Monday Afternoon Math: Gerrymandering

December 8, 2025 by

Good afternoon! Gerrymandering was in the news this week, when the Supreme Court declared that gerrymandering for political purposes is totally fine. So this seems a good moment to give the briefest of introductions to gerrymandering. The idea behind it is that districts can be divided up in such a way as to maximize the chances of one party winning. There are sometimes some rules, like that regions have to follow natural boundaries, but the name itself comes from an 1812 cartoon (probably by Elkanah Tisdale) making fun of a redistricting in Massachusetts that led to a salamander-like voting district. The bill allowing the district had been signed by Governor Elbridge Gerry

Gerrymandered districts have a reputation for having strange shapes, as in the cartoon above, but even with more regular shapes there can be a lot of maneuvering that can be done. For example, suppose you have this area and you want to divide it into 5 districts, with the idea that each district is connected, but you happen to know that people living in the gold areas will reliably vote Gold and people living in the Purple areas will reliably vote Purple.

You could split it into districts horizontally, ending up with 2 Gold reps and 3 Purple reps. This is called packing, because all the districts of one color are packed together, although it happens equally between the two parties.

But you could instead split it into vertical districts, ending up with 5 Purple reps. This is called cracking, because the yellow districts are split up so that they don’t end up with any power.

It’s also possible to split it up so that there are 3 Gold reps. This is also cracking.

From a mathematical perspective it can be difficult to determine if a district is gerrymandered for a particular purpose, unless the people announce it. But if people did want to avoid gerrymandering for any purpose, that is something that math can help with. There are articles like “How Math Has Changed the Shape of Gerrymandering” by Mike Orcutt from Quanta Magazine, and sites like the Institute for Mathematics and Democracy, as well as “The (very) tricky math of detecting gerrymandering in election districts” by Keith Devlin with Ellen Veomett

This will be the last Monday (sometime) Math until late January. I hope that everyone has a peaceful end of 2025!

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Monday Morning Math: Nikolai Ivanovich Lobachevsky

December 1, 2025 by

Good morning! Today marks the first day of the twelfth month of the year (or, as the name suggests, the tenth month of the year, but that’s only if you begin your year in March like an Ancient Roman). It’s also the birthday of Nikolai Ivanovich Lobachevsky, who was born in Russia in 1792.

Nikolai’s father, a clerk, died when Nikolai was seven, so Nikolai, his two brothers, and his mom moved to Kazan. When Nikolai was approaching his teenage years the Russian Emperor Paul I was murdered, which meant his son Alexander became Emperor, and Alexander was big into Educational reforms. He founded several universities, and THAT meant that when Nikolai finished high school, there was one right in town that he could and did attend. He’d planned to become a doctor, but ended up becoming interested in math instead, earning a bachelor’s degree, a master’s degree, becoming a professor, and eventually becoming rector, all at Kazan University. Along the way he married Lady Varvara Alexejevna Moisieva and they had a bunch of kids (eighteen, according to one of his sons, although sadly most did not live to adulthood).

Nikolai liked geometry, and one of the questions of the time was about whether or not it was possible to prove Euclid’s fifth postulate. That’s the one that says, essentially, that if you have a line, and then a point that isn’t on the line, that there is exactly one parallel line that goes through that point, but some people wondered if it would actually happen automatically because of the other postulates, or if it really was something that would have to be assumed. People did already know that on a sphere there weren’t any parallel lines (where lines turn out to be Great Circles that cut the sphere in half), but spherical geometry violated several of Euclid’s postulates so that wasn’t itself a proof.

Rather than try and prove it, Nikolai Lobachevsky decided to assume that the fifth postulate didn’t hold, and instead developed a geometry where there could be multiple parallel lines through that point. This geometry, which is often called hyperbolic geometry because one way of describing it uses a hyperbolic paraboloid, turns out to be a perfectly good geometry, which meant that Euclid’s fifth postulate was indeed a postulate and not something that would happen automatically.

(Image in the public domain from Wikipedia)

Unfortunately for Nikolai, his geometry was not accepted right away. Also unfortunately, he was in poor health when he retired, and died in 1856 in poverty. Triply unfortunately, his name is often associated with a catchy song by Tom Lehrer (based on Danny Kaye’s “Stanislavsky”) about a person who learned the value of plagiarism, although the use of “Nikolai Ivanovich Lobachevsky” in the refrain was chosen because of its meter and not because of any concern about plagiarism with the man itself. But in good news, the geometry he explored was not only eventually accepted but lauded, and it is not unusual to hear it referred to as Lobachevskian geometry.

Sources: Mactutor and Wikipedia and more Wikipedia

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Monday Morning Math: The first US PhD in mathematics

November 24, 2025 by

Good morning!  ‘Tis the season to scurryfunge*, and so this entry will be short.  With Thanksgiving happening this Thursday, it got me thinking about the First Thanksgiving in the region that became the United States, which got me wondering about the First PhD in Mathemathematics in the US. It is surprisingly recent, if you consider the early 1860s to be recent.  On the other hand, PhDs themselves are almost that recent in the United States: Yale was the first university here to offer a PhD.  The program took two years, with the first PhDs being granted to Eugene Schuyler, Arthur W. Wright, and James Whiton in 1861.  The areas were…well, that’s not clear.  The PhDs weren’t assigned to any particular field, so the claim of what areas they were in — Wikipedia says  philosophy & psychology, physics, and classes (respectively) — is just guesswork based upon their future careers.  The physics PhD in particular could well be considered to be a mathematics PhD.  Arthur Wright’s dissertation was “Having Given the Velocity and Direction of Motion of a Meteor on Entering the Atmosphere of the Earth, to Determine its Orbit about the Sun, Taking into Account the Attractions of Both These Bodies” (gotta love those long titles!), which sounds like it could be either Physics or Math.  He went on to a career in Physics (hence the claim that that is what his degree was in), but disciplines weren’t always clearly delineated, and his advisor was most likely Hubert Newton, who was in fact a mathematician.  

A year later, in 1862, a PhD was awarded at Yale to John Hunter Worrall. The title of his thesis is unknown, at least by me, but he did go on to be a math teacher, and many sites list him as the first person to receive a PhD in math from a school in the US.  Still, the case for Wright is interesting, and there’s a article by Steve Batterson about it in the March 2008 issue of the AMS Notices, so we’ll consider both Arthur Wright and John Hunter Worrall  as the first, each with an asterisk*** as appropriate.

I hope you all have a wonderful Thanksgiving!  🍂   

*According to Susie Dent, who is an actual lexicographer**, scurryfunge is an old word that means to rush around and clean up right before guests arrive.  But I can’t track down an early citation – it’s not in Merriam Webster, and the Oxford English Dictionary doesn’t define it that way – the earliest quote with scurrifunge,  in 1789, just means to scrub.  Nonetheless, scurryfunge/scurrifunge is a great word, and having a bit of a moment.

**If you had to look this up, you should thank a lexicographer!  Lexicographers are the people who put dictionaries together.

*** The word asterisk comes from the greek asteriskos/ αστερίσκος , meaning “little star”. So says Merriam Webster.

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Monday Morning Math: Bernoulli

November 17, 2025 by

The other day, TwoPi shared the following meme with me:

I laughed, but the truth is, I mix up the Bernoullis all the time. So here’s a quick primer of the first few generations for anyone who, like me, has trouble keeping track. There’s also a whole family tree from Wikipedia, from which I am adopting the spelling and pretty much all the info I’m writing down here.

  • Jakob and Johann – the original brothers! They were born in the mid 1600s. Older brother Jakob wrote the book known in English as The Art of Conjecture and there’s a bunch of stuff, including a series of numbers (called Bernoulli numbers), named after him. Younger brother Johann was a big fan of then-new Calculus, and also taught then-young Leonhard Euler.

    There were other siblings, too – this is not a complete family tree.
  • The Original Brothers both had kids named Nikolaus (after Jakob and Johann’s dad) who did mathy things, and younger brother Johann had two more sons who studied math too – Daniel and Johann II. Of this batch of brother-cousins, it is Daniel who appears to be most prominent. Bernoulli’s principle (related to the Law of Conservation of Energy) is named after Daniel. These kids were all born more or less around 1700.
  • Johann II – the son of younger brother Johann – also had a bunch of kids who also studied math, generally born in the mid 1700s. Their names are Johann III, Daniel II, Nikolaus III, and Jakob II. THIS is why it is so hard to keep track of all the Bernoullis. I suspect they weren’t all going around signing their names with II and III either.

Those are the main initial generations, but the family tree kept growing and, fortunately, introduced a few new names, like some Leonhards and Carls. I hoped there would be a book entitled The Bernoulli Women but, alas, a search only turned up wristwatches for women from the Bernoulli company. Named after Daniel I.

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Monday Morning Math: Happy Birthday dy/dx!

November 10, 2025 by

Tomorrow is the 350th birthdy of the symbol dy/dx! At the least, in the sense that it was first written down. And actually it was dx/dy, but still worthy of celebration.

Gottfried Leibniz was living in Paris at this time, studying quadratures, which is a kind of fancy term for areas, and writing them down in a manuscript that was eventually published. His use of dx and dy was a bit of a progression – he’d been starting to use similar notation – and the use of dx/dy at first literally meant a ratio. This means that it might be more accurate to say that the beginning of this notation was more of a time period than a single date, but, still, if you want to pick a moment, this is a reasonable one.

My favorite part of this story isn’t just the notation itself, which remains in use 350 years later, but how it was first used. He writes, essentially, “Let’s see whether dxdy is the same as d(xy), and whether dx/dy is the same as d(x/y).” In other words, he’s using the notation as he examines whether taking the derivative of a product or quotient is as straightforward as taking the derivative of a sum or difference.
[Answer: No.]

So have a slice of cake and a slice of calculus in celebration of a pretty good notation!

Sources: A History of Mathematical Notations by Florian Cajori (Section 570 in Volume II), although I did also look at a 1920 translation of Leibniz’s manuscripts, which you can read here.

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Monday Morning Math: The Scottish Book

November 3, 2025 by

About 90 years ago a group of mathematics would gather and talk about mathematics at a coffeehouse called The Scottish Café which, despite its name, was not in Scotland but in Lwów, Poland, which, despite its name, is not in what is now Poland, but what is now Ukraine.

These mathematicians would sometimes write in pencil on the marble tabletops, but those would be cleaned each day so eventually – it’s thought to be either Łucja or Stefan Banach – suggested getting something to write down the ideas. Stefan Banach, who was one of the mathematicians who met there, bought a big notebook and left it with the headwater at the café, and then the math people would request it when they wanted to write down a new problem or add some notes. This notebook became known as The Scottish Book.

As a book of problems, it only lasted a few years: the late 1930s were not a good time in that part of the world, and the 193rd and final entry in the original book was from May 31, 1941 (a problem about matches in a box). But the lore lived on. One of the problems, Number 153, became particularly well known because Stanisław Mazur offered a live goose to whoever would solve it. In 1972 the Swedish mathematician Per Enflo solved the problem and Mazur did in fact give him a goose.

Public domain photo from the goose-giving ceremony

Not all problems came with poultry: Terence Tao solved one of the problems in 2017 and received a jar of honey mead, and in 2021 when Dmitry Ryabogin solved Number 19 — a question from Stan Ulam wondering if “a solid of uniform density which will float in water in every position” has to be a sphere – by showing that it doesn’t, he didn’t get any farm products at all. He did, however, get the President’s Excellence Award from Kent State.

Sources: MacTutor and Wikipedia and a post by Terence Tao and an article at Kent State.

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Monday Morning Math: Marie-Hélène (Lévy) Schwartz

October 27, 2025 by

Today’s Monday Morning Math celebrates  Marie-Hélène (Lévy) Schwartz on what would be her 112th birthday.  She was born on October 27, 1913, in Paris, France, and her dad Paul Lévy (and his dad, and various other relatives) was a mathematician.  She went to high school at the lycée Janson de Sailly, where she met  Laurent Schwartz, who eventually became her husband (and eventually became a Fields Medalist as well – he, too, was a mathematician). They became engaged in 1935, but before they could get married, Marie-Hélène caem down with pulmonary tuberculosis and had to go to a sanatorium for rest and recovery.  

She was there for three years.

In happy personal news, when she got out she and Laurence were able to get married, but in unhappy personal news, this was France in 1938, a rough place to be for anyone, and the couple being Jewish and also Trotskyist (followers of the political ideology of Leon Trotsky) didn’t make them any more safe.  Laurence fought in the war for several years, and both survived.

One war and two children later, Marie-Hélène finished her thesis (Formules apparentées à celles de Gauss-Bonnet et de Nevanlinna-Ahlfors pour certaines applications d’une variété à n dimensions dans une autre — Formulas related to those of Gauss-Bonnet and Nevanlinna-Ahlfors for certain applications of one n-dimensional manifold in another) and began teaching and the University of Paris, then the University of Reims Champagne-Ardenne, and eventually and for the longest time at the Faculté des Sciences de l’Université de Lille.    She did research for many years, significant enough that a conference was held in her honor in 1986,  and lived to be nearly 100 years old, passing away on January 5, 2013.  Quoting Jean-Paul from MacTutor (the source of this biography)

From the study of the functions of a complex variable to the characteristic classes of singular varieties, Marie-Hélène Schwartz’s mathematical journey has followed a well-defined route, braving all the difficulties encountered along the way. This presentation is not intended to write all of Marie-Hélène Schwartz’s work but to show how her results follow this route. We can in fact distinguish in her mathematical journey four periods whose themes successively cover the functions of a complex variable, Ahlfors theory, the Poincaré-Hopf theorem for singular varieties and radial fields, and finally the characteristic classes of singular varieties.

[This quote J-P Brasselet, Hommage à Marie-Hélène Schwartz: un aspect de l’oeuvre mathématique de M-H Schwartz, Gazette des Mathematiciens 138 (2014), 61-71.]

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Monday Morning Math: Square Packing

October 13, 2025 by

Good morning! Two weeks ago I wrote about Kathleen Timpson Ollerenshaw. The work that she did for her doctorate was related to packing. As she writes in her autobiography (quoted on MacTutor):

Critical lattices relate to whole numbers in two or more dimensions and lead, by geometrical methods, to solutions concerned with ‘close packing’, for example, how best to stack tins in a cupboard or oranges in a box.

Today’s problem is a spin off from this. If you have some unit squares, and want to put them in a larger square, how big does the larger square have to be? Some answers are straightforward: If you have 1 unit square it fits in a 1×1 square. If you have 4 unit squares, they fit in a 2×2 square. But what if you only have 2 or 3 unit squares? Can they fit into something smaller than a 2×2 square?

It turns out they can’t. Even just 2 unit squares take up too much space if you rotate them around, so you’re stuck with a 2×2 square. But that rotating does help if you have 5 unit squares (image below in the public domain by Amit6).

The 5 unit squares can fit into a square that is about 2.7 units on each side — technically it’s 2+(√2)/2. Woo hoo! But 6 squares isn’t so efficient – it fits into a 3×3 square with room left over, and nothing smaller will do. The same goes for 7 or 8 squares, and then 9 squares fit perfectly into that 3×3 with no room for anything else.

With these small numbers I already find it interesting that it’s actually pretty unusual that you can rotate one and fit it into a smaller square.

Based on this pattern, though, you might guess that if you have 10 squares then you can do some rotating to get something smaller than jumping all the way to 4×4, and you’d be right, as a creative commons picture adapted by Maksim shows. These 10 blocks fit into a square that’s only about 3.7 on each side — technically it’s 3+(√2)/2.

We’ll try one more. What about 11 squares? Can those fit into something smaller than a 4×4 square?

Good question. Very good question. The answer is yes, but exactly how much smaller is still unknown. Walter Trump, a German high school teacher, found an efficient packing for 11 squares back in 1979, where they fit into a square that was only 3.87 on each side as seen in his creative commons picture, but there’s a chance that it could be smaller, closer to 3.79. Looking at Wikipedia and Wolfram MathWorld, not only are there results from the past 25 years, making this very much a current project, but overall there are papers published by well known mathematicians (Fan Chung, Paul Erdős, and Ronald Graham) but also results published by not-so-well known mathematicians: Walter Trump mentioned above, and John Bidwell, who found a solution for packing 17 squares into a square of side 4.67 while he was an undergraduate by rotating different amounts.

Happy Monday!

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Monday Morning Math: Unknotting!

October 6, 2025 by

Good morning! Today’s idea comes from alumna CP, who shared the news that a conjecture in Knot Theory has been solved, and by “solved” I mean “disproven” because in math it’s just as exciting to prove that a conjecture is false as to prove that it’s true. Sometimes it’s even more so.

The conjecture has to do with something called the Unknotting Number of a knot. If you have a knot, the Unknotting Number is the number of switches that you have to make in order to be able to untangle the knot into a loop. For example, this diagram below (by Hyacinth, in the public domain) shows a knot — technically the trefoil knot — with a Unknotting Number of 1, because it takes just one switch to be able to untangle it.

Fun Fact #1: Do you see how there were 3 crossings in the original knot, so 3 places that could have been switched? The Crossing Number is the smallest number of crossing that a knot has, and it turns out that the the Unknotting Number is always less than half the Crossing Number (according to Wikipedia).

Fun Fact #2: If you combine two knots, then the Unknotting Number of the combination can’t be any bigger than the sum of the Unknotting Numbers of the two knots. So this picture below (public domain) shows two knots, which each have an Unknotting Numbers of 1

and if you add them, the new knot (CC by SA 3.0) has to have an Unknotting Number of at most 2.

Fun Fact #3: But it might not be exactly 2! I mean, this one is, but for the last 90ish years or so, it’s been thought that if you combine knots then the Unknotting Number will be exactly equal to the sum of the two original knots. And people REALLY thought this was probably true. It seemed to be.

But then mathematicians Susan Hermiller and Mark Brittenham rigged up a bunch of computers to try and prove it, and that didn’t work after 10+ years and one small fire, and then they tried to disprove it, and lo and behold that DID work! A knot and its mirror image, which each had an Unknotting Number of 3, combined to give a knot with an Unknotting Number of 5 rather than 6.

The story of how they solved it is written up in “A Simple Way To Measure Knots Has Come Unraveled” in Quanta. If you’re more of a visual learer, then you can watch Matt Parker demonstrate it below!

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Monday Morning Math: Kathleen Timpson Ollerenshaw

September 29, 2025 by

Good morning! It’s a beautiful morning here, and starting to really look like fall.  No surprise that October begins this week.

October 1 is also the 103rd birthday of one Kathleen Timpson Ollerenshaw, a mathematician that I had not heard about until recently, and the more I learned about her the more intrigued I was.

Kathleen Timpson was born near Manchester, England, on October 1, 1922, and had one older sister, Betty. Kathleen was born with otosclerosis, which can impact bone growth in the ear and often leads to hearing loss in and of itself, and when she was 8 years old she became almost completely deaf after an illness. The good news for her is that the University of Manchester nearby had a program that trained teachers to work with students who were deaf, and so Kathleen learned how to lip read [which, at the time, was viewed as the best method for communicating with the most people].

She was very good at math from a young age, despite pushback from her high school where they didn’t think that she needed to study so much because, as a woman, the only reason to lean math was to teach and, as a deaf person, they didn’t think she could teach anyway. Her response was to threaten to leave the school and then, when finally allowed to take math classes, to ace them.

After high school she went to Oxford (where, during the interview, she never mentioned being deaf in order to avoid prejudice against her perceived abilities). She graduated from Oxford in 1933 and for the next several years worked as a secretary (covering for her sister when Betty had a child) and hanging out with the family of her then-fiancé Robert Ollerenshaw, a classmate from elementary school who had become a friend and partner.

In 1936 Kathleen began working at the Shirley Institute in Manchester, which was a research center for cotton production. Her job as a (self-taught) statistician there was to design waterproof material that could be used by the army:

It was moreover a matter of geometry — pure mathematics — a nice problem that had a neat and successful solution. The requirement was that rain falling on a tent or coat should run directly downward and not soak through the woven fabric. It fell to me to devise a weaving pattern so that this could be achieved with cotton. (From her autobiography, as quoted in MacTutor.)

Kathleen continued to work at the Shirley Institute for several years, stopping only after the birth of her first child. Not long after, she wrote a paper solving a problem about lattices (related to packing), which turned into several more papers, which turned into a doctorate. She taught, she promoted education, she got involved with politics (later serving as Lord Mayor of Manchester), she found a general solution to the Rubik’s cube, she published papers on magic squares (which are grids a bit like a Sudoku board, where the rows, columns, and sometimes diagonals add to the same amount), she served as founding member and later president of the Institute of Mathematics and its Applications. And along the way she became a Dame Commander, which is like a knighthood.

Kathleen passed away on August 10, 2014, near Manchester, England.

The 2007 photo below by Sim0n (Creative Commons) shows Dame Kathleen at the Manchester Astronomical Society, where she was an honorary member, because the biography above only touches the surface of all that she did!

Sources: MacTutor, Wikipedia, The Guardian, and The University of Manchester. Ideally her autobiography “To Talk of Many Things: an autobiography” would have been one of the sources too, but I haven’t (yet) read it.

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Monday Morning Math: Fractions

September 22, 2025 by

This past weekend was our reuntion weekend, and at our department open house we ended up looking for neat math patterns in the dates of a few upcoming weddings. We ended up with (prime)/(cube)/(prime) for one, and (twice a prime)/(prime)/(twice a prime) for another, both of which had a nice palendromic feel. Notice that we use “/” to separate the dates.  That’s also a symbol that we use for fractions, which might make you wonder about how different people represented fractions!

  • 4000 years ago in Mesopotamians (now Iraq): People grouped numbers by 60s, so something like 1 with a small space and then another 1 could mean 1 (60) +1=61, but it could also mean 1(3600)+1(60)=3660.  It could also go in the other direction and mean 1 and 1/60, or even 1/60+1/3600!  So they were using a kind of decimal system (but with 60s instead of 10s), but they didn’t actually tell you where the decimal point was.  A similar method was used in China about 1500 years ago, although that method was grouping by the more-familiar-to-us tens instead of 60s and without the ambiguity of not knowing where the decimal point was. 
  • 3500 years ago in Egypt: People drew an oval over a number to indicate its reciprocal, so a 3 with an oval would be 1/3.    This meant that all their symbols — except for 2/3, which had a special symbol — were things like 1/2, 1/3, 1/4, 1/5, etc.  If they had a different kind of fraction, like 2/5, they would break it down into smaller pieces, so 2/5 would be written as 1/3+1/15. 
  • 1500 years ago in India: People wrote fractions as two numbers in a column, so 2/5 would be a 2 over a 5, but without the bar that we use today.
  • 800 years ago in Morocco: The mathematician Abu Bakr Muhammad ibn Abdallah ibn Ayyash al-Hassar added a horizontal fraction bar! He seems to have been the first to use the bar, and not long after, Fibonacci used this notation.  Interestingly, after the printing press began to be used in Europe, the fraction bar was dropped again for a while. 
  • 300 years ago in England: The diagonal line line for a fraction like 1/4 appears in the ledger book of Thomas Twining (he of Twining Tea), though he probably wasn’t the originator.

(Sources: MacTutor and more MacTutor.)

Those last three are what lead to fractions as we in the US write them today, although of course we also can use a decimal point similar to the first example.   I confess I like the idea that the diagonal slash is only a few hundred years old, and so relatively modern. 

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Monday Morning Math:

September 15, 2025 by

ello everyone! The fall semester is well underway, and it’s time for Monday Morning Math!  I had planned to make a quilt for my desk this summer, and while I can say that I created a lot of other quilt tops/lemmas, the problem of the Desk Quilt remains unsolved.  Stay tuned for more info there!

I don’t have my usual post about one thing, but wanted to start MMM so I have a Top 3 Math things to share for the week. 😀

  • If you have any students in grades 6 through undergrad in your life, you can share about the Association for Women in Mathematics Essay Contest!  The Association for Women in Mathematics and Math for America cosponsor an essay contest each year for students in three categories: Grades 6-8, Grades 9-12, and Undergraduate. Contestants interview a woman or an individual from an underrepresented gender identity in a mathematical career about their life and work, and then they write and submit a short biographical essay based on their interview (500-1000 words). You can find out more at https://awm-math.org/awards/student-essay-contest/: contest rules, an FAQ, submission instructions, etc.

    Please enter! We’d love to read your essay (and your friends’ essays, and your siblings’ essays, and…). There are no restrictions on the nationality or gender of contestants.

    The deadline is February 1, 2026. Submissions will open on November 1, 2025 on https://www.mathprograms.org/.
  • Whether or not you have students in your life, you can visit the site Lathisms.org to learn about a new mathematician each day of Hispanic Heritage Month (Sept 15-Oct 15).  The first mathematician is Andrés Felipe Galindo Olarte, who works with kinetic equations.  (That is, equations related to motion.)
  • And if you are looking for a game to play, might I suggest Dive?  Batman (the one who posts here, not the superhero. Unless they are one and the same.) shared this with me a few weeks ago, and I don’t know if Thanks is right, because it’s a littlee addictive.  But it’s math, so yay!

Finally, I’ll close with some “Who Would Win”  by Math with Bad Drawings.  (These are from Facebook, but their site says their work is under   Creative Commons Attribution-NonCommercial 4.0 International License.

Who Would Win #1

Who Would Win #2:

Who Would Win #3:

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Monday Sometime Math: 5 5 Rah Rah Rah!

May 5, 2025 by

Hello everyone!  Today is a special day, if you like the number 5 – it’s 5/5/25!  Or, if you prefer, 5/5/5*5.  

Confession: It’s been 25 years since all the Y2K stuff, and I still don’t feel comfortable writing the year with only two digits.  But 5/5/45*45, while otherwise pretty neat, doesn’t pack the same punch.

AAANNNNYYYYWAY, to celebrate the 5-ness of the day, here are 5 fun facts about 5!!

  1. The number 5 is a Fibonacci number, which means it shows up in the sequence 1, 1, 3, 5, 8, … where each number is the sum of the previous two. 
  2. The number 5 is a  Catalan number, which means it shows up in the sequence 1, 1, 2, 5, 14,…   

    That “…” seems a little presumptuous, since I had to look up how to get the next number.  Here’s one way: Notice that the first two numbers are 1 and 1, and if you write that sequence forwards and backwards and do a multiply-and-add thing, you get  1*1 + 1*1 = 2.

    Now the sequence is  1, 1 ,and 2 and if you do the forwards-backwards then multiply and add thing again you get 1*2 + 1*1 + 2*1 =5, the next number and also our Number of the Day.

    And now the sequence is   1, 1 , 2 ,5 and combining them gives  1*5 + 1*2+ 2*1 + 5*1 = 14.


    The Catalan numbers come up certain kinds of counting problems, like how many ways there are to write possibly-nested parentheses in a sentence

    If you don’t have any parentheses, there is 1 way to put them in a sentence: not doing it.  (That’s a terrible example, but it explains the first 1)

    If you have one set of (), there’s only 1 way to write it: ()
    If you have two sets of (), there are 2 ways to combine them:  ()() or (()).
    If you have three sets of (), there are 5 ways to combine them: ()()() or ()(()) or (())() or (()()) or ((())).  

    Moving on!
  3. The number 5 is a prime number, because the only factors are 1 and 5.  But it’s also a Fermat prime, because it can be written as 22^n+1 for some n (in this case, 22^1+1.  There are only 5 known Fermat primes.
  4.  The number 5 is also a Sophie Germain prime number, because not only is 5 a prime number, but 2*5+1 is also prime!
  5. And, finally, a pentagon has 5 sides, and a regular pentagon can be drawn with just a ruler and compass!  So can the triangle, square, and hexagon, so that might not sound impressive, but the regular heptagon with 7 sides can’t be drawn this way, so it really is pretty special that a pentagon can.

I hope your day is now 500% better.  Or at least 5%. 😀🌟

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Monday Morning Math: Fractal Dimension

April 21, 2025 by

Good morning! Today’s bit of math includes a bit of nothing. Well, sort of. We’ll start with a question: If you start with a square that is one unit on each side, subdivide it into 9 smaller squares and remove the middle one, subdivide the remaining squares into 9 smaller ones and remove the middle one, and keep going, what is the size of what is left?

Image by Johannes Rössel

The area of this fractal (the Sierpinski carpet) is less than 1, which would be the size of the filled-in square.

It’s less than 8/9, which would be the size of the square with just that big middle square removed.

It’s less than (8/9) of (8/9), which would be the size after you’ve removed the next round of squares.

It’s less than (8/9) of that (8/9)2, which is what would be left after you’ve removed more middle squares

In fact, no matter which whole number n you might pick, the area is less than (8/9)n, which means that it has no area at all. And that is really weird.

It seems like something 2-dimensional should have at least some area. The problem is that this fractal isn’t quite 2 dimensional. One way to think of dimension is to think of how a measurement multiplies if you enlarge an edge. For a 3-dimensional cube, if you triple the edge, the new volume is 3x3x3=33 as much as the original. For a 2-dimensional square, if you triple the edge, the new area is 3×3=32 as much as the original. For a 1-dimensional line, if you triple the edge, the new length is 3=31 as much as the original. But for the Sierpinski carpet, if you triple the edge, the new carpet will only be 8 times as large as the original because of the giant hole in the middle. And if you solve the equation 3x=8, the value of the dimension x is about 1.89, or a little less than 2. Meaning that it isn’t 2-dimensional at all – there are too many holes. So many holes that if you were to look closely, it’s basically all holes held together with cobwebs. Not good for measuring area, but very good for saving on ink if you were to print out a giant one.

Happy Monday!

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Monday Morning Math: Agnes Denes

April 7, 2025 by

A few weeks ago I went to the Museum of Fine Arts in Budapest, Hungary, and they had an exhibit on Agnes Denes (Dénes Ágnes), who lives in New York now but was born in Budapest. Her website describes her art as follows:

Investigating science, philosophy, linguistics, psychology, poetry, history, and music, Denes’s artistic practice is distinctive in terms of its aesthetics and engagement with socio-political ideas. As a pioneer of environmental art, she created Rice/Tree/Burial in 1968 in Sullivan County, New York which, according to the renowned art historian and curator Peter Selz, was “probably the first large scale site-specific piece anywhere with ecological concerns.”

As the description above suggests, several of Denes’s pieces are large scale and environmental: the exhibit I saw showed film from a 1982 project in which a wheatfield was created in lower Manhattan. But many of her pieces are small and, most significantly for this site, explorations in mathematics. She has several pieces based on triangles, both small (including a series based on Pascal’s triangle) and large

pyramid with plants

(the photo above is by JasN, Creative Commons license and there are more photos from a related exhibition at the Socrates Sculpture Park)

A video about a similarly shaped piece shared that she worked with a mathematician to get the curvature right, but unfortunately I couldn’t find more about the mathematician or the math.

Another series was based on projections. Even as I type this I’m aware of the difficulty of writing about an artist without showing her mathematical pieces, and yet copyright laws suggest I can’t post the photos I took of her work. Fortunately, she’s had pieces exhibited in several museums, and so you can see images of her work at the Metropolitan Museum of Art and there is a (nearly one hour) interview on YouTube where she talks about her pieces, including about 5 minutes on the projections starting at about 12:45:

I hope you enjoy this artist and her works!

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Posted in Art and Math, Featured Mathematician, Monday Morning Math | Leave a Comment »

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