Right now my project is drying so I don’t have a final photo yet. Here it is before I got it wet. (It still has the water-soluble transfer paper on it in the photo.)

Update: Here’s the photo now that I’ve removed the water soluble transfer paper.

Earlier today I typed “embroidery math” in Google and a site called Threaded Theorems popped up. Nary a second passed before I was on the site. With a website name like that! I saw the instructions for embroidering the aperiodic monotile called the “hat tile.”

Here’s a Scientific American article that explains the math and is very accessible: https://www.scientificamerican.com/article/newfound-mathematical-einstein-shape-creates-a-never-repeating-pattern/

Technically what I embroidered uses a slight variant of the “hat tile,” because that tile requires the shape and its reflection when filling the plane, and this variation in the embroidery does not require its reflection. Just a single tile, repeated over and over again, but creating something that fills the plane but doesn’t have translational symmetry! (The variant tile is known as “the spectre.”) And this is modern math! It only was discovered in 2023!

Here’s the website with instructions: https://www.threadedtheorems.com/aperiodic-tiling

I literally know nothing about embroidery so I’m learning and doing things on the fly. In this project the instructions had me learn the backstitch and satin stitch.

And if you want more technical definition of aperiodic tilings, and some really pretty pictures, here’s the wikipedia page!

I tried to get it to work when it arrived, but for some reason, I couldn’t get the drawings to work out. (The instructions that it came with were okay but there were bits that could have been much clearer.) However I found this box in my closet a few days ago, and decided to try again. And it worked!

Here are some drawings I’ve created:

You can see the machine in action in this youtube video [starting at time 4:53]:

Of course I wanted to somehow mathify this. In fact, I’ve never even mathified sprigraph before (though I’ve seen some people derive the equations, but I’ve never tried to do it on my own)! I decided to jump right in and see: can I come up with equations for the final graph, given all the components of the machine?

It was harder than expected, and I had moments of frustration and felt like I wanted to quick. It took me a couple days. I first started doing all this algebra on paper, and realized that was not going to work because the equations got gross. So instead, I went to Desmos and everything became much easier.

Without going into too many details, here’s my setup in Desmos, and a picture of the hypnograph, so you can see the connections:

I have point B spinning around a circle/gear (and you can change the location, the rate of rotation, and size of the circle/gear), Similarly I have point P spinning around a second circle/gear (with the same things changeable). The arm that connects B and P is fixed at point B, and slides at point P (like with the machine). (I’d watch the video above to see the difference between the fixed point and the sliding point. In that video, the fixed point is on the red gear and the sliding point is on the green gear.)

Initially, I had Desmos plot just the pen location. It was creating some really pretty drawings, but it didn’t mimic what I saw on the hypnograph itself. The hypnograph images always have some sort of symmetry, and my final result didn’t always have that symmetry.

I posted my Desmos and a video of the machine working on Bluesky to see what I was missing. I knew I just needed fresh eyes to show me what I was blind to. And diffgeom clocked it (and dandersod agreed)!

Not sure this is the only issue (and have not checked your model), but the paper is rotating relative to the table (and therefore relative to the rods holding the pen), while on your screen it appears not to be…?

YEESH! I forgot to have the paper rotate as the pen was drawing! ACK! But thinking about that hurt my brain. I came up with one solution, and tried to implement it, and that failed. But with some perseverance, I came up with a solution and I think it works! [1]

Here’s an image from my Desmos sheet to show you what it produced:

And if you want to play with the Desmos sheet, change the gear sizes, the length of the two arms, and the speed of the gears and rotating paper to see the various results, here you go! Click on the random image below I made in Canva, and press play. It’ll look like garbage at first as it’s drawing and then suddenly you’ll see how cool it looks!

Now I should qualify to say I’m not 100% sure this is perfectly correct. But I wanted to share what I had because I’m proud of it. And it was fun! And if I don’t write it down, I’ll never have a record of working on this project!

Archiving for later: These youtube videos [one, two] are great for getting a sense of the machine and how to modify it so it works the best.

[1] A bit of an aside/archive that won’t make sense to anyone not in my head: Essentially I realized I was doing a hack for polar coordinates but not allowing r to be 0. So it was graphing the right equations but having all negative r graph as positive r. When I realized that, I finally figured out what I needed to do. I made the center of the spinning paper (0,0). That had to be fixed. Then I had the coordinates for the pen. So I put those coordinates through a rotation.

The pen location was at this point:

So to make the equivalent marking on a moving paper, I multiplied it by the rotation matrix (where the rotation matrix was filled with values that had the wheel rotate at a certain rate):

Which got me the pen location after it had been rotated! In my Desmos page, that turns out to be:

Honestly I was really proud of this solution! And what I love about all this work is it incorporates all the stuffI teach in precalculus… polar coordinates and rotation matrices FTW! We stopped teaching much about vectors, but I used a bunch of vector thinking in this also!

When thinking about the quilt, I started thinking about Truchet tiles and how those could create some beautiful patterns for quilts. And then the pictures of Truchet tiles reminded me of a series of posts that come on my feed regularly by the Hitomezashi Bot on Bluesky, created by Alien_Sunset.

So I went to that bot, which led me to the creator’s website (which is here), and finally I found the Hitomezashi page here. That page led me to this Numberphile video with Ayilean (a creator who I adore):

And also to this paper by Katherine A Seaton and Carol Hayes on the mathematics behind these figures. In that paper, the authors wrote:

A method of specifying hitomezashi designs in shorthand form using binary strings was employed by the first author when she proposed an activity for the Math Art Challenge organized by Annie Perkins (Perkins, Citation2020). The premise of the Challenge was to provide, each day for 100 days, an activity which could be done during lockdown or remote learning with materials on hand. Hence it was suggested that hitomezashi designs could be drawn on graph paper, if sewing materials were not available; some participants chose to render them using their preferred software. Three ways to choose the binary strings were suggested: to intentionally create regular patterns by repetition (which we explore in this paper), by flipping a coin (thus making aleotoric art) or as a form of steganography (e.g. by using the ascii representation of letters).

The Math Art Challenge hitomezashi activity created a lot of interest on Twitter; subsequently, a Numberphile video featuring Ayliean MacDonald took the drawing idea to an even wider audience and explored the steganographic and aleotoric aspects (Haran, Citation2021). This video came to the attention of Defant and Kravitz (Citation2022), who have subsequently proved a number of results about arbitrary hitomezashi designs…

And FULL CIRCLE! When I got here, my heart was twitterpated because the authors talked about Annie Perkins and her 100 Day math-art challenge. I KNOW ANNIE! And I love Annie’s work so much! And so I found her math-art challenge from Day 14.

Just from googling “hitomezashi math” there are a ton of hits. I love seeing how the simple setup of the hitomezashi activity can lead to some really interesting patterns, observations, and questions/conjectures.

So what did I do? I HAD TO MAKE ONE MYSELF!

I made 16 horizontal lines, and 16 vertical lines, on my linen — just using a ruler and pencil.

To decide the way I would stitch the horizontal lines, I chose to do what Ayilean did in her Numberphile video and encode a phrase:
MATHISJOYFULART!
0100100100101000 [every consonent is a 0, every vowel is a 1]

And vertically, I used rainbow dice given to me by a math-teacher-friend-and-colleague to generate 16 numbers and I got:1011110100111001 [every even roll is a 0, every odd is a 1]

And then I had to sew! I was basically figuring this out on the fly. How to tie knots, how to end a row of stitches, how to efficiently thread the needle, how to avoid knots. And I knew I wanted the back of the embroidering to look nice too!

And then we get to the end! The final product!

Clearly I’m not amazing at making my stitches yet. And I think I might want to use a thicker floss or even some cotton yarn next time. I saw some patterns made with light string on dark cloth, and I think those looked beuatiful too. But I’m really proud of how I just decided “hey, can I do this? I think I can!” and just made it happen!

From thinking about quilts to Truchet tiles to hitomezashi, my mathematical life today was very full.

Update with some links I want to save:

I found this link to coloring in Hitomezashi patterns: https://openprocessing.org/sketch/871770

A Hitomezashi generator (regular and isometric): https://hitomezashi.com/

A Hitomezashi generator where you enter words (like in the Numberphile video) to generate the pattern: https://public.tableau.com/app/profile/rincon/viz/HitomezashiMakeArtwithMaths/Hitomezashi

A Hitomezashi generator where you can enter binary strings: https://www.forresto.com/sashiko
[I like this because then you can see what the front and back are by changing the strings to their dual]

So here’s what she sent me (from here)

The question that I’m attempting to figure out is: what is the radius of the small circle in relation to the length of the side of the square? And you’ll see from the text she sent me, there is the answer.

We found the puzzle curious and we thought the figure was lovely, so we decided to make a piece of stained glass math-art for it. (We’re trying to teach ourselves how to make math-based stained glass art. As you can see from the picture below, we’re still very much novices!)

I spent over an hour on the problem this morning. I played on Geogebra to get the contours of the problem itself to help me understand the constraints. And then since I didn’t have a better approach, I started brute forcing it. I came up with equations when I put everything on the coordinate plane (using calculus to find slopes!), found some tangent lines, and looked for intersection points of these lines. I knew it wasn’t efficient, but I thought it would at least help me see how things would shake out I was hoping lots of terms would drop away…

Alas! The algebra got messy, and then I thought I’d have a system of equations that could be solved.

To be frank, I’m pretty sure in the mess of algebra, these equations are probably not correct. And I couldn’t really solve them anyway. (If you’re wondering what a, b, and c are, they stand in for slopes of the three lines (pink, light blue, green).

I’m officially annoyed and ready to give up. At least for now, or for a few days, or maybe forever. BUT the reason I’m posting this is that I’d really love a solution. Why is the radius of the circle 4/33s (where s is the side length of the square)? What is the approach to figuring this out?

So if you like geometry puzzles, like these Sangaku, or the puzzles of Catroina Agg (https://bsky.app/profile/catrionaagg.bsky.social), help me out!

UPDATE: I posted about this online and here are some of the assists I’ve gotten. I was going to look at them today but I’m not feeling super well.

From Facebook:
Bowen: “Check out Descartes’ Circle Theorem…” [here]
From Jason: “I am sitting across from Bowen this morning and UNBEKNOWNST TO ME, you sniped us both in parallel. Unbelievable. I solved it on paper, got all excited, and then saw that Bowen– SITTING ACROSS FROM ME– has already supplies that link above. Anyway, I’m not going to let him Liebniz my Newton here. I did some work on paper independent of him and will publish it with pictures in the comments below.

I went to https://en.wikipedia.org/wiki/Problem_of_Apollonius and used these equations:

The algebra I algebra’d.

I mean, I did not write the work in a way that was designed for other people to write, so there are a LOT of “S” letters that might either stand for “the numerical length of the side” or “the name of the solution circle”, depending on context.”

From BlueSky:

From diffgeom: If we invert the plane ((r,θ) -> (1/r,θ)) about the point where the three large circles meet, each maps to a coordinate line; the small circle maps to a circle whose center and radius look easy to find, and inverting again should give the radius?
Will give this a try…

I replied: Omg- my brain!!! Wow! I don’t know if I’ve ever inverted the plane and I didn’t even think to look at this problem through polar. I’m so intrigued!!! Essentially you’re saying the diagram will look easier once we do the inversion, we find the answer in the inverted world, then put it back?

diffgeom replied: Possibly a fun stop on your travels to learn about inversion: https://noai.duckduckgo.com/?t=h_&q=cylindrical+mirror+art&ia=web

[At this point, I looked up and watched this amazing 27-minute Numberphile video that explains some of the basics of inversion and got me excited: https://www.youtube.com/watch?v=sG_6nlMZ8f4]

tcorica replied: So, I’ve captured everything in a Desmos sheet EXCEPT how to take the final circle and un-invert it to find the uninverted center and radius. Any hints? I’m avoiding “reading up” on inversion, but I will if needed!

https://www.desmos.com/calculator/rzkrhnudqe

P.S. I flipped the image across the horizontal so that the given circles had centered on an axis, making it easier to write the polar equations in terms of r. @samjshah.bsky.social

tcorica also shared: I saw the responses suggesting “inversion” – i.e. recasting the problem by writing equations r= and then rewriting them as r’=1/r=. I’ve still managed to avoid looking inversion up , but I’ve done some thinking and experimenting with it – it’s a surprising and cool technique.
I’ve found a reasonably nice solution to this using inversion. Because the answer is a weird looking fraction, I’m wondering if perhaps there actually isn’t a pretty geometric solution to this pretty problem!
As an introductory exercise, I found it useful to consider the problem of finding the intersection of the circle with radius P and center at (P,0) with the circle of radius Q and center at (0,Q). E.g., show that the POI lies on y=(P/Q)x. Tons of approaches possible, inversion being one. 3/3

diffgeom replied: My approach: A circle of radius r at distance R from the center of inversion has points at distance R±r on a ray; inverting in the unit circle gives points at distance 1/(R±r), so the radius is (1/2)(1/(R-r)-1/(R+r)), or r/(R^2-r^2). (IIRC, the distance to the center is half the sum, R/(R^2-r^2).)

nsato7 also commented after diffgeom:

As @diffgeom.bsky.social points out, inversion takes care of the problem nicely. (And many of these Sangaku problems were solved by inversion.)

The three arcs invert to three lines. There are two circles that are tangent to these three lines; we want the “top” one.

mqb2766 shared: Similar to the other posts, a modern version of this problem is gps/trilateration with the distances from the known centers being 2-r and 1+/-r so 3 quadratics in 3 unknowns – the center and the radius. You get a trivial solution of r=0 as well as the stated one.

https://en.wikipedia.org/wiki/Trilateration

noswald also shared (unrelated to the others): My inelegant solution would be to impose a coordinate system on the figure and set up three equations involving the coordinates of the inscribed circle and its radius. We know the distance from the center of the circle to a corner is s-r, to one side’s midpoint is s/2-r, and to another side is s/2+r

From Email

My friend Japheth W sent me his written work, which he said I could archive here, noting “It’s not quite publication ready, but probably understandable to math teachers who are experts at reading even more disorganized student work.”

It is a “choose your own adventure” model where we have lots of options of how to engage. I focused my attention on working on the problem sets and attending zooms from various research mathematicians who were talking on their research related to the problem sets. Because of my summer schedule this year, I didn’t dedicate myself in the same structured way that I did in 2023 when I felt like I mined a lot of the content. Here, I did some deep dives into much (but not close to nearly all) of the content, and I wasn’t able to make all the zooms or create/give an end-of-program presentation. Still I feel like I got a lot out of it.

Participants communicated on a Zulip site (like a discord server). I wanted to archive some of the links that participants shared on that site, since it will force me to go through everything posted, and also collect all the links I think might be helpful in the future where I can access them all in one place. It’s a bit of a hodgepodge of things.

Articles/Books and Interesting Problems:

One participant posted: “I’m interested in making ways for people outside universities to engage with current maths research (recently I put together this puzzle x article that folks here might be into!)” [and here are all of their science/technology articles] 

Someone asked for a good book to teach themselves some number theory, and someone else responded with the book written by a math tiktok-er I like! It’s free online, which is awesome, called “Number Theory and Geometry.” A quick skim makes it feel accessible, which doesn’t surprise me from their tiktoks! Another highly recommended “An Illustrated Theory of Numbers” which isn’t free, but the preview looks really lovely and up my alley since I love concrete and visual learning.

One interesting problem that was shared was the Sword of Knowledge: “A sword of knowledge can slay the Dragon of Ignorance, who has three heads and three tails. With one stroke, slayers can chop off one head, two heads, one tail, or two tails; however, the consequences lead to a variant of the structure. In order to successively slay the Dragon of Ignorance, Students must exercise strategic analysis when fighting this creature.” And what’s really lovely is that the website has a lot of great “math fair problems” which seem like fun, rich, and investigative problem solving tasks.

Living Mathematicians/Mathematicians Stories

Annie Perkin’s “Mathematician’s Project“

Living Proof: Stories of Resiliance Along the Mathematical Journey

Meet a Mathematician:     https://www.meetamathematician.com/home

Her Math Story:  https://hermathsstory.eu/

Mathematically Gifted and Black:  https://mathematicallygiftedandblack.com/

Indigenous Mathematicians:  https://indigenousmathematicians.org/

Latinx and Hispanics in Mathematics: https://www.lathisms.org/posters

The PD organizer also shared: “Here are a couple of short videos related to this thread; they recently popped up in my social media. They are clips of interviews with two Oxford undergraduates studying maths. I particularly like what Ellie says in Part 2 about “You realize that you’re right for being confused, and that anybody at any stage who acts as if they understand what’s going on with maths has no idea what’s going on.” Sometimes it’s our students who are trying to actually understand what’s going on are the ones we think are “struggling”.

Part 1: https://www.youtube.com/shorts/B2O6S_BWkyI
Part 2: https://www.youtube.com/shorts/3cie4VnHeC0“

Math Communities:

Another participant is heavily involved with the Julia Robinson Mathematics Festival (JRMF).  They shared about two online math teacher community that grew out of this world they are involved in:
“JRMF Community Math Circle: After running math festivals all over the world, JRMF pivoted in March 2020 to offering a weekly online math circle.  I joined as a volunteer for a couple of months and then as an employee for a year.  Being part of a community of people with similar interests all over the world was a silver lining to the pandemic for many of us.  
When JRMF went back to doing in-person events, some of us volunteered to keep the math circle going as a monthly event.  We are always happy to have more volunteers.  The math circle meets on weekends at 11 am PT, and for the two weeks prior to each math circle, we hold 90-minute trainings – exploring the topic and discussing pedagogy for it.  You are welcome to come to observe a math circle and/or come to a training to find out about it.  Even if you join as a volunteer, there is no obligation to volunteer any particular month – you just sign up for when you can.
We are currently on a 2-month summer break but will resume in September.
Puzzles & Pedagogy: The other is a weekly event that is kind of like a math circle for adults.  It’s called Puzzles & Pedagogy and is run by Gord Hamilton of Math Pickle.  Topics vary in level and the balance between puzzling ourselves and discussing pedagogy also varies.  It meets weekly on Thursdays 12:30–2:00 PT and is also free and no-obligation.  
Both Math Pickle (MathPickle.com) and JRMF (JRMF.org) are also great resources for teaching – at any level!”

In response to that, another participant shared that they found an online Math Teacher circle, CAMI (Community of Adult Math Instructors). I’ve always wanted to participate in a math circle, so I’m hoping to join one of them! Here’s a flyer they link to, with a bit of information.

LaTeX:

For using LaTeX in MS Word, here’s a 1 minute video on how to do that. And here’s a comprehensive beginner’s resource for LaTeX and a place where you can draw a symbol and it spits out the LaTeX code for that symbol.

The PD organizer also gave these resources: “Overleaf is an online platform for writing documents in LaTeX. It’s like Google docs but for LaTeX. It also has a lot of document templates, very handy. It’s free for individuals, but there’s a subscription fee if you want to collaborate with others. https://www.overleaf.com/

In actual Google docs, you can go to Extensions and the Add-ons and install something called Auto-Latex Equations. This will let you type LaTeX formulas within your Google doc.

If you’re on a Mac, you can (like me) use TeXShop to compose LaTeX documents. It’s a free downloadable program. https://pages.uoregon.edu/koch/texshop/

If you just need a snippet of LaTeX as an image, there are a number of website that can help you out. Here’s one: https://www.quicklatex.com/

Also worth knowing: if you have Keynote, LaTeX is a native feature! Just go to Insert and then Equation, type your LaTeX (no dollar signs required), and hit Insert, and you get a beautiful math expression that is scalable, etc.”

Soap Bubbles: These were one of the research focuses for the program

In one of the presentations, a researcher shared how he made the soap bubbles to illustrate his ideas: “To make these planar soap bubbles, you just need a shallow pan of some kind. I used the lid of one of these, because it’s what I had lying around. A baking sheet with raised edges, for example, should also be fine.

Then you need a sheet of acrylic. I got one from a picture frame. It needs to be big enough to balance on the raised edges of the shallow pan, so that it’s raised off the bottom a little (like, a cm or 2). Then you add water and dish soap to the pan, and use your hand to spread some on the underside of the acrylic. Put the acrylic on top of the pan and use a straw to blow bubbles in the gap between the pan and the acrylic. That’s it! If the bubbles pop when they hit the acrylic, spread more bubble mix on the underside.

I find that the bubbles are very visible in person, but they don’t show up well on cameras (or on zoom). If that’s what you need, adding a bit of food coloring to the bubble mix helps.”

Quanta article “‘Monumental’ Math Proof Solves the Triple Bubble Problem and More” (the bubble problem was one of the big problems we were working on this summer)

As part of this professional development, we started filling in a spreadsheet with various areas- perimeter combinations we could create. This is a screenshot of collaborative sheet we came up with:

I played around with this problem for hours and hours. I decided I wanted to focus on this question: If you have a given Area of n, what is the Maximum Perimeter you could get, and what is the Minimum Perimeter you could get?

I was relatively easily able to prove that for an Area of n, that the Maximum Perimeter you could create is 4+2(n-1). Here’s the argument in a nutshell. Each time you add on a new square to an existing figure, you either are adding 0 to the perimeter, 1 to the perimeter, or 2 to the perimeter. (This is easy to show.) So to get the Maxmimum Perimeter, you want to be adding the most to the perimeter each time. So if you start out with a single square (n=1) with a perimeter of 4, each time you add a square, you’re adding at most a perimeter of 2 to the figure. One easy way to create this figure is to just add squares like this:

This is where the 4+2(n-1) comes from. You start with a perimeter of 4, and add 2 every new square you add to it.

It took me a while to figure out the Minimum Perimeter. I am pretty certain I have it, but I’m still working on proving it in an airtight way.

Basically, the way to construct the figure with a minimum perimeter is to create the largest square you can, and then append the remaining blocks to it. Below I have a series of images to show what the figures could look like for an Area of 25, 26, 27, …, 36, and also the Perimeters. I used s in the image (instead of 5) to show how this generalizes for any square number.

Some observations:

(1) Notice that after you start with that perfect blue square, each time you add a square (so for an area of s^2+[1, 2, 3, …, s]) you’ll always get the same perimeter of 4s+2. At this point, you’ll have an s by (s+1) rectangle.

And then when you continue on adding more squares to this rectangle (so for an area of s^2+[(s+1), (s+2), (s+3), … (2s+1)]), you’ll always get the same perimeter of 4s+4.

(2) At the end, when you’ve finished this process, you’ll have a new area of s^2+2s+1, which simplifies to (s+1)^2. And the perimeter of this is 4 times the side length of the square, which is now s+1. So this makes sense as 4s+4.

Here’s a chart I made which shows how the perimeter goes up after you hit perfect squares… Notice we know the perimeters of perfect squares, so for example at A=16 (a 4 by 4 square), the next four numbers will have a perimeter of 18 and the four numbers after that will have a perimeter of 20. That’s then when we hit the next perfect square of A=25.

So far, we have this:

Take your Area of n and find the largest perfect square the goes into it. Call that s.
If the remainder is 0, then you have a minimum perimeter of 4s+0.
If the remainder is 1, 2, 3, …, s, then you have a minimum perimeter of 4s+2.If the remainder is (s+1), (s+2), (s+3), …, (2s), then you have a minimum perimeter of 4s+4

However I didn’t want a formula in terms of s. I wanted an explicit and closed formula in terms of n. I wanted a direct answer to the question: if you have an Area of n, what is the Minimum Perimeter in terms of n? And I got it…

It’s kind of ugly looking, but hopefully you can see the motivation. The first part is just 4s. The second part is the correction factor (of +0, +2, or +4). The numerator in the ceiling function calculates the remainder. When you look at the entire ceiling function*2, it will give you 0 when you have a reminder of 0. It will give you 2 when you have a reminder of 1, 2, 3, … , s. And it will give you 4 when you have a remainder of (s+1), (s+2), (s+3), …, (2s). I was pretty proud when I was able to use my table above to figure this out. I checked using Desmos to confirm!

At this point, I’m confident about these bounds for the Perimeter of a figure with Area n…

Now I’m stuck on the very last thing… I am confident with my formula for the Minimum Perimeter, but I haven’t proved it.

I think I have a way to show that if the Area is a perfect square (so n^2), then the Minimum Perimeter always occurs when the squares are arranged to form an n by n square. However at the moment I can’t seem to move forward from there.

In any case, that’s where I am with this problem!

Let’s say I have two circles that “intersect at 120 degrees” like in this diagram below (if you zoom in enough at to the intersection point of the circles, the two circular lines are approximated by the tangent lines I drew, and the tangent lines form 120 degrees to each other):

It turns out that there is a THIRD circle which can also intersect with these two circles and form 120 degree angles. Here are some different configurations that showcase this.

The question I’ve been dealing with is this: given the two initial circle radii (the red circle and blue circle, in the pictures above), can I come up with a formula for the radius of the third (green) circle?

Here is the applet I made to play around with these circles [https://www.geogebra.org/classic/kaynpexw].

I’ve been working for a few days to find the answer. I felt at times I was going in circles (no pun intended), and indeed I was. When I got the answer, I was astounded.

It was much more elegant than I anticipated! Everything simplified in a beautiful way. And my work was so ugly… I mean, I was using the Law of Sines and the Law of Cosines and the sum of angles formula for sine… so after taking a break from the problem, I went back to see if I could do it much more simply, now that I had played around with the problem and better understood the contours of it. I could!

First, I had to show that the angle formed by the radii to the intersection point had to be 60 degrees.

Here’s my work for that, using the fact that the tangent lines to the circles form right angles with the radii.

I next added in the third circle into the picture. Here’s a very not-pretty applet I made [https://www.geogebra.org/classic/wysfbqwn]. Drag point E and see what happens… I’m getting all possible green circles that intersect the red and blue circles at 120 degrees. There are an infinite number of them!

If you just want to see the picture, click through the images below to see what happens as I change the radius of the green circle to gradually bigger and bigger and bigger.

There are an infinite number of green circles that work! However, only one of the green circles also intersects at point G, the other location where the red and blue circles also intersect. So that’s the green circle we want. It took me a hot second to reason out why point E has the lie on the same line as C and D. I’m just trying to get this posted, so I’ll add that reasoning in later if I have time.

At this point, for me, the problem involves focusing on the triangle highlighted below:

I’m going to look just at the triangle now. We showed that the angle between r1 and r2 is 60 degrees. An identical argument can be used to show that the angle between r2 and r3 is also 60 degrees.

To keep things straight in my head, I highlighted the things we know in yellow. We are given r1 and r2, and we know the two angles are 60 degrees. We just have to find r3, which is the radius of the third green circle!

At this point, I think there are lots of things one can do. In my original work, I was using lots of Law of Sines and Law of Cosines. But in my revised work, I didn’t need any of those things. I decided to rotate the triangle.

And the reason I did this is to put everything on a coordinate grid. From here on out, my work just fell out nicely! On the diagram below, I found the equation of the purple line. And then I wrote the equation of the line that forms r3 (what I’ll call the green line). And then I found the intersection point of the purple and green lines (the green point). And then I found the distance from the origin to the green point.

And that formula below is precisely what I found in my messy, convoluted first attempt that took hours and wasn’t as elegant.

Writing this formula in reciprocal form, we get either the first or second formula:

I remember from when I taught multivariable calculus that the measure of curvature of a circle is the reciprocal of the radius. So in that formulation, the curvature of the third circle is the difference in curvature of the other two circles.

Beautiful!

I shared it with two friends in my department because they both really love combinatorics like me. One gave me some really nice ways to think about it and set me upon a nice path forward. But before I could go there, the other (the illustrious James Cleveland-Tran) shared with me his thinking. It was amazing. I wrote in my last post I love when things I can’t explain (in mathematics) feel like magic, because it means I get to learn something cool to figure out what makes it work. And when I do, when I understand something so well that the magic suddenly feels mundane (I think I used the word “obvious”), that is a wonderful moment. Because that means I fully grasp something. As James shared his thinking with me, I had that moment. Of course this had to be true. It doesn’t involve anything so deep I had to work to get it. It felt simple, and simply put, beautiful. I’m going to share that reasoning here.

So to start, I want to remind y’all that the things being factorialized and multiplied in the denominator all add up to the numerator. I am going to use a specific example to illustrate this and work through the ideas:

The whole proof hinges on this idea: a product of n consecutive integers will be divisible by n! Since James showed me this, I’ll refer to this as the James’ Theorem.

So let’s look at this example more closely. I’m going to expand out the numerator.

Now I’m going to group the numerator into groups of consecutive integers. Of course you can see we can do this in different ways. Here are some of them!

By James’ Theorem, we know the three consecutive blue numbers in the top of the fraction will evenly divide the 3! in the bottom of the fraction, the four consecutive red numbers in the fraction will evenly divide the 4! in the bottom of the fraction, and the five consecutive yellow numbers will evenly divide the 5! in the bottom of the fraction. And thus we’re done.

Now how do we know James’ Theorem is true?

Okay, I officially hate my life. GOSHDARNIT ALL TO HECK! As I was typing up the rest of this post, showcasing what I thought was a wonderful, deep, clear understanding of something that felt so obvious, I realized I don’t understand it fully. ACK!

And so, again, I’m in the realm of magic again. This isn’t the first time. This happens to me a lot with really interesting problems. I almost always discover a mistake in my reasoning when trying to write up an explanation with simplicity but also being comprehensive. And that’s exactly what happened here. (This is one reason why explaining my reasoning clearly to others is so important in my math process, and why I love kids talking about their mathematical thinking with others! It forces critical self-reflection.)

So here’s where we’re at. I do fully accept that if James’ Theorem is true, then we’re done! But I had a misconception that makes me feel a little embarrassed looking back at things.

I was thinking for each consecutive 5 integers, for example, had one of those integers divisible by 5, a different one had one of those integers divisible by 4, a different one had one of those integers divisible by 3, a different one had one of those divisible by 2, and the last remaining one had one divisible by 1.

Like I’m illustrating here:

But… yeah… that’s not always the case… Here’s the text I sent my friend James…

I’m now wondering about the factorial proof. I agree that if we know that the product of k consecutive integers were divisible by k!, the proof is done. But it’s not obvious to me anymore.

If we have the five numbers 27, 28, 29, 30, 31, I had convinced myself that one was divisible by 5, a different one was divisible by 4, a different one was divisible by 3, a different one was divisible by 2, and the last one was divisible by one. In my mind they had to be different ones. But that doesn’t work here.

And I thought they had to be different because in this case for example when they’re not, you then have to show after you divide one number out you might have to then divide a second number out and I don’t know why the second division necessarily works. Like in this example… after dividing out the 5 and the 4 and the 3… the 2 would have to divide into the 30 or 28. But the 28 already had 4 divided out so it wouldn’t be able to be divisible by 2 (what you’re left with is 7). And it feels magical that the 30, after being divided out by 5, happens to still then be divisible by 2.

In other words, how do I know the 2 will still be able to be divided into the 30 or 28?!?

Of course, I know with this specific example it can divide into the 30 (even after the 5 divides into the 30)… but how do I know if I did this with another set of n consecutive numbers, I wouldn’t hit a place where after you’ve divided out a bunch of numbers, you’re left with a number in the denominator that won’t go into the numerator?

Here’s my current state of things. I believe James’ Theorem and know it will unlock this problem. But I don’t get with any depth why it’s true. So I’m super happy that the original question posed by my student has been reduced to something “smaller.” I also think I might be at the point of wanting to google this to see what I can discover. I am still going to hold off for a bit longer, but I’m not sure how much longer.

We’re studying combinatorics (the art of counting) in precalculus. And we were seeing things like

pop up as answers to counting questions. And each of these evaluates to an integer.

The student asked: “BUT WHY? Why do these happen to evaluate to an integer?”

To give credit to what we’ve been doing in class, we spent about 3-4 days building up a deep conceptual understanding of why these would be the answers to particular counting scenarios, starting with first principles [1]. And we knew, since they were answers to counting problems, and that the number of outcomes to our given scenarios had to be an integer, that these had to evaluate to an integer. I would argue we have formally proved using a conceptual argument they had to turn out to be an integer. I think this is quite elegant! So on a mathematical level, we’re done. We’ve proved it.

However I liked this question a lot because I remember having it myself when I was in high school. And when I was teaching combinatorics for the first time. And randomly every so often since. I absolutely am certain I have thought about this before, and I also am certain I have no memory if I ever answered the question for myself or not. (My mind is like a goldfish.

So I pose it to y’all. On a purely algebraic level, how can we prove the following will be an integer:

Or I’d be okay with an even simpler version: how could we prove (again, purely algebraically, or with number theory, but with no reference to combinatorics) that the following will be an integer:

It’s harder than it looks, at least for me upon initial glance. I mean, I’m me, and needed to ground myself, so I started with a concrete example:

And I did the “canceling” and I saw we have some stuff that happens to divide evenly…

Yay, we see the 6 in the numerator divides out with the 3 and 2 (or the 6 divides out the 3… and 8 divides out the 2). But it feels like happenstance. And the more I wrote up examples, and I saw random things divide out from the denominator into various numbers in the numerator without rhyme or reason, the more happenstansical it all felt. It didn’t leave me to the next step in a proof…

The more examples I did in this very concrete and simplistic way, the more magical it felt. That I could find the right numbers in the numerator to cancel out with the numbers in the denominator — and still be left with an integer at the end! I loved the feeling of magic in math, because it means there’s something that is happening that I don’t understand, that’s ripe to be mined! And the moment I make something magical into something “obvious,” that’s when I know I’ve fully understood something. Right now, I’m in the magical phase.

In any case, I was thinking maybe I need to move forward with a proof by contradiction… or even induction?

I’m not sure if any of this made sense. But if it did, I’d love any thoughts in the comments!

[1] I have a very unique way of introducing combinatorics. At some point I should blog about it! But it basically reduces many, many combinatorial problems to the same type of problem. By the end of our unit, students don’t see “combinations” and “permutations” as different things that are related to each other. We zoom out and end up seeing them as the same thing. And moreso, my approach even lets them solve more complicated problems that combinations and permutations can become a bit more challenging to use. :)

Although I haven’t blogged in forever, I wanted to at least outline a few things tonight that I can return to. But it’s late and I’m tired, so I’m only going to do a few tidbits.

There were two sessions that got the mathematical side of my brain whirring.

Ryan P gave a talk on “A Rainbow of Random Digits,” where he went from 1D to 2D to 3D to think about a interesting problem. I thought what he showed was beautiful, especially for the 1D and 2D which I could see using with students in a sort of ongoing independent study/investigation. For the 1D question, he asked us if we had two dowels, of length 1 and length 2, and we split the larger dowel into two pieces, what is the probability that the (now three) dowels could form a triangle. A nice fun introductory question with various approaches to answering it. For the 2D question, he asked as a warm up: if you had two numbers chosen randomly between 0 and 1, (a) what is the probability that the sum is greater than the product, and (b) what is the probability that the sum of between 5 and 6? Lovely scaffolding, and the first question requires a little bit of calculus, which is fun. And then the full 2D question (which can be answered using some calculus, but I think it’s even more beautiful without it): if you have two random numbers between 0 and 1, what is the probability that the first non-zero digit of their ratio is a 1? a 2? 3?… 9? Totally fascinating 2D probability space, with lots of “triangle slivers.” To whet your appetite, here it is:

And totally not my intution, but the probability of getting a 1 (33.3%) is greater than the probability of getting a 2 (14.9%) which is greater than the probability of getting a 3 (10.2%), etc. The calculation involves some fun infinite geometric series. All of this was new to me. I also have no idea why getting a 1 digit is more likely than getting a 2 digit which is more likely than getting a 3 digit, etc. My intuition–which was way off–made me think all digits would be equally likely. Now honestly, I don’t know how teachers can actually build something like this problem into their normal classroom practice, but I do know that I’d love to work with a student to get them from the statement of the question to the answer in a lovely set of independent investigations, and some well-thought-out hints to guide. (And maybe chatgpt to write some simple code to do an initial simulation.) My friend and colleague suggested that maybe the distribution might be related to Benford’s law.

The second session that got the mathematical side of my brain whirring was by Bryan S. He had first learned about Conway’s “Rational Tangles” a few years ago, and wanted to present it to us. Wonderfully, I had first learned about these from Conway himself when I was in high school attending Mathcamp (and Conway was a guest lecturer). Conway was an electric speaker — and this one lecture of his imprinted itself on my mind. Now skip forward to this year. I had students work on “Explore Math!” projects and one worked on knot theory. I mentioned in my feedback to one student I had a really cool knot theory-adjacent thing I learned and I could show her. She responded saying “yes, please!” Of course it’s been years since I learned Rational Tangles. And it’s like the universe said “Oh, let me bring Bryan S. to you to remind you about all the nitty gritty of it.” And he was fabulous — a marvelous instructor who somehow managed to convey the excitement, weirdness, inquiry, all in a single short session. The crux of the setup is that you have two ropes held by four people:

There are two moves: T(wist) which has the front right person and back right person switch positions, where the front right person brings their rope under the back right person’s rope, and R(otate) where the four people just rotate clockwise. It turns out that by doing seqences of moves like this, such as TTRTRTTRT (etc.), you can get a pretty tangled tangle in the middle of the two ropes. One question — the main one we talked about — is if you can do a series of Ts and Rs to “undo” the tangle and get back into the original position of just two untangled ropes. Amazingly, a few Mathcampers created a digital version of this twisting and rotating and it took me about 20 minutes today to find it even though I knew it existed and I had played with it before. Here it is!

Another session I went to was Chris B’s Estimathon! I participated in my first estimathon at the Park City Math Institute many years ago (and again, a couple years ago). I hated it both times, for a few years. First, I hate estimating. I love thinking and calculating — but Fermi problems? I get really annoyed because I feel I rarely have the adjacent information that can unlock the problem. Second, the other people (both on my team and on all the other teams) were very, very competitive. I prefer a cooperative board game over a competitive board game, and in this, I think I feel similarly. That being said, I really enjoyed doing the estimathon with Chris and our other math conference participants. It was fun because the other people on my team were chill about it, and also let me sort of work alone at times when I got obsessed with a problem. Here are two example questions we were tackling:

Yes, I went to #5 because you can calculate that. Here’s what the scoring sheet looked like:

So you get to guess a minimum value and a maximum value for the range that the answer is in. And your score is ceiling(maximum/minimum). In other words, take the maximum value, divide it by the minimum value, and round up. So min: 2000, max: 3000 would yield a score of 2, and min: 2000 max: 4000 would also yield a score of 2…. but min: 2000, max: 4001 would yield a score of 3. Your goal is to get the lowest score. At the end, Chris gave us the absolute best idea (which I think he got from his colleague Emily). You have kids find out numbers for something that they are passionate about or would be an expert in. It could be “the number of pokemon” or “how many grandchildren does Sameer’s father have?” And then later in the year, you could create an estimathon out of these numbers — where kids have to see what they know about each other and their passions. I love this as a way for people to get to know each other.

There were two additional sessions that I attended, which were about students and math, and I loved both. First was by Jenny W, Lauren B, and Kevin J, and reminded me of the “5 practices” (https://www.nctm.org/Store/Products/5-Practices-for-Orchestrating-Productive-Mathematics-Discussions,-2nd-edition-(Download)/). It’s about using Desmos to highlight and discuss student thinking and to uplift student brilliance. Although I’ve seen many, many presentions on the 5 practices over the year (especially at PCMI), this was a great reminder of a lot of the things I don’t know, and teacher moves I’ve stopped flexing.

The second was by Lauren B called “I am, We are, You are.” It highlighted a few things to me. First, there’s a gap between the demographics of people who teach mathematics (and their identities) and the population of students who learn math (and their identities). She posed a question (of which she thinks the answer is yes): “Is there a way to expand identity in a math class?” I think this is a great question to chew on — and not easy. Especially if you take away classes like statistics or data science from the mix. We played with a super engaging Desmos activity which gave us choice on which data sets to plot against each other, and the fun part was guessing what the scatter plot would look like before we saw it. And she had a quote from Rochelle Gutierrez (who I’ve met before briefly!) which I couldn’t copy down quickly enough, but went something like “Do I have to be a better you in this classroom, or can I be a better me?” This is a student asking the question — essentially saying “Do I have to mold myself to be a miniature version of you, the teacher, to succeed in this room we’re in together?” I also thought that had a lot to chew on… in terms of what we expect from students and the culture that we build together.

Okay, it’s now 9pm and I’ve been at this for way longer than I intended, so to sleep I go.

***

It’s the next day and the conference has finished and I wanted to archive — briefly — the remaining sessions.

First, Hollylynne L. gave the keynote talk “Data Science is Everywhere and For Everyone.” Some takeaways for me are that there are a few large organizations (like NCTM, NCSM, ASA, etc.) that are collaborating to create a united data science position. It’s drafted, and about to be adopted, and has four guiding principles: (1) data science is contextual and multidisciplinary, (2) data science is an investigative process, (3) data science understanding and experiences are for everyone, and (4) data science educators must develop and practice ethical uses of data. The presenter shared her experience ethnographically observing data scientists for 9 months and what traits they exhibited. Then she shared with us that she feels like data science needs to be presented to kids with larger data sets — in terms both of cases but also in number of attributes shared (e.g. not just a survey with one or two questions, but a survey that has a ton of questions!). As an expert on CODAP, she shared how to fluently use it to show data, ask questions, and then interrogate the data. That part was inspiring, and in only a few minutes showcased the power of CODAP (codap.concord.org). Her talk also got me thinking about how our department has over the years shoved all of data science/statistics to our AT Statistics courses as we were making room for everything else we need to teach. Lastly, at the end, she shared a resource I want to follow up on called InSTEP. It is a free online site [https://instepwithdata.org/public/] that is designed to get teachers ready to teach data science and statistics, and it sounds like you learn lots of content, pedagogical moves, and you learn to use various tech tools but primarily CODAP (which is what I want to learn). So yay!

My next session was by Reed H and was an invigorating conversation on Standards Based Grading. He presented a “post mortem” of him implementing SBG in his precalculus classroom for the first time, sharing why he made the switch but also the tradeoffs that occurred. Although our school is moving in a different direction, I was still curious to see how the SBG conversations were going — and it reminded me how much I liked SBG even though it took me 4 years until I had refined it to the point where I could run it fluidly in my standard calculus classes. Reed’s own observations, and the conversations we had as small and large groups, also reminded me of my own path to SBG, and how I now know there is no single flavor of SBG that is going to work for all, because its success is dependent on so many cultural and institutional factors. And there is no magic bullet that is going to make it suddenly easy.

My last session was by Verónica Z and Doru H and was on Linear Programming and Other Means of Optimization. The presenters shared three ways to do optimization without calculus. First, the standard linear programming. The second was something called the “simplex algorithm.” Honestly, I got very confused at this point, because it isn’t in any way intuitive and I think that part of the presentation was meant for people who knew the method. So I ended up stopping my notes and writing “very confused” on my page. (I did find this example that went through the algorithm that I’m curious to read though, to get the crux of the algorithm, but don’t think it will explain “why” it works.) Finally, we learned the TOPSIS algorithm which was just invented in 1991 (“Technique for Order of Preference by Similarity to Ideal Solution”). New math! And the presenter, Verónica, did a really cool job of showing us how to make a decision of which new phone to buy (out of a choice of three) if we were looking at two variables: picture quality and battery life. It’s such a simple algorithm that — at the highest level — has you develop two new “fake” phones that exist — the best phone and the worst phone — one with the best of the traits of the three phones that exist and one with the worst of the traits of the three phones that exist. We plot all three phones and the best “fake” phone and worst “fake” phone. And then we find the “distance” from each of the three phones to the best and worst “fake” phones, and use those distances to rank the phones. Details are in my notes, but I loved learning new math in the universe, and new math to me!

Lastly, the conference was raffling off math art, and although I didn’t win, a new friend did, and she saw how much I coveted them and offered me hers. I demurred and then eventually accepted.

***

Personal Note: Speakers for the most part shared their slides with attendees. So I’ve downloaded the sessions I went to and saved them on my google drive. But I don’t know if they are officially public, so I’m just linking to them here for my own easy access.

It would be called something like “Reading and Writing about Mathematics.”

I’ve always been obsessed with reading books that aren’t textbooks about mathematics. I have almost half a bookshelf filled with these books. I love (when I have time) reading articles about modern mathematics in Quanta magazine. I’ve sometimes formally incorporated reading books about math in my classes in the distant past (a variety of books and articles in multivariable calculus; a book called Weaponized Lies: How to Think Critically in the Post Truth Era in my Algebra 2 class). And for many years, I’ve reached out to kids and set up many math book clubs, where we meet over lunch for a few different times to read books about math. I even was once interviewed about this, and wrote an article about this.

So why not formalize this into a class?

The biggest benefit I can see for the class would be introducing students to what modern mathematics is, so they don’t leave thinking mathematicians are just doing precalculus or calculating integrals in their ivory towers. Just like kids in physics will learn that the physical world is weird and wild when they are introduced to the ideas of quantum mechanics and relativity (even though they aren’t delving into the nitty gritty) in high school, I’d like a way for kids taking this class to learn that the mathematical world is weird and wild… and most importantly: human. A course like this will humanize math for students.

Right now I’m reading Jordan Ellenberg’s book Shape: The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else [here], with 5 students and 2 other teachers. As we’re reading it, there are parts that intrigue students but we need to parse it out together for it to fully make sense. One example was in this book, Ellenberg talked about the question “How many holes does a straw have?” and he brings us into topology and understanding how holes can have various dimensions. As we parsed this, I thought: oh, reading this and then also doing some mini-problem sets on the math could help kids understand things at a deeper level, or confirm their understanding of what they read. We could do our own simulations of random walks, in the section involving that idea. We could parse out the idea of the “necklace problem” which we’re going to talk about at our next meeting, which I’m sure the kids won’t understand because they aren’t drawing things out but more passively reading. In other words, we’d be able upgrade a passive reading of the book to an active reading of the book with a mini-problem set that brings ideas that might not fully make sense to life.

Additionally, there are books or articles that we could read that would help students understand the social construction of mathematics (hello anything on the controversy of the ABC conjecture and Mochizuki’s “proof”).

We could learn about the origin of ideas that seem to have existed forever or that we take for granted, but actually had to be developed (e.g. the idea of “0” or the controversy over calculus being on firm footing). And mind-blowing ideas like how all of mathematics almost fell down with the work of one mathematician, Godel. And we can learn about marginalized and overlooked people. There are some really great children’s books on mathematics, so we could read some of those as we begin the class! Maybe complement that with some excerpts from Douglas Hofstader’s Godel, Escher, Bach, a newspaper article about a modern mathematical breakthrough, some math poetry, a formal mathematical paper that has come out in the past 10 years and is hailed as one of the most important discoveries, and one more piece that’s maybe more traditional, so kids can see the wide variety of ways people write about mathematics.

And kids would think about how the writing that they’re reading is effective to reaching their intended audiences, or how it isn’t effective and what would make it better.

And kids could pick ideas or people to learn about, do their own research, and write their own popular mathematics writing. We can workshop it and publish a little journal at the end of class with our pieces.

I could reach out to English teachers to learn how they facilitate conversations about books so that it doesn’t go stale (we aren’t doing the same thing every time we discuss some of the reading). I’m sure there are universities that offer similar classes, so I could see what else is out there. And I know MIT has a graduate program in science writing that I could draw ideas from.

Okay, I’m done brainstorming for now. I just wanted to get all these ideas out before I forget them. I might update this post with additional ideas as my mind percolates. As I said… most likely nothing will come of this. But I could see having a really fun summer trying to put this course together.

NOTE: The deadline has passed to propose this course for next year. But it would take me a long time to develop the course outline anyway, so I could try to design it this upcoming summer and submit it for the following year.

I’m not sure this is comprehensive, but this is a list of books I’ve worked with kids/faculty in various capacities on [but it is not all the popular math books I’ve read]

*=I’ve done this for a student (or student and faculty) book club
**=I’ve led/organized a book group with just teachers on this
***=I’ve had students individually read this (to discuss with me in an independent study or for a math project)

*Anna Weltman, Supermath [here]
*Edwin Abbott, Flatland [here]
**G. Polya, How to Solve It [here] 
*Amir Alexander, Infinitesimal [here]
***Charles Seife, Zero: The Biography of a Dangerous Idea [here]
*G.H. Hardy, A Mathematician’s Apology [here]
***David Leavitt, The Indian Clerk [here]
*Steven Strogatz, The Calculus of Friendship [here]
**Steven Strogatz, Infinite Powers [here]
*Edward Frenkel, Love and Math [here]
*Robert Kanigel, The Man Who Knew Infinity [here]
***James Gleick, Chaos [here]
***Yoko Ogawa, The Housekeeper and the Professor [here]
*Margot Lee Shetterly, Hidden Figures: The American Dream and the Untold Story of the Black Women Mathematicians Who Helped Win the Space Race [here]
***Hiroshi Yuki, Math Girls talk about Integers [here]
***Hiroshi Yuki, Math Girls [here]
*Jordan Ellenberg, How Not To Be Wrong: The Power of Mathematical Thinking [here]
*Jordan Ellenberg, Shape: The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else [here]
*Daniel Levitin,Weaponized Lies: How to Think Critically in the Post-Truth Era [here] — the first third
***Ben Orlin, Math With Bad Drawings [here]
*Ben Orlin, Math Games with Bad Drawings [here]
***Paul Lockhart, A Mathematician’s Lament [here]
***Cathy O’Neil, Weapons of Math Destruction: How Big Data Increases Inequality and Threatens Democracy [here]
*Hannah Fry, Hello World: Being Human in the Age of Algorithms [here]
***Allison K. Henrich, Emille D. Lawrence, Matthew A. Pons, and David G. Taylor, eds, Living Proof: Stories of Resilience Along the Mathematical Journey [here]

The movies I saw using the pass:

1. Mission: Impossible – Dead Reckoning Part One
2. The Lesson
3. Oppenheimer
4. Past Lives
5. Indiana Jones and the Dial of Destiny
6. Shortcomings
7. Joy Ride
8. Haunted Mansion
9. Barbie
10. Easy A
11. Harry Potter and the Sorcerer’s Stone
12. Jules

I think my favorite was just a random movie I had never heard of and went to see to pass the time — The Lesson. And I thought the best was Oppenheimer.

I wanted to see what a great deal this was for me. For the basic calculations, most movies cost between $15.50-$18.99 depending on the time of day (and all movies on Tuesdays are only $9). For simplicity’s sake, I’m going to say the total cost to me with the Movie Pass over the course of the month was going to be $30+$1.89m (where 0<m<31) for the price of the pass and the cost of the service fee per movie.

And without the movie pass, it would have been around $17m (since I saw some movies during the day, and some movies at night).

So the break-even point was seeing less than two movies! I spent a total of $52.68 with the pass to see 12 movies, while I would have spent out of pocket $204 if I didn’t have the pass and saw all 12 movies. I know money doesn’t really work like this, but in my mind I like to pretend I saved $151.32!

Another way to think about this is to calculate the price per ticket:

So at 12 movies, it was like each movie only cost me $4.39. (I was saving $12.61 per ticket!) And the more movies I would have watched, the cheaper each ticket would have been!

But of course, this isn’t totally realistic. The thing about the Alamo is that you have at your disposal when you go there delicious foods and drinks. When I got the pass, I vowed I wasn’t going to always get something to eat or drink — and it would be a rare occurrence. But… life happens and I ended up getting something to eat on 5 of the 12 times I went. And I calculated the total and it came to $91.13 in food/tips.

So in reality, I spent a total of $143.81 at the movies, and it was in total $11.98 per movie on average.

Psychologically, it’s a little bit sneaky. You think “oh, I can see so many movies for $30.” Then there’s that $1.89 fee per movie that isn’t a lot but adds up. And of course the impulse food purchases is something Alamo surely is counting on. So I can go around telling people (and myself) “I saw 12 movies for $30″… when the reality is that what I should be saying is “I saw 12 movies and spent $143.81.”

Overall, was it worth it?

Absolutely.

Each week in July, he provided problem sets for us to work on. Justin kindly allowed me to share these: week 1 problems, week 2 problems, and week 3 problems. Each week’s problems were related to a chapter in the book that formed the backbone of the course: Theorems of the 21st Century by Bogdan Grechuk.

This book’s author went through articles published in 2000-2010 in the prestigious Annals of Mathematics and found (at least to his opinion) the most important results, and wrote a section trying to explain to ____ precisely what that result was and if possible, the general argument used to get that result. The reason I used the blank in the previous sentence is because I am a little confused who the intended audience of the book was. What was great is that I — a math teacher who majored in math two decades ago — could follow along some of the sections. And we had a number of participants who were middle school math teachers who were playing along too.

Justin picked interesting and accessible sections of the book for us to read (one per week) and built his problem sets around those sections.

• Week 1: Section 3.2, Representing 1 as a Sum of Reciprocals of Selected Integers.
• Week 2: Section 2.13, Transforming Convex Bodies into Balls.
• Week 3: Section 10.2, On Divisibility Properties of Dyson’s Rank Partition Function.

I got really hooked on the problems. They reminded me of doing super fun “morning math” at PCMI, where we got to discover new things, think in different ways, and remember the frustrations and elations of learning new math. I did hit some walls, but also had some insights I was really proud of. Since I wanted to take my hand at practicing LaTeX after many years, I typed up my work each week.

For personal archival purposes, I’m uploading my writeups for each problem set here:

We had a choice of reading the section from the book the problem sets were based on before or after working on the problems. Mostly, I chose to work on the problems first, and then after finishing the problem set to the best of my ability, work on understanding the section from the book.

In addition to working on these independently, all the participants had a Discord that we were invited to to chat. I had never used Discord before, but it was pretty neat. We introduced ourselves there, and not all people used it to talk, but some of us were sharing insights and asking questions about the problems there.

Each week, we had a Zoom. Justin offered it twice each week because everyone has such different schedules in the summer. The first part of each Zoom was Justin talking to us about what it means to do research, and be a mathematician, in the 21st century. It was interesting to see some of the charts and data he collected about number of publications, number of PhDs, and gender and minority representation (or underrepresentation, as the case it, obviously). Then we broke up into breakout rooms and discussed the problem set questions.

Additionally, we were invited to read and summarize a different section in the book — whatever section felt accessible and interesting to us. I really enjoyed this part! The first week, I chose a section and then got into a long discussion of something I didn’t understand on the Discord with a few others. I honestly think I found an error in the book — not in the math, I’m sure, but in how the theorem was described in the book.

The three sections I chose to read and try to understand were:

• Section 8.7: ”A Negative Answer to Littlewood’s Question about Zeros of Cosine Polynomials.
• Section 10.2: ”On divisibility properties of Dyson’s rank partition function.”
• Section 2.5: ”The Optimality of the Standard Double Bubble”

(You can read my summaries of what I understood from each section at the bottom of each problem set I turned in.)

Additionally, Justin invited mathematicians to talk to us on a Zoom each week. The truth is, this is one thing I didn’t participate in. I’m so mad at myself. I talked myself out of it, because I secretly was so nervous of being the only person in the Zoom and then having nothing to say. An irrational fear, but it won out.

Lastly, in the fourth week, Justin didn’t have problems and readings. We each were asked to create a 2-5 minute presentation on anything related to the course. Some wrote up what they learned. Others created activities for their classroom based on something they discovered mathematically or pedagogically. Some delved more into the readings. I did that.

Again, for archival purposes, here’s my presentation on “partitions” and “rank”. However just note that it’s designed for someone who read the reading.

The structure was sort of a choose-your-own adventure. You could choose to do the problems or not. You could choose to engage in the Discord or not. You could choose to the Week 4 presentation or not. I was curious how I’d feel about this being a remote PD. I’m so over Zoom that I almost didn’t sign up because of that. But it turned out that for someone like me, who wanted to do (almost) all the things, this all threaded together to make a really wonderful mathematics-filled July.

Justin didn’t charge for the course, which was great because I signed up after my school year ended so I probably couldn’t have gotten funding from my school. But in line with his values, he instead requested if we could, to donate $25 to one of these organizations.
African Institute for Mathematical Sciences (AIMS) – donate
Bridge to Enter Advanced Mathematics (BEAM) – donate
Enhancing Diversity in Graduate Education (EDGE) – donate
Julia Robinson Mathematics Festival (JRMF) – donate
Latinx and Hispanics in the Mathematical Sciences (Lathisms) – donate
Mathematically Gifted and Black (MGB) – donate

I chose two :)

I loved doing math over the summer, and having some light structure to guide me! And learn a bit about what has been happening in mathematics more recently in a different way than reading Quanta articles.

Below is a graph of stitches I need to make to create any given row. So for example in Row 16 and Row 17, I have to make 45 stitches.

So to find the total number of stitches from row 1 to row n, I have to take the integral of this function.

So to find the total number of stitches in the entire shawl, I’ll take the integral from 0 to 168.

And two things become apparent. First, we get the answer we expected from the calculations in the previous post: 30,072 stitches. Second, we have a Riemann Sum that looks like a bunch of mini-rectangles, but since we’re doing so many rectangles, it almost looks like we’re taking the integral of a line and estimating it using a Riemann Sum!

So what line is this?

The line is y=2x+11. Why? Every 2 rows of stitching should increase by 4 stitches. So each row we move up should increase 2 stitches. And the +11 comes from the fact that the first row has 13 stitches! So 2(1)+11=13.

Let’s check a little more to ensure we’re thinking properly! The number of stitches forrow1+row2=13+13=26. And when we take the integral of 2x+11 from 0 to 2 gives us an area of 26. (We can intuit that visually from the graph or algebraically by calculating the integral.) And similarly, the number of stitches for row3+row4=17+17=34. And when we take the integral of 2x+11 from 2 to 4 gives us an area of 34.

So let’s say g(x)=2x+11 is a function that gives the number of stitches in row x.

So if we want the total number of stitches from row 0 to row n, which we’ll call T(n), we have to integrate this function:

Whaaaa? Oh right! That’s precisely the equation we got in the previous post by the arithmetic sum! It’s the cumulative sum of stitches, not how many stitches per row.

And so let’s see how many stitches are needed in the first triangular section, second triangular section, third triangular section, etc…

Oooh this matches (as we’d expect) what we saw from the previous post:

Lastly, I was curious when I’d be halfway done with the Shawl. Since there were 30,072 stitches in the shawl, I wanted to see what row would give me the halfway stitch. It is row 117.245. Since each triangular section takes 12 rows to knit, that would be when I’m knitting the 9th triangular section of the shawl!

I was also thinking we could answer this using a basic geometric idea. When we finish the first triangular section, it has area A. Then when we finish the second triangular section, it has area 4A. Why? Because the second triangle is similar to the original triangle but with all sides twice as large. Thus, the area increases as the square! When we finish the third triangular section, it has area 9A. And so forth! So after the 14th triangular section, we have an area of 196A.

So the total area of the whole shawl is 196A. So half the area would have an area of 196A/2=98A.

This would happen after we finish the 9th triangular section (an area of 81A) and the 10th triangular section (an area of 100A). Yay, our geometric understanding matches our algebraic understanding!

Fin. (For now.)

Now I was at the library knitting, after writing a college recommendation and needing a breather… and I started wondering seeing some nice math in what I was doing. So I wanted to recreate some of that thinking here and extend it.

Before starting, you should know how it is knit. Here’s what I had at the library:

The way this is knit isn’t across the top like you’d think. No, in fact, you go along the triangular edge, from the left side to the apex to the right side — like the yellow lines I added in show. Then you flip it over and do a basic stitch on the back the entire way. So you’re doing two rows of stitches, one row on the front, one row on the back. So to finish “one full” repeat of the shawl, you go back and forth along the triangle. The four places with DOTS are special places. where you ADD a new stitch the first time you stitch across the triangle, and then you “lock in” those stitches when you stitch across the back. So in total, for each back-and-forth, for every two rows, you’re adding in a total of four new stitches.

And that makes sense — you have to add stitches every two rows — because your triangle is getting bigger!

I then did some counting of my stitches, and found the following:

I saw I had 64+1+64 stitches for where I was at in my knitting process.

I also noticed that I was on my 60th row of knitting. (The top measures how many rows I’ve knit… And from the center, I have 12*5=60 stitches.)

For my 59th and 60th row of knitting, I had 64+1+64 stitches.

So for my 61st and 62nd row of knitting, I’d have 68+1+68 stitches (remember I add 4 stitches every other row)

For my 63rd and 64th row of knitting, I had 72+1+72 stitches.

I love that I see an arithmetic series here! So I came up with this gem!

And indeed, it works!

So for row n we have this many stitches:

Now. Okay, let’s make things easier and only use even row numbers… then we get 2(row number)+9.

An obvious question for me is: how many stitches will be in this whole thing?

Well, looking at the original grey Boneyard Shawl picture, I see we have 14 triangles, and then a simple border.

Let’s ignore that simple border and only consider the 14 triangles. Each triangle requires 12 rows of knitting! So at the end, we’ll have done 14*12=168 rows.

In the last row, we’ll have 2(168)+9=345 stitches

So we have row1+row2+row3+row4+…+row167+row168 stitches.

This is 13+13+17+17+…+345+345

There are many ways to add up this arithmetic series. I get 30,072.

To find the sum to an even number of rows, let’s say we want to add up to row n. Then we have a total number of stitches of:

Whoa! It simplifies nicely!

When I’m knitting, I think about each set of 12 rows as one “triangular section.” And obviously, each set of 12 rows takes longer to knit than the previous set since there are more stitches. Let’s figure out how many more!

I wanted to see how much longer each section would take to knit than the section before. Each stitch takes about the same amount of time, so the second triangular section would be 564/276=2.04 times longer… Looking at the chart, we see:

And this makes sense to me. Although each section is taking longer to knit than the previous section, as we get near the end that ratio gets closer and closer to 1. This is because we’re only adding a fixed 288 stitches each time we create a new section, compared to the previous section.

And of course since the number of stitches in the first section is so small that adding 288 to that is a relatively large proportion of stitches,, while the number of stitches in the final sections are so large that adding 288 to that is a relatively small proportion of stitches.

Okay, let’s now get to time! I did a quick test of my knitting and it takes me about 4.651 seconds on average per stitch.

For the whole shawl it will take me:

30,072 stitches * 4.651 seconds / stitch * 1 hour / 3600 seconds = 38.85 hours.

But as I’m knitting, I think in terms of each triangular section. It’s too daunting to think in terms of the whole thing. So this is how much time it will take me-ish to do each section:

Okay. As I was doing all of this, I saw so many connections to calculus… derivatives and integrals! Maybe I’ll blog about that at some point… but hopefully you can see hints of those things popping up.

UPDATE: I wrote a quick followup for the blogpost showing some of the calculus connections: Knitting Math — Part II

[1] If you’re wondering… I’m really enjoying knitting this! It’s my first shawl, and I thought the pattern looked a little too advanced for me. In fact, it is the perfect first shawl! Stephen West has a tutorial video here. I’d say if you’re thinking of tackling a shawl, this is a great one to tackle!