Playing with Borromean rings

We’ve seen three references to Borromean rings in the last few days. None of the references had anything to do with each other, but taken together . . . well, I figured we had to do a project.

The first reference was in our new book about knots:


The second was in the newly released Numberphile video with Tadashi Tokieda

The release of this video was sort of a double coincidence since we just saw Tokieda give a talk at MIT last weekend. Our project based on that talk is here:

Tadashi Tokieda’s “World from a sheet of paper” lecture

The third was in a George Hart video that Laura Taalman tweeted out today. The video is from 2012 and I can’t believe that I’d never seen it before. The boys are excited to try out some of the programs he mentions for a new 3d printing project. Can’t wait ūüôā

After watching George Hart’s video we started our short little project. The first thing that I wanted to do was see if the boys could figure out how to orient the Sierpinski tetrahedron so that it looked like a square. They were able to do it and there was even a surprising (and totally accidental) twist!

We have a 3d printed Sierpinski Tetrahedron thanks to Laura Taalman’s amazing Makerhome blog:

The Sierpinski tetrahedron on Laura Taalman’s Makerhome blog

Here’s our talk about the shape:

Next we talked about Borromean rings. It was a fun challenge to the boys to make the shape out of the “tangle” that comes with Colin Adams’s “Why Knot?” book. I loved the way the boys worked together to figure out how to make the shape:

So, a fun coincidence seeing three different references to Borromean rings in the last couple of days. It was fun to turn all of those references into a little project for the boys.

Using Laura Taalman’s 3D printed pentagons to talk math with kids

Saw this incredible tweet from Laura Taalman a few days ago:

She’s created 3D print versions of all of the known pentagons that can be used to tile the plane!! The 15th pentagon was discovered in 2015, and that discovery is discussed by Evelyn Lamb in this amazing article:

There’s Something about Pentagons by Evelyn Lamb

I thought that the pentagons and tiling patterns would be really interesting for kids to see, so I spent the day printing the “cookie cutter” version of all 15 of them. Once the printing was done, we started talking about pentagons:


I had each of my kids pick a pair of pentagons to talk about – my son picked #8 and #13 (as numbered on Taalman’s program linked above). Here’s what he noticed about these pentagons:


Next, my older son talked about what he noticed about pentagon #5 and pentagon #9 – I was happy to hear him speculate about the possible tiling patterns formed by the shapes.


When I was printing the pentagons I happened to notice that #5 and #6 looked pretty similar. It turns out that there are two different ways to tile the plane with these shapes, so I did print out these tiling patterns. Luckily my son had picked #5.

It was fun to see that the 2nd tiling patter wasn’t totally obvious to the kids, but they figured it out eventually.


We wrapped up by playing around with the interface Taalman made for the pentagons. It is amazing to be able to see and play with these patterns. The boys played with it for 20 minutes after we finished filming ūüôā


So, a super fun project. The great thing about 3D printing (and the thing I can’t say thank you enough to Laura Taalman for teaching me) is that holding these shapes in your hands leads to great conversations!

10 fun math things from 2014

I’ve been paying attention to math a little more in 2014 than I have in previous years and thought it would be fun to put together a list of fun math-related things I’ll remember from this year:

(10) Dan Anderson’s “My Favorite” post

Dan asks his students to talk about things they would like to learn more about in math class, and the students talked about subjects ranging from topology to diving scoring. I was really happy to see the incredibly wide range of topics that the kids thought would be interesting. Beautiful post by Dan and a fantastic list of topics chosen by his students – this one made a big impression on me:

Dan Anderson’s “My Favorite” post

My initial reaction to Dan’s post is here:

A list Ed Frenkel will love

(9) Laura Taalman’s Makerhome blog:

We bought a 3D printer early in the year and it allowed us to do a bunch of math projects that wouldn’t have occurred to me in a million years. Most of those projects came either directly or indirectly from reading Laura Taalman’s 3D printing blog. As 3D printing becomes cheaper and hopefully more available in schools, Taalman’s blog is going to become the go to resource for math and 3D printing. It is an absolute treasure:

Laura Taalman’s Makerhome blog

An early post of mine about the possibilities of 3D printing in education is here:

Learning from 3D Printing

and one of our later projects is here:

Klein Bottles and Möbius Strips

(8) Numberphile

It has been nearly a year since Numberphile’s fun infinite series video hit the web. I know people had mixed feelings about it, but I loved seeing a math video spark so many discussions:


I’ve used so many of their videos to talk math with my kids, I’m not even sure which of them to pick for examples. Here are two:

Using Numberphile’s “All Triangles are Equilateral” video to talk about constructions

Some fun with Numberphile’s Pythagorean Theorem video

(7) Fields Medals and the Breakthrough Prizes

Erica Klarreich’s coverage of the Fields Medals over at Quanta Magazine was absolutely amazing. Two of her articles are below, but all of them (including the videos) are must reads. Her work her made it possible for anyone to meet the four 2014 Fields medal winners:

Erica Klarreich on Manjul Bhargava

Erica Klarreich on Maryam Mirzakhan

A really cool opportunity to understand the work of one of the Fields Medal winners came when the Mathematical Association of America made an old Manjul Bhargava’s paper available to the public. I had a lot of fun playing around with this paper (that he wrote as an undergraduate, btw). It made me feel sort of connected to math research again:

A fun surprised with Euler’s identity coming from Manjul Bhargava’s generalized factorials

The Breakthrough Prizes in math didn’t seem to get as much attention as the Fields Medals did, which is too bad. The Breakthrough Prize winners each gave a public lecture about math. Jacob Lurie’s lecture was absolutely wonderful and a great opportunity to show kids a little bit of fun math and a little bit about the kinds of problems that mathematicians think about:

Using Jacob Lurie’s Breakthrough Prize talk with kids

I’m glad to see more and more opportunities for the general public to see and appreciate the work of the mathematical community. Speaking of which . . . .

(6) Jordan Ellenberg’s “How Not to be Wrong”

Jordan Ellenberg’s book How not to be Wrong is one of the best books about math for the general public I’ve ever read. I have it on audiobook and have been through it probably 3 times in various trips back and forth to Boston. My kids even enjoy listening to it – “consider the set of all integers plus a pig” always gets a laugh.

One of the more mathy takeaways for me was his discussion of infinite series and what he calls “algebraic intimidation.” Both led to fun (and overlapping) discussions with my kids:

Talking with about Infinite Series

Jordan Ellenberg’s “Algebraic Intimidation”

(5) The Mega Menger Project

The Mega Menger project was a world wide project that involved building a “level 4” Menger sponge out of special business cards. We participated in the project at the Museum of Math in NYC. The kids had such a good time that they asked to go down again the following weekend to help finish the build.

Menger Boys

It was nice to see so many kids involved with the build in New York. It also made for another fun opportunity to explore the math behind the project a little more deeply:

The Museum of Math and Mega Menger

(4) People having a little fun with math and math results

For some family fun, check out the new game Prime Climb:

our review is here:

A review of Prime Climb by Math for Love

Also, don’t forget to have a little fun when tweeting about new and important math results. Like Jordan Ellenberg tweeting about the solution of an old Paul Erdos conjecture:

Erica Klarreich’s Quanta Magazine article on the same result was just published yesterday by coincidence:

Erica Klarreich on prime gaps

For me the math laugh of the year was Aperiodical announcing the results of an 8 year search confirming the 44th Mersenne Prime:

(3) Evelyn Lamb’s writing

Evelyn Lamb’s blog is a must read for me. I love the wide range of topics and am pretty jealous of her incredible ability to communicate abstract math ideas with ease. Her coverage of the Heidelberg Laureate Forum was sensational (ahem Breakthrough Prize folks, take note!). This post, in particular, gave me quite a bit to think about:

A Computer Scientist Tells Mathematicians How To Write Proofs

My thoughts on proof in math are here:

Proof in math

Away from her blog, if you want a constant source of fun and interesting math ideas just follow her on Twitter. For instance this tweet:

led to a great little project with the boys:

Irrationality of the Square root of 2

(2) Terry Tao’s public lecture at the Museum of Math

On of the most amazing lectures that I’ve ever seen is Terry Tao’s public lecture at the Museum of Math. I don’t know how it had escaped my attention previously, but I finally ran across it about a month ago. What an incredible – probably unparalleled – opportunity to learn from one of the greatest mathematicians alive today:

Explaining a few bits of his talk in more detail led to three super fun projects with the boys:

Part 1 of using Terry Tao’s MoMath lecture to talk about math with kids – the Moon and the Earth

Part 2 of using Terry Tao’s MoMath lecture to talk about math with kids – Clocks and Mars

Part 3 of using Terry Tao’s MoMath lecture to talk about math with kids – the speed of light and paralax

(1) Fawn Nguyen’s work

When one of the top mathematicians around is tweeting about projects going on in a 6th grade classroom 2000 miles away, the world is working the right way!

Fawn is producing and sharing some of the most interesting math projects for kids that I have ever seen, and I’m super happy that her work is getting recognized. She’s probably inspired more than 20 projects with the boys, and I can’t wait for the next 20 in 2015. Here are two from this year:

Fawn Nguyen’s Geometry Problem

A 3d Geometry proof without words courtesy of Fawn Nguyen

If you have even a passing interest in fun, exciting, and generally kick-ass math projects for kids – you have to follow Fawn.

Banach-Tarski, Hilbert curves, and infinite sets

I have my kids write short reports every day on chapters they select in Cifford Pickover’s amazing “Math Book.”¬† (sorry I don’t know Latex well enough to format the title correctly).

These reports give them a chance to see fun math outside of the standard stuff covered in their school books.¬†¬† Last week my younger son stumbled across the section on the Banach-Tarski theorem and it really intrigued him.¬† I finally got around to talking a little more about that theorem with the boys today, though it obviously isn’t the easiest subject to cover with younger kids!

The first thing we talked through was the two different statements of the theorem.  A short, and excellent as usual, summary of the two theorems can be found on the Cut the Knot website:

I covered the the two different statements of the theorem and moved on to a much easier to understand example of infinite sets – why there are the “same number” of positive integers and positive even integers.

With the example with integers and even integers showing us how to compare infinite sets, I moved on to showing them that a line segment of length 1 has the “same number” of points as a line segment of length 2.¬†¬†¬† The ideas in this proof at least let you see how one object could somehow be the “same size” as something that seems to be twice as large.


The next thing we talked about was how we could see that a line segment of finite length could have the “same number” of points as an infinitely long line.¬†¬†¬† We approach this idea using stereographic projection:

Next we moved on to 3D and I showed them that the sphere has the same number of points as the plane.¬† The idea here was also to look at stereographic projection, though luckily for this example we have a special prop designed by Henry Segerman that we found on Laura Taalman’s 3D printing blog:

Goes without saying that holding the model in your hand is quite an improvement over a sketch on the board!

So, by this time we’d seen that a line segment has the “same number” of points as the whole line, and a sphere has the “same number” of points as the plane.¬† Now we show something really amazing – a line can fill up a square, and hence the plane.¬† That means that a line segment has the “same number” of points as the whole plane.¬† Wow.

The approach here took much longer than what is on camera.  We found this great website that gave a tutorial on Hilbert Curves:

We also found some space filling curves on Laura Taalman’s blog:


So, the 5 minutes on camera was actually preceded by a couple of hours of printing and drawing Hilbert curves on our own.  It made for a really fun morning:

Lots of people to thank for this one РClifford Pickover,  Kerry Mitchell, Alexander Bogomolny, Laura Taalman, and Henry Segerman.  So glad to have resources like theirs online to help kids learn about this kind of fun math.

** Addendum **
After finishing up this post we were playing around with our 3D printed Hilbert curve and took it off the base.  After we did that, we found that we could stretch it out into almost a line.  Cool!!  I think it helps kids get a better feel for the fact that it is all one long line segment twisted up into a curve:



Learning from 3D Printing

About a month ago Patrick Honner linked to this video about a 3D printer at his school:

What I found particularly intriguing about this video was the potential educational uses of 3D printing.¬† Though I’d seen articles here and there about 3D printing, the focus always seemed to be what you could make rather than what you could learn. ¬† The educational possibilities in the Brooklyn Tech video convinced me to get one. ¬† We’ve had it for a little over week and are really having fun learning how to use it.¬†¬† One of the best resources we’ve found so far is this amazing blog by James Madison University math professor Laura Taalman, aka @mathgrrl:

By coincidence, the Mathematical Association of America just this week released this video where Taalman talks about some of her experiences with students and 3D printing.  Her example about printing a set of Borromean rings was particularly fascinating to me.

Mostly as a result of playing around on the MakerHome blog, we’ve printed several different knots, a Sierpinski tetrahedron, a bunch of different polyhedra and some really neat hinged shapes (and this list is just what’s in front of me on the kitchen table right now!):


We’ve also made a few things on our own after learning from some of the instructions on the MakerHome blog as well as from this helpful video from Wolfram:

For example, from those two sources, and lots of trial and error, we were able to print out a hollowed out cube that illustrates the “Prince Rupert Problem” – a cube is actually able to pass through a second cube of the same size:

Rupert Side By Side

Rupert In

Much like the Brooklyn Tech and Taalman videos suggested, printing this example is filled with great ways for kids to see some interesting math.¬† I’m really excited to find more fun projects to do with the boys.¬† I think this is going to be a great tool to help them understand some pieces of math that may have previously been a little out of their reach right now.

Using Manipulatives for a fun twist on Dan Meyer’s geometry problem

About a week ago Dan Meyer posted an interesting geometry problem on his blog and asked for some open-ended feedback.  Over the next few days he provided some additional thoughts / content and the resulting discussion was fascinating.

I’ve written two prior blog posts about the problem and an extension that was particularly interesting to me.¬† Thanks to an Evelyn Lamb post on Twitter today I saw something new that reminded me of Dan’s problem and made me think of a way to introduce a similar problem to younger students using manipulatives.

Each day since the end of August¬† @mathgrrl has been posting a 3D printing project on the MakerHome blog.¬† The March 2nd post was on Haberdasher’s Problem:

These 3D prints from the Haberdasher problem give younger students a way to gain access to Dan’s problem without the need to know any geometry formulas at all.¬† We just need to remove the circle from the original problem and replace it with an equilateral triangle.¬†¬†¬† The new problem would go something like this, but, of course, I’m not suggesting to use this specific abstract mathematical language with younger kids:

Given an arbitrary point P on a line segment AB, let AP form the perimeter of a square and PB form the perimeter of an equilateral triangle. Find P such that the area of the square and triangle are equal.

Here’s what I think I’d do (and I would love to have been able to run through this with my own kids, but a stomach bug had other ideas, unfortunately):

(1) Given the kids these manipulatives and let them play around:



It is pretty amazing that you can form both a square and an equilateral triangle from these hinged shapes.¬† I think it would be fascinating to hear kids who haven’t had much exposure to geometry discuss the various relationships between the shapes (including concepts of area and perimeter) using whatever mathematical language is appropriate for them.¬† The goal isn’t precision or geometry formulas just discussion.

(2) The next would be to introduce the new version of Dan’s problem without the mathematical formality.¬† Several of the commentors on Dan’s original post suggested the idea of cutting a string to form two shapes of equal areas.¬† That formulation of this new problem would work really well here, I think.¬† There’s a new and pretty interesting math challenge at this step relating to scaling.¬† If the string happens to be exactly as long and the combined perimeters of our square and triangle, the kids already have the solution in front of them.¬† However, if the string is a different length, the problem hopefully will lead to¬† a neat discussion about ratios and scaling.

By funny coincidence the correct cut of the string divides it into two pieces roughly equal to 47% and 53% of the original string’s length – almost exactly the same as the two pieces for the original square and circle problem (with a twist!).¬† Maybe that fact would provide an interesting extension, too.

As I said in my prior posts, Dan’s found a really fun problem!

Fun with M√∂bius Strips

The interesting math topic that I saw in a couple of different places over the last two days involves Möbius strips, and a question from my youngest son about the Möbius strip this morning led to a fun conversation.

The thing that got him thinking about Möbius strips was this 3D printing page about the Umbilic Torus:

We followed the directions on this page yesterday evening and printed a small triangular cross section M√∂bius strip.¬† It is a really neat shape and my younger son had a lot of fun playing with it yesterday before he went to bed.¬† I hope we can figure out how to print an actual Umbilic torus, but we don’t yet have that level of 3d printing skill just yet.

After he went to bed I saw this wonderful video posted by Steven Strogatz on twitter:

I was really happy to see Strogatz’s post for a couple of different reasons.¬† First, several months ago someone else had posted about the double M√∂bius strip in association with a Martin Gardner celebration and I’d not been able to find the link again.¬† Unfortunately I’d misunderstood the setup so it was nice to see where I’d gone wrong.¬† Second, having just spent half an hour talking about 3D M√∂bius strips the exercise in this video was a great natural extension (which I’ll probably do with the boys this weekend).

With that background, I was really happy to have my son ask this question today when he got up:

“What happens if you do two half twists instead of one when you make a M√∂bius strip?”

That led to this fun little discussion which shows a surprising result from geometry:

This is one of my favorite examples in math!

Of course, no discussion of M√∂bius strips is complete without showing Vi Hart’s amazing “Wind and Mr. Ug” video:

Such a fun topic for kids.