Have a happy day!

Monday Morning Math: Gloria Ford Gilmer

February 7, 2022 by

Our mathematician today is Dr. Gloria Ford Gilmer, a pioneer in ethnomathematics (the study of the relationships between mathematics and culture).  Gloria Ford Gilmer was born in Baltimore Maryland, in 1928.  She studied mathematics at Morgan State University in Baltimore, where she was a student of Clarence Stephens, and where she earned her bachelor’s degree in 1949.  Two years later she earned her master’s degree in math at the University of Pennsylvania.

She did ballistics research for the US Army, but soon turned to teaching.  She taught both high school and college students, eventually earning a PhD in curriculum and instruction at Marquette University.  Most of Dr. Gilmer’s research was in ethnomathematics.  She was particularly interested in finding mathematics in everyday places, and is known for her mathematical analyses of the braiding patterns in African American women’s hair. 

Dr. Gilmer was active in many professional organizations, and was a “first” for many of them. She was the first African American woman on the board of governors for the Mathematical Association of America, and the first woman to give the Cox-Talbot Address for the National Association of Mathematicians. In 1985 she, along with Ubiratan D’Ambrosio, Gil Cuevas and Rick Scott, co-founded the International Study Group on Ethnomathematics (ISGEm); she served as the organization’s president for 11 years. 

Dr. Gilmer passed away only a few months ago, on August 25, 2021.  The recently established American Mathematical Society’s Claytor-Gilmer Fellowship is named in her honor.

No picture here because I couldn’t find one without copyright restrictions, but you can see one on the site Mathematically Gifted & Black, where she was an honoree last year.  Every day in the month of February the site recognizes a mathematician – check out the 2022 honorees!

Sources:

Monday Morning Math: Pythagoras

January 31, 2022 by

This week’s mathematician is someone you have probably heard about, although it turns out very little is known for certain. 

The mathematician is Pythagoras.  

Pythagoras was probably born in Greece, on the island of Samos, over 2500 years ago (570 BCE, plus or minus a few decades).  His mother was from that island, and his father Mnesarchus was a merchant.  Pythagoras would travel with him sometimes when he was a child, and when he was an adult he studied mathematics with Thales, another now-famous mathematician.   

After many years (decades) of study, Pythagoras formed a group known as the Pythagoreans, who followed a strict vegan diet and believed that everything was essentially a number.  There was an inner circle of mathematicians and an outer circle, and there is some indication that the groups were equally welcoming to women and men. It is not possible to distinguish who proved any one result (because secrecy was the name of the game rather than publishing), but there were important results from this group related to music and geometry.  

Two of the most significant results attributed to Pythagoras are the Pythagorean Theorem (written in geometric terms that a square on the hypotenuse of a right triangle has the same area as the sum of squares on each of the two legs) and that the square root of 2 is irrational (written in geometric terms that the side and the diagonal of a square are incommensurable, meaning there’s no teeny tiny amount that fits into both an integer number of times.  There are stories that someone figured this out and was killed, either for figuring this out or for telling people outside the Pythagoreans, but like the rest of what I’ve written this was a story – everything we supposedly know about Pythagoras comes from reports well after his death, so at best this is educated guesswork.

Pyth.jpg
Etching on the wall of Peckham Hall, our math and science building. Not seen is the QED in the lower right corner, as it was also in darkness when I took the photo just now

Finally, the video below, from BBC learning, is under 5 minutes and fun to watch, though it does start off with an excited “Pythagoras!” that will get the attention of anyone around you.

Other sources:

Monday Morning Math: 24

January 24, 2022 by

Instead of a mathematician today, we’ll look at the number 24.  (Thanks to Kevin Laley for the suggestion to revisit  Adam Spencer’s Book of Numbers: A bizarre and hilarious journey from 1 to 100!  Wolfram Alpha was also a nice resource.)

  • 24 =4! 
  • 24 can be written as 2^2+2^2+4^2, which is the same thing as 2^2+2^2+2^4 and also 2^2+2^2+2^{2^2}
  • 24 is the smallest number that can be written as a sum of 2 primes in three different ways.  (Can you find them?)
  • It is possible to draw a regular icositetragon (that’s a 24-gon and yes I had to look it up) using a straightedge and compass.  I don’t think it would be too hard, either – I’d start with a regular hexagon, which you can draw using equilateral triangles, and which happens to be part of the homework assignment for the Geometry students this past weekend.
  • There are 24 hours in a day.  This apparently comes to the United States from a path starting with Egypt (via Babylon, where the hour was then divided into 60 minutes) although China also had 24 hours in a day. 
  • There are 24 letters in the Greek alphabet, alpha to omega.  Omicron is the 15th letter, in case you were wondering.
  • Pure gold is 24 karat.   If you want to make fancy desserts like on The Great British Baking Show then you want  your gold to be 22-24 karats.