# Our math year in review

I’m sick and not working today but instead have been sort of day dreaming about all of the math the boys and I worked on this year. The sheer amount of absolutely great ideas that people are sharing via blogs, twitter, or otherwise makes it incredibly easy to find fun projects. Here are some of my memories from 2016.

Again, though, I’m sorry that this likely reads so poorly. Despite being so sick it was fun to write this and think back through the year.

(1) Sharing math that I saw from professional mathematicians

Two of my favorites in this category came from Bjorn Poonen, Eugenia Cheng and Laura Taalman:

The project we did after seeing Poonen’s problem about n-dimensional spheres was the most viewed math project for kids on my blog this year:

Bjorn Poonen’s n-dimensional sphere problem with kids

A neat video from Eugenia Cheng inspired me to revisit an old post from Laura Taalman and do a project on “tiling pentagon cookies”

At the risk of failing to mention lots of people in my current Dayquil-induced state, I’m also incredibly grateful to these professional mathematians who have inspired tons of our projects with the math they’ve shared:

Jim Propp

Steven Strogatz

Evelyn Lamb

Dave Richeson

and one of the neatest things that happened to me all year was when Joel David Hamkins created an amazing “fold and punch” activity based on an activity that I found in some old material from “Family Math Night” at my younger son’s school:

Math for nine year olds: fold, punch and cut for symmetry

(2) James Tanton

I have to give him his own category because the work he is doing to share math with kids (and everyone, really) is astonishing.

This project on cutting Mobius strips that I saw in his book “Solve This” is one of the most incredible math projects I’ve ever seen:

An absolutely mind-blowing project from James Tanton

Here are all of our project inspired by Tanton – you can basically pick one at random and have an amazing math conversation with kids (though if you don’t want to pick at random the candy dividing one is really cool!):

Project inspired by James Tanton

(3) Laura Taalman and Henry Segerman’s work in 3d printing

The work that Taalman and Segerman are doing with math and 3d printing is stunning. I mentioned one of projects inspired by Taalman above – there are dozens’s more:

Projects inspired by Laura Taalman

Segerman published a new book about math and 3d printing this year.  I was incredibly lucky to be able to bring the boys to a talk he gave about his work:

Projects inspired by Henry Segerman

(4) Sharing new and / or popular math ideas with kids.

Erica Klarreich and Natalie Wolchover are doing amazing science (and especially math!) journalism work at Quanta Magazine. Oh to have had writing like theirs around when I was a kid!

I’m so happy to see pieces like Klarreich’s article on Maryna Viazovska’s sphere packing result

In fact, when I asked my older son what his favorite math memory was from 2016 he said it was learning about sphere packing. Yay!

Wolchover’s article about hyperuniform distributions blew me away and led to a really fun project with the boys:

Using a Natalie Wolchover article to talk about hyperuniform distributions with kids

I don’t think it is possible to overstate the importance of Klarreich’s and Wolchover’s writing. They are going to influence a generation of young mathematicians and physicists.

Another fun math-related item that got a lot of attention this year was Sugihara’s “ambiguous cylinder”:

We really had fun playing with this shape and I want to give a special thanks to Dave Richeson and Brenda Landis for sharing a 3d print of the shape.

Playing with Sugihara’s “ambiguous cylinder”

(5) The sphere packing problem reminded me of the new PBS Infinite Series work that Kelsey Houston-Edwards is doing.

Holy cow are these videos amazing! Here’s just one example:

Wwe are one behind because of the holidays, but each of Houston-Edwards’s videos has inspired a really fun project. Her videos are great tools to use to share math with kids.

Projects inspired by Kelsey Houston-Edwards

(6) Three projects from twitter that completely blew my mind:

There are actually a couple of projects that Simon Gregg’s tweet inspired. The main picture is this one (which always has weird embedding problems, so sorry it isn’t aligned correctly):

Prepping for this project to make sure that we could do it with our Zometool set was really fun, too:

A neat post from Simon Gregg

Much like the James Tanton “cutting a Mobius strip” project above, the idea is to try to guess what the shape is going to look like when you unzip it!

This was a great project with the boys and I also used it for a talk to a high school math camp at Williams. If you play with the Desmos program below your mind will be blow, too 🙂

(7) We played with more math-related art, too:

Paula Beardel-Kreig’s “Puff Boxes” were incredibly fun:

Playing with Paula Beardell-Krieg’s Puff Boxes

And Henry Segerman’s 3D Printing book introduced me to the work of Bathsheba Grossman, and we explored several of her creations:

Our projects with Bathsheba Grossman’s work

(8) Our Zometool work

I know there are lots of ways to spend money on math-related games, books, and toys in general. Building up a good Zometool set is my #1 recommendation. The opportunities to play and learn and study are endless!

Here are all of our Zometool projects:

Our projects with our Zometool set

I particularly recommend the bubble projects and Nesting Platonic Solids

(9) Dan Anderson’s Gosper Curves

When I asked my younger son what his favorite project from this year was he said that it was playing with the Gosper Curves. He really likes fractals!

We got a nice surprise in April when Dan Anderson sent us some laser-cut versions of the Gosper island shape:

Playing with Dan Anderson’s Gosper Curves

We did a few more Gosper-related project (including a Zome one) which are here:

Our Gosper projects

And, of course, we did a million projects inspired by ideas that Dan shares on twitter:

Our Projects Inspired by Dan Anderson

(10) Patrick Honner’s Pi Day exercise

On March 14th Patrick Honner shared a fun little “Pi Day” exercise:

This terrific project inspired me to try it out in 4 dimensions. That led to a fun multi-day project with my older son as we search for which 4-dimensional platonic solid was the most spherical (according to Honner’s definition).

This project combined ideas from geometry, Zometool, and 3D printing.

Here’s a collection of the projects:

Patrick Honner’s Pi Day Exercise in 4 dimensions

Honner’s “pi day” exercise is a perfect example of why I love all of the sharing of math ideas that people are doing these days. Not in a million years would I have come up with an idea like that – luckily he did, though, and it turned out to be a really fun way to explore more than just 3d objects!

It really was a great year in math for us. Can’t wait to see what 2017 brings.

# A project inspired by an AMC 12 octagon problem

The problem pictured below from the 2003 AMC 12 gave my son some trouble:

We talked through it together a few days ago, but I thought it would be fun to try to do an octagon-inspired math project today.

We started with the problem and then talked a bit about a 3d print we found on Thingiverse:

Next we took a look at a version of the 3d printed shape that we made from our Zometool set. You can’t make a regular octagon with a Zometool set, and the fact that our shape didn’t have a regular octagon led to a good discussion:

For the last part of the project we tried to find the volume of our truncated cube.

# A hand waving approach to a problem posted by Cut the Knot

Saw this tweet yesterday:

It was a fun problem to think about and the two solutions on the site use the Stolz-Cesaro Lemma, which is basically l’Hospital’s rule for sums.

Through the various Christmas preparations yesterday I was wondering if there was a simple way to see why the limit exists in the first place. What follows below isn’t a rigorous proof (or even close to one!) but instead how I convinced myself that the limit probably does exist.

Since seeing Tim Gowers “live blog” his solution to an old IMO problem, I’ve been interested in occasionally sharing the solution process rather than polished solutions to problems. Two examples of problems I’ve used for that idea are below:

A Challenge / Plea to math folks

A challenge relating to a few problems giving my son trouble

So, for the problem at hand, here’s my “hand waving” approach to convincing myself that the limit even existed:

We know that:

$\lim_{x\to\infty} E_n = \lim_{x\to\infty} (1 + \frac{1}{n})^n = e$

and that

$\lim_{x\to\infty} H_n = \lim_{x\to\infty} (1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n}) \approx \ln{n} + \gamma$

So I’ll approximate the difference we are considering like this for large $n$:

$(E_{n+1})^{H_{n+1}} - (E_n)^{H_n}$

$\approx e^{H_{n+1}} - e^{H_n}$

$\approx (e^{H_n})(e^{\frac{1}{n+1}} - 1)$

$\approx (e^{\ln{n} + \gamma}) (\frac{1}{n+1} + \frac{1}{2(x+1)^2} + \ldots )$

$\approx (e^{\gamma}) * n * (\frac{1}{n+1} + \frac{1}{2(n+1)^2} + \ldots )$

$\approx e^{\gamma}$

The fact that this hand waving approach arrives at the “right” answer is just a coincidence as I’m playing pretty fast and loose with limit rules. But, at least I now have some indication that this strange (and lovely!) $\infty - \infty$ limit might actually exist.

Just for fun here’s what the expression looks like for n up to 1,000,000:

Definitely a fun little problem to noodle over 🙂

# Sharing a “visual pattern” triangular number identity with kids

Saw a fun tweet last night from Matt Enlow:

Here’s the underlying tweet since it doesn’t show up in wordpress:

Shortly after seeing the tweet my younger son and I were playing Othello. The combination gave me the idea for today’s project.

We started by talking about the triangular numbers and why consecutive triangular numbers might sum up to be a perfect square. My older son’s idea of how to think about triangular numbers was computational rather than geometric.

Now we moved to the Othello board and looked at the geometry. My younger son found two different geometric ideas which was fun.

Finally, I gave the kids a challenge to try to find another geometric version of the identity. This question was a bit more challenging that I intended it to be, but we eventually got there and even saw how our new picture related to the sum formula that my older son used in the first video:

# An AMC12 algebra problem that gave my son trouble

The problem below gave my son some trouble this morning:

When he got home from school we talked about it in more detail and it seemed to make more sense for him than it did this morning. The problem is a nice introductory algebra / quadratic problem:

Next I showed him a similar solution, but where “x” represented a different number:

Finally – just for a completely different way of looking at the problem – I wanted to show him a way that we could use the choices to help us find the solution. This is sort of cheating, but he was very confused by the problem this morning and I wanted to show him a way to get a little un-stuck when you are stuck.

Also, we got interrupted by the guy servicing our furnace – so sorry the video jumps in the middle 🙂

# Writing 1/5 in binary

I’ve spent the last couple of days talking about binary with my younger son. We were inspired a bit by Kelsey Houston-Edwards’s latest PBS Infinite Series video on binary. It has been a fun little review.

Tonight we talked about how to write 1/5 in binary. I didn’t really know how the conversation would go, but it ended up being a nice little arithmetic review.

We started talking about the problem and he settled on the idea that we needed to find a number that would equal to 1 when we multiplied by 5. That got us going on the arithmetic review since that idea works in any base.

Now we had to figure out now to divide 1.000000000…. by 101 in binary. This long division problem gave us an opportunity to talk about subtraction (and borrowing) in binary:

The last step was multiplying the number we thought was 1/5 by 101. Once again this was a great opportunity to review some basic ideas about arithmetic and multiplication.

So, an unexpectedly fun project! We learned what 1/5 was in binary and had a nice review of subtraction, division, and multiplication along the way 🙂

# Sharing Kelsey Houston-Edwards’s binary video with kids

Kelsey Houston-Edwards released a new math video last week:

So far we’ve been able to use all of her videos for great weekend projects. This video had a fun little surprise because we’d seen the problem she talks about in a (seemingly) totally different context – an old magic set! Once we dug out that out magic set from under my younger son’s bed we started the project 🙂

Before jumping to the challenge problems, we looked at the old magic trick:

Finally, we tried to answer the two challenge problems from Houston-Edwards’s video. I’m sorry this got a little rushed at the end – I’d not noticed that we were out of batteries! We finished with about 10 seconds to spare!

The two challenges are great problems for kids to think through – the boys found a few interesting patterns even though the relationship with powers of 2 was a little hard for them to see.

# Talking about “The Cat in Numberland”

Last we did a couple of projects based on Kelsey Houston-Edwards’s video about infinity:

Sharing Kelsey Houston-Edward’s Infinity video with kids

Extending our project on Kelsey Houston-Edwards’s infinity video

I got a comment from Allen Knutson on the 2nd project recommending using “The Cat in Numberland” to talk about infinity with kids. I ordered the book immediately and had the boys read it a few times this week. We got around to talking about it this afternoon.

Here’s their initial reaction to the book:

In the last video we I asked the boys for 3 ideas from the book that they wanted to talk about. They chose:

(1) When “Hilbert’s Hotel” is full, how do you fit one more person in?

(2) How about fitting in 26 more people?

(3) When you take away half the people how can the hotel still be full?

Here’s the explanation for part 1 – the idea here shows one strange thing about infinity!

Here’s part 2:

My older son got a little confused by the numbering of the hotel rooms in this video. The numbering of the rooms is hardly the main point, but it is nice to be able to review / revisit some counting ideas in this unusual context:

For part 3 we had a nice conversation about how you can form a bijection between the counting numbers and the non-negative even integers. That conversation went pretty fast so I asked the boys to each find another bijection and got really lucky when they picked two pretty cool ideas – powers of 2 and prime numbers.

The last movie ended with a question about whether or not the primes were infinite. This was also hardly the main point of the project, but turned out to be a fun way to end the conversation today.

So, thanks to Allen Knutson for pointing me to the book and to Kelsey Houston-Edwards for the Infinity video which has now led to three fun projects with the boys!

# A fun coincidence with an Eduardo Viruena creation

I got some great feedback from Eduardo Viruena on the project we did with one of his math designs:

A short project inspired by a Holly Krieger tweet

One of his other designs he pointed me to was this one:

A small stellated dodecahedron approximated by dodecahedra

Here’s his picture:

I printed it over the course of the day (took about 6 hours) and showed it to my younger son when he got home from school. Here’s he described the shape, including noticing one very interesting pattern that he thought would form an Archimedean solid:

It turns out that the shape he saw would indeed be an Archimedean solid. In fact, it the exact solid we did a project on a few weeks ago!

Here’s that project:

Revisiting our Zometool Snowman

Which was inspired by this tweet from Eli Luberoff:

The Snowman is still up in our living room (which I’ll attribute half to coincidence and half to laziness . . . . ) so we looked carefully at the two shapes:

Amazing what kids notice when they look at mathematical objects!

# Playing with Sugihara’s “ambiguous cylinder”

The amazing video below came out over the summer:

My favorite memory of the video is my 5th grader seeing it for the first time and saying “dad, that mirror is terrible.” Ha!

Over the summer we printed a few of the shapes and played around with them, but ended up giving them away. Today I was reminded of the shape by this tweet from Dave Richeson:

I wasn’t sure if the print he showed in his tweet was available, so I found a different one on Thingiverse and started the print so the boys could play with the shape again tonight:

Ambiguous Cylinder Illusion by Make_Anything on Thingiverse

As luck would have it, though, Brenda Landis did put the print from Dave’s tweet on Thingiverse: