A fun fractal project: exploring the Gosper curve

Over the last few days I’ve been preparing a project with the boys based a great fractal geometry example I found in this wonderful book:


We finally got going with that project this morning. The starting point was watching this Vi Hart video which gives a “proof” that \pi = 4:

After we watched the video we sat down to talk about the strange result and what they thought was going on. They seemed to gravitate to the idea that the jagged edges were causing problems, but the fact that the zigs and zags were getting smaller and smaller – and would eventually have a height of 0 – was still a bit confusing:

After talking about the Vi Hart video I introduced the kids to the Gosper curve by showing them the figures in the book that inspired this example. We also made use of an amazing program that Dave Radcliffe shared when I asked for a little help on Twitter:

Playing around with this program really helped the boys see the first couple of shapes in the sequence that eventually leads to the Gosper curve. I definitely owe Dave a big favor!

The next part of the project was to build the first couple of shapes that lead to the Gosper curve out of our Zometool set. The initial hexagon was easy, obviously, but the shape at step 2 gave them a little difficulty. In the video below they talk about building the shapes and then explore a connection between the hexagon from step 1 and the shape from step 2. The fun part here is that the boys saw some of the important connections that lead you to the Gosper curve.

Next we built a level 3 shape. It was lucky that we had the program from Dave Radcliffe since that allowed the boys to a little more confident that they had the right shape. It is interesting to see the 6 new level 2 shapes surrounding the original level 2 shape. Too bad our living room isn’t big enough to make a level 4 shape!

One interesting comment from my younger son is that he thinks that as we increase the size we’ll get closer and closer to a shape that looks like the original hexagon.

For the final part of the project we used our 3d printer to make 7 of the (approximate) Gosper curves. Here’s the shape we used from Thingiverse (our shape is the 2nd of the three shapes, but I can’t get that one to link properly):

The Gosper Curve on Thingiverse (I printed the middle one)

The punch line for the project is the same punch line that caught my attention in the book – when you increase the linear size of the Gosper curve by 3, the area inside the curve increases by a factor of 7 rather than a by a factor of 9. Everything that the boys have learned about scaling up to this point is that area scales as the square of the linear factors, but fractals have a different property. Pretty amazing!

Also, sorry for not explaining the analogy between the two boundaries right. Felt as if I was wrong as I was explaining it, but didn’t see what I got wrong until just now.

As a fun end to the project, I showed them Dan Anderson’s modification of the Gosper Island shape – sort of a combination of the Sierpenski Triangle / Menger Sponge shapes and the Gosper Island:

So, a fun project giving the kids an introduction to fractal dimension – a concept that I never would have guessed could be made accessible to kids. Really happy to have had the luck of running into this fun idea last week.

A super fun fractal project for tomorrow

It all started with a Steven Strogatz tweet from a few weeks ago:

It wasn’t the Lorenz attractor that caught my eye, though, it was the purple book behind it with the curious title: Fractals Chaos Power Laws.

With a little bit of free time on my hands yesterday I finally decided to track down that book. It wasn’t too surprising to find out that the MIT library had it (though it was sort of a surprise to see that it was cataloged in the quantum field theory section!). So . . . off to Cambridge.

I spent the afternoon reading through the book and was absolutely blown away. Among the things that caught my attention was this fractal example in the first chapter (you’ll have to scroll up a bit to see the full example):

A cool example of how fractal dimensions work

It looked like a really fun example to talk through with the boys, too. The more I thought about it the more I liked it, and it also seemed like being able to hold the objects in your hand would make the project even better. One slight problem on that front, though, was that since we are in the middle of a move I don’t have access to the computer I normally use to make the 3D printing files . . . .

A little shout out to twitter produced some helpful suggestions and basically a miracle save from Dave Radcliffe:

The code he shared is going to be a fantastic resource for the project tomorrow, but the bit of super amazing luck was that the shape actually had a name – the Gosper Island.

A quick check on Thingiverse revealed that there was already a couple of different choices for prints. I chose this one:

The Gosper Curve on Thingiverse (I printed the middle one)

Here’s the picture of the finished set of 7 Gosper curves (each one took about an hour to print):

The plan that I’ve sketched in my mind for the project tomorrow goes something like this:

(1) Start with the “standard” proof that \pi = 4 to show that strange things can happen when you have infinite jagged edges. Something along the lines of this old Vi Hart video:

(2) Next jump to the hexagons. We can make the first few iterations of the Gosper curve with our Zometool set. Something like this, but maybe the next iteration, too:


(3) After making a few iterations ourselves, we’ll jump to the program that Dave Radcliffe shared to see how the shape evolves with a few more iterations.

(4) Finally we’ll play with our set of 7 3d printed (almost) Gosper curves and find the surprising scaling result – multiplying the linear size of the figure by 3 scales the area by a factor of 7 rather than the expected scale factor of 9. Just like the \pi = 4 “proof” the infinite jagged-ness is causing something unexpected to happen.

I’m super excited to try this all out tomorrow!

Would appreciate any help on this problem

We are in the middle of moving and I just for now don’t have access to my compute that has Mathematica. I hadn’t really worried too much about it since we really only use it for 3D printing projects, but guess what . . .

I was reading through this book in the library today:


and saw this amazing example near the beginning (probably have to scroll up to the top of the page in the link):

Interesting fractal hexagon example

So, what I’d like to do is make a simple 3d print of the fractal hexagons. Maybe 3 or 4 iterations would be enough. Maybe 5.

But, without Mathematica I’m not going to be able to do it! Pretty sure this shape isn’t going to come together as an stl file in Excel 🙂

Anyone got any ideas – this looks like a really cool project and I’d love to play around with it this weekend.


What’s do you see that is similar / different

Traveling this morning, but would have loved to have been able to ask the boys some notice / wonder and same / different questions about these two things this morning:

First Dan Anderson’s amazing geometry video / program:

Second, the famous bowling ball pendulum:

A lucky save courtesy of Khan Academy

We are in the process of moving right now and I discovered last night that I accidentally moved the math books up to the other house. Whoops!

Because of that goof up by me, this morning I had my older son play around with spheres, cylinders, and cones on Khan Academy. Turned out to be a lucky break as those exercises revealed a little gap in his understanding that I hadn’t noticed previously.

The exercises themselves aren’t necessarily super special or anything, but they were different enough from the exercises in the Art of Problem Solving book to reveal this little gap. The problems he was working on were from this section:

Khan Academy problems on volume

The slight difference between this problems and the ones in our Introduction to Geometry book was that (most of) the problems asked you to round the answer to the nearest integer. One of the answers was something like \frac{100 \pi}{3} My son first rounded 100 / 3 and then multiplied by \pi on this one.

It was interesting to see the difficulty he had seeing \pi as a number rather than as a symbol. I’m happy that these problems from Khan Academy helped me discover and address (hopefully!) this issue.

Here’s the last problem we did this morning.

So, a fun and instructive morning for both of us. Lucky to have a positive outcome arise from leaving the math books in the wrong house!

Kate Owens on concrete vs. abstract

Saw a fantastic sequence of tweets from Kate Owens yesterday:

I think about (and struggle with) similar ideas constantly and try to make sure that I show the boys some abstract ideas from math regularly.

One of the pieces that got me thinking about this subject more carefully is Numberphile’s interview with Ed Frenkel from last year:

I love Frenkel’s “painting the fence” analogy, and I super duper love his simple idea: “So how do we make people realize that mathematics is this incredible archipelago of knowledge?”

Since seeing this interview I’ve been trying to pay much more attention to the math that is being shared both in popular culture and online, and I try to mold some of these abstract mathematical ideas into fun projects for the boys. It doesn’t always work, but for the most part I think they have enjoyed these projects which are far from what they see in their school math books.

One of the first tries at sharing some abstract math was using what Jordan Ellenberg calls “algebraic intimidation” to talk about some infinite sums, including the sum made famous by Numberphile’s video last year:

1 + 2 + 3 + . . . . = -1/12

Jordan Ellenberg’s Algebraic Intimidation

Also last fall I happened to find a link to the public lectures that mathematicians have delivered at the Museum of Math in New York. Terry Tao’s lecture – “The Cosmic Distance Ladder” – inspired three projects with the boys:

Three projects from Terry Tao’s MoMath lecture

Another MoMath Lecture – this one from Bryna Kra – turned into a fun project with snap cubes and angry birds!

Using Bryna Kra’s MoMath lecture with my kids

Along the same lines as the MoMath lectures, Jacob Lurie’s public lecture after winning the Breakthrough Prize is a beautiful way to share some profound mathematical ideas with kids:

Using Jacob Lurie’s Breakthrough Prize lecture with kids

Recently I’ve seen two amazing pieces of math that University of Colorado professor Richard Green has shared on Google+. The remarkable problem in the tweet below even attracted the attention of Tim Gowers – so click to Green’s post to see that comment!

Another great piece of math to share with kids from Richard Green

Of course it isn’t just math professors sharing great ideas. Fawn Nguyen is a constant source of inspiration:

A 3D Geometry proof with few words courtesy of Fawn Nguyen

and I cannot wait to give the project she shares in this blog post a try:

Fawn Nguyen share’s John Conway’s rope game

Tina Cardone shared a neat geometry problem last fall that turned into a fun 3D printing project for us:

A cool geometry problem shared by Tina Cardone

(our blog has probably 20 more 3d printing projects inspired by Laura Taalman, Steven Strogatz, Patrick Honner, Evelyn Lamb, and others!).

So, I was glad to see those tweets from Kate Owens yesterday. They made me think finding more ways to share some abstract math ideas that kids don’t usually get to see. I was also happy to see that other people are thinking about how to share more abstract ideas with kids and students, too. I really do think that people learning math not only benefit from seeing these abstract ideas, but that they also really want to see them.

Shortly after I saw the Ed Frenkel interview last year Dan Anderson shared a wonderful project that he did with his students – Just click through to Dan’s blog to see the list of topics the kids chose to see why I really believe that kids want to see more of these ideas:

A list Ed Frenkel will love

Looking forward to the next fun idea I find online 🙂

Exploring some special triangles with Zometool

Yesterday I started in on the “special triangles” section of our Prealgebra book with my younger son. I also happen to be studying some simple 3-dimensional figures with my older son. Since we were going on a little trip today I thought it would be fun to try to combine the two topics into a fun little Zometool project.

One difficulty is that (I’m pretty sure) you can’t make a 30-60-90 triangle with the Zometool struts. No matter. If you use blue struts for the 1 and 2 lengths, you can use the yellow struts for the \sqrt{3} length even if they don’t plug in properly.

We started by looking at two simple triangles – a 45-45-90 triangle and a 30-60-90 triangle. After the kids finished building them I asked them what they noticed:

The next challenge was to build and study a tetrahedron. We found that the height of the pyramid would be yellow struts, but we couldn’t find the right length. In this project we tried to see if we could calculate that height. This part of our project combines 3D geometry, 30-60-90 triangles, and the Pythagorean theorem:

The last part of the project was going through the calculation of both the height and the volume of our tetrahedron. Understanding the 3d geometry was a tiny bit confusing for my younger son, but the Zometool set definitely helped him see the geometry a little better. I think that if we only had the picture on the whiteboard he would have really struggled to calculate the height of the pyramid.

The last part of this project was also a nice way to get a little practice manipulating the square roots that come up all over the place in the height and volume calculations.

So, a fun impromptu project where the Zometool set helps us get our arms around both 2d and 3d geometry. Along the way we got a little practice calculating with square roots and even figured out the volume of a tetrahedron made out of green Zometool struts. It is nice little coincidence that the boys happen to be studying similar subjects right now 🙂

What learning math sometimes looks like: a 30-60-90 triangle struggle

By chance I’m going through some similar topics with my older and younger son right now. My older son is learning about 3d shapes for the first time and my younger son is learning about 2d shapes. Both are having a few struggles. Nothing super surprising, but I’m finding it really interesting to watch.

The topic with my younger son today was “special triangles” which was mainly a discussion about 45-45-90, and 30-60-90 triangles. We encountered the problem in the videos below in the exercises at the end of the section. Quite a challenging problem on the first day you see 30-60-90 triangles, but the good news is that he had lots of ideas. Though he needs a little help keeping track of all them, what is cool is that by the end of this video these ideas have led to a picture that solves the problem in a way that I was not expecting at all.

Now that he has the right picture, finding the lengths that he’s looking for just comes down to the Pythagorean theorem. The tilted picture gives him a bit of trouble initially, but we eventually get things oriented the right way and compute the side lengths:

Having walked to the end of this problem, I thought it would be worthwhile to go back and have him put all of the ideas together from the start. Hopefully that gave him a chance to reinforce some of the connections he made.

At the end of his solution I showed him an alternate way to compute the lengths using a different 30-60-90 triangle.

So, a good struggle and hopefully a productive one. As I always say in this “learning math” posts – learning math is definitely not a walk in a straight line!

Another great piece of math to share with kids from Richard Green

Saw this really cool post from Richard Green over the weekend:

I love unsolved problems that kids can understand! In this case what really jumped off the page was that there were so many different directions to go when sharing this problem with kids. I picked the first three ideas that came to mind and used them for a fun little project with the boys this afternoon.

Sorry that this one goes a little longer than usual, but you’ll see the kids remain totally engaged (and fascinated) all the way through. So much fun!

I started by simply sharing Green’s google+ post with them:

The first project based on this unsolved problem that I thought would be interesting to kids was looking to see if they could find numbers that were written with just 0’s and 1’s in base 2 and in base 3. To a mathematician this probably doesn’t seem to be that interesting of a problem, but the kids found it to be pretty neat. They were really excited when they discovered the pattern!

The second project I thought would be interesting took about 10 minutes. The idea in this part of the project is to see if we can find a pattern in the way to convert numbers from base 2 to base 4. It took a while for the kids to see the pattern, but they were really happy when they found it. Again, the connection here probably isn’t really that surprising to mathematicians, but it is amazing to watch kids see it for the first time:

The last project that I thought the kids would find interesting was finding the probability that a number written in base n would have just 1’s and 0’s. To simplify the project we just looked at 3 digit numbers. The kids had some really great ideas here and we got to explore a couple of different ideas and patterns.

At the end of the second video in this part we returned to talking about the ideas in the original problem.

As I said at the beginning, I love sharing unsolved math problems that kids can understand. The really nice thing about the problem Richard Green shared is that there are lots of neat properties of base number arithmetic that are closely connected to this problem. Talking through some of these properties is a fun way for kids to explore math, and maybe even get a tiny little glimpse of mathematical research. Definitely a fun afternoon 🙂

Practicing fractions while learning some introductory geometry

Saw a nice introductory geometry problem in our Prealgebra book today. It is problem #7 on the 2005 AMC 8

2005 AMC 8 problem number 7

Here’s the problem:

Bill walks 1/2 mile south, then 3/4 mile east, and finally 1/2 mile south. How many miles is he, in a direct line, from his starting point?

I like a couple of different things about this problem. First, there are multiple ways to solve it, and each provides a little different insight into the geometric situation. Second, it provides a nice opportunity for a little fraction review, since you’ll likely encounter adding, multiplying, and maybe even dividing fractions in the solution.

Here’s my son’s approach to the problem:

His initial solution involved using two different triangles to find the length. I was interested to see if he could do it with just one triangle. This new solution involves drawing in two new lines, or maybe just rearranging the initial picture. I wanted to walk through this second solution because I think it is instructuve, but a little harder for a kid learning geometry to see.

It took a while, but we got there. One of the stumbling blocks was understanding what happened to the distances as we moved some of the triangles around.

I really love problems like this one. It gives a great opportunity to cover a new topic from a few different angles and also gives you an opportunity to sneak in a little review of an old topic. Definitely a fun morning.