Carnival Of Mathematics 206 – July 2022

Welcome to the 206th Carnival and my third time hosting the event.

To see past entries in the Carnival Of Mathematics and future scheduled hosts, please visit The Aperiodical.

I am honored to again host the Carnival of Mathematics! I learn so much from hosting; things I usually wouldn’t be exposed to are jam packed into every Carnival Of Mathematics post. Be sure to dig into the archive!.

Here are the entries. Enjoy!

Math Vs Culture

by Storm Bear Williams

A short little rant about all of those forgotten mathematicians from antiquity that hardly ever get the credit they deserve.

Mom Does Math

by Dr. Alyssa J Foss

I’m on the life long journey to recover from math-phobia. Which is why I have thrown myself into the math world head first! Next to momming and mathing, writing is my favorite thing. So, I’m trying to do the blog thing. This is my first post from September.


Editor’s Note: Her blog, Math Rehab, is really well written. Very worth the read!

What Is the Pascal Matrix?

By Nick Highham

In mathematics, particularly matrix theory and combinatorics, a Pascal matrix is a (possibly infinite) matrix containing the binomial coefficients as its elements. It is thus an encoding of Pascal’s triangle in matrix form. There are three natural ways to achieve this: as a lower-triangular matrix, an upper-triangular matrix, or a symmetric matrix.

Here Professor Higham steps his way through an example.

This article is part of the “What Is” series, available from and in PDF form from the GitHub repository


Editor’s Note: For more on Professor Highham; Twitter, Wiki, Google Scholar, Blog.

Ivan Guo: Financial models of the future

by Dr Ivan Guo

How can a 240-year-old logistics problem be used in quantitative finance? Dr Ivan Guo’s research lies predominantly in the areas of stochastic control and financial mathematics. In this interview with the Sydney Mathematical Research Institute, Ivan describes how stochastic transport theory applies in financial maths and how financial models are applied. He also debunks some misunderstandings about his field.

What does craiyon/DALL·E mini ‘think’ mathematics and mathematicians look like?

By Peter Rowlett

“You may have seen DALL·E mini posts appearing on social media for a little while now – it’s been viral for a couple of weeks, according to Know Your Meme. It’s an artificial intelligence model for producing images, operating as an open-source project mimicking the DALL·E system from company OpenAI but trained on a smaller dataset.”

Peter Rowlett presents a good introduction to DALL-E and offers several awesome examples.


Two Saturday Morning Breakfast Cereal Cartoons

by Zach Weinersmith

The first fun cartoon is titled “Incomplete”. LINK

The second is titled “Mathematics” and we can all feel this one down in our bones! LINK

Editor’s Note: You can find out more about the author here: Twitter, Facebook, Tumblr, Patreon


by Sneak Thief

One of my (many) interest areas is in algorithmic music composition. But it was not until @CarnivalOfMath mentioned that content didn’t need to be blogs, that I thought to submit this.

This piece is one where I use the Golomb ruler to determine the bar position, and length, of notes in a composition. I wrote a short piece of JavaScript which creates the data, incorporate it with a MIDI library which exports a MIDI file, that in turn can be loaded into a sequencer. I then assign each note to specific sound, based on their duration and what my ear tells me is good. (Being a synth-based composer, some sounds change a lot over time and are therefore better for long notes.)

The notes are always determined by me, a human, to match a particular key signature (to they sound in tune) and varied according to previous trial runs of the algorithm. So, for example, if notes of length 10 and 11 do not appear simultaneously I can assign them the notes E and F which (normally) do not sound good together. Similarly, I try to ensure that the start and end of the composition include notes which give an element of “resolution” which is prevalent in most western music. (Just because it’s based on maths, doesn’t mean it has to _sound_ that way!)

Finally, I sprinkle additional sounds generated by a different algorithm.

Click the image within the link to see a representation of the music, and 
you’ll clearly see the ruler being used.


From Alice To The Mad Hatter: The Road To A PhD

If you spend just a few minutes on Twitter’s #PhD you will see a range of emotions displayed by those on the path to earning a PhD.

These brave individuals all seem to start out as shining little newbies, fresh out of the newbie bag – with wonder and star-struck innocence in their eyes, eager to start on their journey to increase human knowledge and be known as Doctor.

As they chase the white rabbit of their academic dreams, they soon realize they are spiraling down – not down a rabbit hole, but a deep, festering abyssal where dreams, happiness and wonder are exchanged for stress, tears, fear and abject frustration.

They battle multiple complications and obstacles, all coming at them at once, from all directions. Supervisors seem to produce the most anxiety and dread. Errant supervisors, abusive supervisors, absent supervisors, the handsy supervisor with a furry fetish – they all are problematic and add to the toil of getting a PhD.

Once you FINALLY make your saving throw against the bad supervisor, you get to read. Well, kinda read. Maybe read. You get to read all you want if you have access to Elsevier, if not you either have to get real lucky and find all your relevant research on open servers, find them available on the piratey Sci-Hub or sell a kidney. How science became so closed it has become a real disservice to humanity and modern civilization.

Then the clock gets involved. Days turn to weeks, months to years – and years turn into chapters. You see the end of the tunnel. Light harkens through the darkness, crawling its way to you – showing you a new future… no, fuck that, “major revisions required.”

And just as the funding begins to run out and your Xanax prescription has seen its last refill, you submit FinalFinalFinalThesis-Revision9-Version4-WithChangeNotesSierra.pdf to your supervisor – the second one. The first one took a job at Google. Well, they tell you he went to Google, you can’t exactly recall. You spent a few months in a black out drunk when you heard the news that you would need to reroll the saving throw.




You hear nothing back for a few days.

Panic attacks.

You are now in full Mad Hatter mode. At this point, you are sitting naked, in a room only illuminated by your laptop, tapping violently on the keyboard as you try to explain the scientific method to the QAnon trolls on Facebook.

You wake up on a random Wednesday, fending off a Redbull-induced stroke, to hear that your dissertation defense has been scheduled.

Can this be true? A smell of disbelief fills the room, reminding you you haven’t showered since you submitted.

The door to a conference room opens and you step through to see your second supervisor and other members of the dissertation committee. As the door closes shut you try to recall how you got there.

Time passes.

As the door reopens, the pressure seems to leave the room and the others congratulate you on a job well down. You are now a Doctor Of Philosophy.

This story may not be reality but it seems to be when I read the horror stories of others who have successfully, and sometimes unsuccessfully, traveled this path.

So now, it is my turn. PhD here I come! I have always wanted to earn a PhD ever since I met Carl Sagan. I love science, all aspects of it, all subjects – I cannot get enough. Even as I approach that time where cashiers will give me the “Senior Discount” I continue to be in awe at the things we have discovered. But I could never narrow down what topic to pursue. You can’t just get a PhD in “General Studies,” you have to be startlingly specific. Decades passed and I could never pick something; biology to chemistry, then on to astrophysics, then back to biology… maybe archaeology. I would read a new article or get triggered by advances in understanding of mycelium networks. Who wouldn’t want to do a deep dive on that? It wasn’t until I read the XKCD cartoon titled Purity that I got it.

With all of these subjects that I have an intense interest in, looking at this cartoon made the secluded seem obvious. Mathematics is the base code of the universe and a PhD in math would allow me to peek into the underpinnings of the classic sciences; biology, chemistry and physics. I should have known, but I didn’t. I was too busy in my autism loop process to take the accretion disk point of view.

I have decided that PhD by Previous Publication (PPP) is the best for me. The idea of it doesn’t trigger my autistic predispositions to testing or intellectual judgement. I kinda go at my own pace. I think about my project, write the papers that will take the reader through the story. Rinse and repeat until the body of work is sufficient for defense.

So here I go, one step in front of another, into the future – I mean, Costco to buy a pallet of Red Bull.

Carnival Of Mathematics 195 – July 2021

Welcome to the 195th Carnival and my second time hosting the event.

To see past entries in the Carnival Of Mathematics and future scheduled hosts, please visit The Aperiodical.

I am honored to again host the Carnival of Mathematics! I learn so much from hosting, things I usually wouldn’t be exposed to are jam packed into every Carnival Of Mathematics post. Be sure to dig in to the archive!.

Here are the entries. Enjoy!

Bad Math Memes

By Storm Bear Williams

This is a video by me discussing how crazy I get when I see crazy math memes on Facebook and Twitter. Most are not educational and further separate mathematics from would-be students. We as mathematicians must do everything we can do to get people to EMBRACE mathematics, not shy away from it.

Here’s a proof that Tolkien’s Middle-Earth is not flat

By: @fermatslibrary (Twitter)


A 2021 problem: 20∼21 and 43×47

By: Ed Pegg
Submitted By: Lewis Baxter

Ed Pegg noticed that 2021 = 43 x 47 which are successive primes with 20 and  21 being successive integers. He asked for similar solutions and Robert Israel quickly found the next biggest solution, a number with 36 digits. I (Lewis Baxter) found more than 1500 bigger solutions, the largest having  3011 digits. This month I certified the two primes (which are  20690  apart). Unlike other “titanic” primes they are not the value of some small arithmetic expression.


Who Needs Trig Sub?

By: Patrick Honner

Mr. Honner sent this link in, bragging about what his students came up with. “This was the coolest math my students produce this year,” Mr Honner gushed!

“I’ve taught this topic for many years and never thought of this approach. I’m grateful to have learned something new from my students, who never fail to impress me with their creativity. And I’m glad I gave them time and space to solve what I thought was an impossible problem! When I teach this next time, I’ll be sure to do it again. And I’ll be sure to share this ingenious integration.”


When cubic polynomials have three real roots!

By: Freya Holmér (via Twitter)

Holmer Breaks down how they can be solved using trigonometry. Geometrically, you can visualize it as an equilateral triangle centered directly above the inflection point, where its vertices coincide with the three roots.

Why do perpendicular lines have slopes that are opposite reciprocals?

By: Howie Hua (via TikTok)


Why do perpendicular lines have slopes that are opposite reciprocals? #math #mathematics #teacher #teachersoftiktok

♬ original sound – Howie Hua

Half a year of the Liquid Tensor Experiment: Amazing developments

By: Peter Scholze
Submitted By: Robin Whitty

“Exactly half a year ago I wrote the Liquid Tensor Experiment blog post, challenging the formalization of a difficult foundational theorem from my Analytic Geometry lecture notes on joint work with Dustin Clausen. While this challenge has not been completed yet, I am excited to announce that the Experiment has verified the entire part of the argument that I was unsure about. I find it absolutely insane that interactive proof assistants are now at the level that within a very reasonable time span they can formally verify difficult original research. Congratulations to everyone involved in the formalization!!

In this Q&A-style blog post, I want to reflect on my experience watching this experiment.”


Singmaster’s conjecture in the interior of Pascal’s triangle

By: Terence Tao
Submitted By: Robin Whitty

Kaisa MatomäkiMaksym RadziwillXuancheng ShaoJoni Teräväinen, and myself have just uploaded to the arXiv our preprint “Singmaster’s conjecture in the interior of Pascal’s triangle“. This paper leverages the theory of exponential sums over primes to make progress on a well known conjecture of Singmaster which asserts that any natural number larger than 1 appears at most a bounded number of times in Pascal’s triangle.



Submitted By: Sam Hartburn

#GeometrySketchbook is a hashtag that has been used for a daily maths art 
challenge throughout June. A huge variety of media and art styles have been 
used; if you’re interested in mathematical art you’re sure to find 
something inspiring here.


Is This Some Kind of Code? You Can Solve the …

By: New York Times


SRC: New York Times

“Erik and Martin Demaine, a father-and-son team of “algorithmic typographers,” have confected an entire suite of mathematically inspired typefaces.”

The verb “puzzle” — to perplex or confuse, bewilder or bemuse — is of unknown origin. “That kind of fits,” said Martin Demaine, an artist in residence at the Massachusetts Institute of Technology. “It’s a puzzle where the word ‘puzzle’ comes from.”