Maryna Viazovska’s Sphere Packing talk at Harvard

Last spring Maryna Viazovska solved a fascinating unsolved problem about the most efficient way to pack 8-dimensional spheres. Erica Klarreich wrote a great story about the problem and Viazovska’s result in Quanta magazine:

Reading the article I was surprised to learn that a former classmate of mine – Henry Cohn – played a role in the story. This little connection made me follow the story more closely than I’ve followed other math stories. So, I was super excited when I saw that Viazovska was speaking at Harvard:


Her talk last week was fantastic. I’ve been out of academic math for too long to follow every detail, but I’d say that the talk was pitched at a 1st year graduate student level which meant that I could follow enough of it to get the general idea of what she did.

Also, as an aside, while probably any talk in the Harvard math department has impressive people in the audience, the various math folks sitting around would have reduced me to a pile of jello as a speaker. One sort of surprising thing, though, was that there weren’t a lot of people there aside from the Harvard faculty members. The last time I’d been at a math talk at Harvard was in the mid 90’s when Cliff Taubes was doing a series of lectures on the new Seiberg-Witten theory. Those lectures (I think) were in the same classroom and were standing room only. I’d guess maybe 30 total people were there for Viazovska’s lecture, which was a little disappointing to me.

The talk started off with a couple of small surprises to me. First, it actually isn’t totally obvious how to define the best packing of spheres (in any dimension), so you need to be careful and get the definitions right. Second, the best packing aren’t necessarily nice symmetric lattices.

Next up was a discussion of the background work that Henry Cohn and Noam Elikes (who were both there for the talk) had done. That work included a theorem giving a bound on the sphere packing density that used a function with certain peculiar properties. I think (but am not totally sure) that people knew that this function must exist, but no one knew how to find it or describe it in any simple, closed-form way. Viazovska’s result was writing down the formula for that function in 8 dimensions.

She didn’t show the entire procedure for how to write down the specific function (but very dryly and hilariously assured the audience that she’d checked the calculation at home and it worked so there was nothing to worry about🙂 ), but instead showed a case that she said was a bit easier. The easier case still seemed like an impossible problem to me, though sort of reminded me of this famous problem:

Let x_n be the positive, real solutions to the equation \tan(x) = x. Show that \sum\limits_n1/x_{n}^{2} = 1/10.

Unfortunately I had to leave the talk before the end to get to a dinner, but I’m really happy that I got to see it. I hope there’s more publicity around this result because that students who are interested in math will find it to be fascinating. There are also several nice techniques from Fourier analysis and complex analysis in the result that would make great examples in intro graduate level courses in those subjects.


Anyway, I thought her talk was the end of the story, but I was pleasantly surprised to see a poster for a second talk – this time at BU – earlier this week.


I happened to have a meeting in Boston at 2:30 and was able to take the T over to BU to catch Henry’s talk. His talk was more of an overview of the sphere packing results over the years with a sketch of Viazovska’s result. For me it was a nice complement to the talk at Harvard because now I had a much better feel for the historical context of the results. He also mentioned a little bit more about the work that Tom Hales did on the 3 dimensional result, which is an incredible story all by itself.

I think this talk that he gave at the Berlin Mathematical School this summer is close to the talk he gave at BU:


I wish that I had some good ideas for how to share the 8 dimensional sphere packing results with kids or with the public. Right now I don’t, but it feels like the problem is accessible even if the solution isn’t. For now my best high dimensional project is Bjorn Poonen’s sphere problem:

Sharing Bjorn Poonen’s n-dimensional sphere problem with kids

At least this problem shows off some of the really strange properties of high dimensional spheres.

Anyway, even though I don’t really have any good ideas about how to share this new result, it was a really nice couple of weeks. It was fun getting a little peek back inside of academic math.

Struggling through a geometry problem

My older son picked an interesting challenge problem from Mathcounts for our movie this morning. As with most Mathcounts problems there is a quick way to solve it, but he didn’t see that quick way.

It was interesting watching him work his way through it:


It was tricky for him to see how to proceed although he did have most of what he needed on the board. I asked him to look at some of the angle relationships and that helped him get to the end:


Some mistakes solving a tricky problem about a cone

I made up a problem about a cone last night and asked my older son to take a crack at solving it. He’s been working on 3d geometry lately. It took him 3 tries, and I think the mistakes are really interesting to follow.

First try:


Second try:


Third try:


Dave Radcliffe’s “unit fraction” tweet

Saw a neat tweet from Dave Radcliffe a few weeks ago:

I’d played around with it a bit on Mathematica and the code was still up on my computer screen when we were playing with base 3/2 yesterday, so the kids asked about it.

Radcliffe’s proof is a bit too difficult for kids, I think, but the general idea is still fun to explore. I stumbled through a few explanations throughout this project (forgetting to say the series should be finite, and saying “denominator” rather than “numerator” at one point), but hopefully the videos are still clear.

I started by explaining the problem and looking at a few simple examples:

Next we looked at how it could be possible for a finite sum of distinct numbers of the form 1 / (an integer) could add up to 100, or 1000, or some huge number:

Now that we understood a bit about the Harmonic series, we jumped to Mathematica. I sort of half explained / half skipped over the “greedy algorithm” procedure that Radcliffe uses in his paper. I thought seeing the results would explain the procedure a bit better.

We played around with adding up to 3 and then a couple of numbers that the boys picked.

After playing around with a sum adding up to 3, we tried 4 and the boys got a big surprise. We then tried 5 and couldn’t get to then end!

After we turned off the camera we played around with the sum going up to 5 a bit more sensibly and found that there are (from memory) 102 terms and “n” in the last 1/n term has 142,548 digits!

So, a little on the complicated side, but still a fun math fact (and computer project!) for kids to explore.

Revisiting James Tanton’s base 3/2 exercise

Several years ago we played around with James Tanton’s base 3/2 idea:

Fun with James Tanton’s base 1.5

A tweet from Tanton reminded me about his project earlier this week. I was excited to revisit it and got a double surprise when my older son told me that he actually did it in his 7th grade math class last week! It is nice – actually amazing – to see Tanton’s work showing up in my son’s math class!

An unfortunate common theme with some of our recent projects is that they aren’t going as well as I hoped they would. Still, though, this was fun and I’ll have to spend a bit more time thinking about the last bit – how to write 1/3 using base 3/2.

We started by reviewing base 2 and, in particular, how you can play around with binary using blocks.

Next we looked at base 3/2. I’m sorry that this video runs 10 min – I definitely should have broken it into 2 pieces.

Finally we accidentally walked into a black hole. I assumed that writing 1/3 in base 3/2 wouldn’t be that difficult and that an easy pattern would emerge quickly. Whoops.

Turns out that no pattern emerges quickly, and even playing around on Mathematica for a bit after we turned off the camera we couldn’t find the pattern. The discussion facilitated by the work on Mathematica was great – at least my kids learned that (i) there are multiple ways to write a number in base 3/2, and (ii) there are easy sounding project that I can’t figure out!

I hope to revisit this part after I understand it better myself. Any help in the comments would be appreciated.

I really like this project and am sad that a little bit of stumbling around by us might have obscured the beauty of Tanton’s idea. Hope we’ll be able to revisit it soon.

One that didn’t go so well

Last week my old son had a nice challenge problem in his weekend math program:

An 8x8x8 cube is painted black out the outside and then chopped into 1x1x1 cubes. How many of the 1x1x1 cubes have paint on 0, 1, 2, 3, 4, 5, and 6 faces?

This problem was even accessible to my younger son, in fact, we started the project to by discussing that problem:


So, for today’s project I thought it would be interesting to try out a similar problem in 4 dimensions.

It turned out to be far more confusing to the boys that I was expecting. So, whoops, I guess. I’ll go through this project again with them, but using a very different approach. But, for now, here are the 4 long discussions we had about how to attack this problem.

Mostly this post is a reminder to me that not everything goes as well as you’d hoped it would:





Talking through Matt Enlow’s “truth” tweet with my older son

Saw this really neat tweet from Matt Enlow earlier today:

I thought it would be fun to talk through the list with each of my kids and ask the for examples of each on of the statements. My younger son was home this afternoon so he went first – the project with him is here:

Talking through Matt Enlow’s “truth” tweet with my younger son

Tonight I worked through the 7 parts with my older son

(1) The statement is true, here is a proof:

His ideas involved the area of a triangle and the Pythagorean theorem


(2) I believe the statement is true, here’s why

It took him a while to come up with something, but eventually he mentioned the quadratic formula, which seems like a great example:


(3) My gut tells me that this is true

Here he picked a postulate from Euclidean geometry – a single line passes through any two given points.


(4) I have no opinion as to whether or not the statement is true or false

OMG OMG OMG OMG – I’m not even going to give it away. The best!!


(5) My gut tells me the statement is false

He had a really hard time coming up with an example here.

Eventually he came up with a really interesting example from the quadratic formula – the quadratic equation x^2 + 24x - 1 = 0 has integer roots.


(6) I believe the statement is false – here’s why

He came up with a nice example here – a cube has integer sides and volume 7.


(7) The statement is false – here is a counter example.

He came up with a simple example first -> 1 + 1 = 7. I told him that was too cheap and asked for a second example.

The second example was |x| = -2. He also gave a really nice explanation.


So, a really fun project – as I said in my younger son’s project, I’d love to see lots and lots of kids come up with examples for each of Matt’s 7 statements.

Talking through Matt Enlow’s “truth” tweet

Saw this really neat tweet from Matt Enlow earlier today:

I thought it would be fun to talk through the list with each of my kids and ask the for examples of each on of the statements. My younger son was home this afternoon so he went first:

(1) The statement is true, here is a proof:

He said that he knew how to prove that the square root of 2 is irrational, and got most of the way through the proof!


(2) I believe the statement is true, here’s why

For this one he picked the Collatz Conjecture:


(3) My gut tells me that this is true

He picked some of the infinite series stuff we’ve been talking about lately!! I couldn’t believe that this was his example for something that he thought was true – yay1!


(4) I have no opinion as to whether or not the statement is true or false

Here he picked the question about whether or not the number of twin primes is infinite.


(5) My gut tells me the statement is false

He had a hard time with this and I ended up breaking the video into two pieces because once he came up with a question relating to the Gosper Island, I had to go get them!

What his gut tells him is false is that the perimeter of the Gosper island is finite.



(6) I believe the statement is false – here’s why

Here he picked a really neat example, and one that his class is working on in school. The game is to make integers using only 4 fours. So, for example, 17 = 4*4 + (4/4).

He does not believe that you can make all of the integers using just 4 fours, and he has a nice reason.


(7) The statement is false – here is a counter example.

He didn’t quite understand the statement, but still came up neat idea. He made the statement “you can’t express all integers using binary” and (sort of) showed why that statement is false.


So, a really fun project – I’d love to see lots and lots of kids come up with examples for each of Matt’s 7 statements.

Using Vi Hart’s hyperbolic space tweet with kids

Yesterday Vi Hart tweeted about an amazing program you can use to explore hyperbolic space:

The program, which (I think) came from joint work by Andrea Hawksley, Vi Hart, Henry Segerman, and Mike Stay, is here:

A program to explore hyperbolic space

The youtube video in Hart’s tweet shows her playing with the program and explaining a little bit about the different shapes you see. After watching that video I thought it would be fun for the boys to explore the program without any explanation of what was going on. I was really interested to see how a kid would react to seeing this 3 dimensional hyperbolic geometry for the first time.

Here’s what my older son had to say:


and here’s what my younger son had to say:


This is such a fun program to let kids play with. The boys noticed many strange properties of hyperbolic space without knowing they were looking at a strange new space. It really is amazing to have resources like this right at your fingertips!

Exploring Colin Adams’s “Why Knot?”

I bought a copy of “Why Knot?” last spring and we played with it once:


Playing with Colin Adams’s “Why Knot”

I brought it out again today to explore a bit more about knots with the boys. The goal of the project today wasn’t depth, but rather just exposure to some basic ideas in knot theory. Knot theory is very simple in some ways an incredibly complex in other ways – I just wanted to show the boys a couple of examples of ideas that help us tell two knots (or links) apart.

We started with a basic idea about links and eventually talked about the linking number. The boys had some nice ideas about how you might tell two links / knots apart:


I felt a bit unsatisfied with the first part of the talk so I decided to show the boys a couple of simple examples of the linking numbers staying unchanged when the links move around a bit:


The next topic we looked at was the un-knotting number of a knot. I used the trefoil knot as the example here. First the boys played around with the knot to convince themselves that it was not the same shape as a circle, or trivial not.

Once they believed it was, indeed, different from the trivial knot, we changed one crossing and found that it was now the same as the trivial knot.


The last part of today’s project was looking at the different types of knots with 0, 1, 2, 3, and 4 crossings. This was just a high level overview and I was hoping to hear some simple ideas from the boys about, say, why all knots with one crossing were the same as the trivial knot.

At the end (and off screen) I asked them to make a version of the knot with 4 crossings. That picture is below:


So, a fun project. I’d love to think a little more about how to make knot theory accessible to kids. It sort of feels as though it is a subject that requires more than just one or two projects, though. Right now I’m not really sure what to do next.