# Revisiting the Gosper curve – day 2

Yesterday, thanks to a comment from Jacob Rus, my younger son and I took a fresh look at the Gosper curve:

Revisiting the Gosper Curve – part 2

That project (and today’s, too!) uses some ideas from this site:

The Gosper curve on fractalcurves.com

A few of our prior Gosper curve projects are:

Dan Anderson’s Gosper curves

A fun fractal project – exploring the Gosper curve

Today my older son was back from his trip and three of us built a level 3 Gosper curve from our Zometool set.

Because my older son missed yesterday’s project we started by looking at the Gosper curve on the website linked above:

After the quick introduction we moved to the living room to try to build the curve. In preparation for today’s project we’d built 7 level 2 curves last night. It turned out, though, that we’d made a small mistake in the last movie so in order to build the curve correctly we had to take a short (and not filmed) pause to go back and look at the picture on the computer again.

Finally, we built the complete level 3 curve together (without filming – it probably took 20 minutes). Here’s what the boys had to say about the completed shape and what they learned builiding it. I hope the video quality is good enough to be able to see the yellow Zome struts that we used to mark the points where the level 2 curves connected to each other. My younger son refers to those connections a few times.

# Revisiting the Gosper curve part 1

I got a really nice series of comments about the Gosper curve from Jacob Rus. In particular, he shared this link with me:

The Gosper curve on fractalcurves.com

After looking at the site, I decided to look at the Gosper curve using our Zometool set. Unfortunately my older son was away today, so it was just me and my younger son.

First we looked at the Fractal Curves site:

First we built the level 1 Gosper curve. Here’s what my son had to say:

Next we moved up to level 2. Building the level 2 curve was quite a challenge for my son and we had to return to look at the picture on the computer screen several times. Here’s the 3-part discussion we had about the build:

Now here is the last part where we get to the finish line on the level 2 Gosper curve. I was definitely surprised at how difficult this build was for my son, but there really is a lot going on. Building levels of the Gosper curve is a neat way to learn about fractals:

# Playing with Dan Anderson’s Mandelbrot program

After seeing our introductory Mandelbrot project from yesterday:

An introduction tot he Mandelbrot set for kids

Dan Anderson sent us a link to a wonderful set of Mandelbrot Set-related resources on his programming page:

Today I had the boys play around with the “Mandelbrot Zoom” program:

Dan Anderson’s program “Mandelbrot Zoom”

It is really fun to hear what the kids have to say playing around with the Mandelbrot set.

My younger son was fascinated by the “mini Mandelbrots” that he saw as we zoomed in:

My older son wondered if the Mandelbrot set was connected. This problem actually took many years to resolve (it is) but it was fun to hear what he meant by “connected.”

So, a fun little project just talking about and exploring the Mandelbrot set.

# An introduction to the Mandelbrot set for kids

Last night I was writing about “beautiful math” for kids:

When I asked my younger son what he thought was the most beautiful math he’d seen, he replied “fractals” and specifically mentioned the Mandelbrot set.  We haven’t done a project about the Mandelbrot set, so it seemed like a good idea to talk about it today.

As I say in the introduction below – it isn’t just a pretty picture, there is some really cool math.

So, I started the project by talking (or probably more accurately stumbling) through an explanation of the map that defines the Mandelbrot set.  After that, we worked through a few examples:

Having looked at 0, -1, and 1, we now moved on to looking at some complex numbers. The next numbers we tried were i and -i. It turns out that both of these numbers are part of the Mandelbrot set, but the calculations are slightly (really, just slightly) more complicated.

At the middle of this video we produced a crude map of the Mandelbrot set with snap cubes, and then at the end we discussed a little bit about how a computer program to plot the Mandelbrot set would work.

Now we moved to the computer to study the Mandelbrot set in Mathematica. Luckily Mathematica has a function – MandelbrotSetPlot[] – that makes this part of the project pretty easy for us. In this part we talked a little bit about what happens when you vary the number of iterations, and also what happens when you zoom in.

Determining the coordinates for zooming in was also a nice little mathematical discussion with the boys.

The coordinates to zoom in on turned out to be a more interesting topic than I was expecting. With the camera off we had a long discussion about the coordinates of the location that they wanted to see more carefully. I’m sorry I turned the camera off, actually, but oh well.

I love the discussion and general thoughts from the kids about the shapes we were seeing here. My younger son is right – this really is beautiful math for kids to see!

# MoMath’s “Beautiful Math” collection, and some “beautiful math” for kids

The museum of math has put together a nice (and growing!) collection of videos of mathematicians talking about beautiful math:

Watching the videos it struck me that they were aimed at adults and older kids, but I thought it would be easy to show some beautiful math that younger kids could appreciate, too. I thought of two of my favorite projects with the boys and then asked each of them to tell me what they thought was the most beautiful math that they’d seen.

My two favorites:

(1) The Chaos game:

Our project is here:

Computer Math and the Chaos Game

and my favorite part starts about 2:30 into this video and goes for about 45 seconds:

(2) John Conway’s “Amusical” version of the Collatz Conjecture

Our project is here:

The Collatz Conjecture and John Conway’s “Amusical” Variation

and my favorite part is at the end where we convert the “amusical” process to music. The music starts around 2:00 – prior to that is just explaining Conway’s Collatz process in the Mathematica code. I love it when my younger son says “I didn’t know you could hear 20”:

(3) My older son’s choice for beautiful math – the 4th dimension

We’ve done a couple of projects related to the 4th dimension – here are a few:

Carl Sagan on the 4th Dimension

Sharing 4d-shapes with kids

which had a fun connection to our Zometool / bubble project. Around 1:00 in the video is a great moment – “who knew that bubbles could find the center of a tetrahedron?”

That comment from my son led to this wonderful drawing posted on twitter:

Another really fun higher dimensional problem for kids who know the Pythagorean Theorem is a neat problem I learned from Bjorn Poonen:

Talking through Bjorn Poonen’s N-dimensional Sphere Problem with kids

(4) My younger son told me that he thought fractals were the most beautiful math that he’d seen.

We’ve done several fractal projects for kids, too. Here are a few:

A fun fractal project – exploring the Gosper curve

After this projects, were really luck to receive some laser cut Gosper curves from Dan Anderson to play with:

Dan Anderson’s Gosper Curves

Using the Koch Snowflake to introduce fractals

Using Matt Parker’s Menger Sponge video to talk fractions with kids

One of my favorite moments from these projects happens around 3:00 in the video below as the boys stretch out a 3d printed Peano curve into (nearly) a straight line:

So, I’m really happy to see the Museum of math putting together a collection of mathematicians talking about beautiful math. I can’t wait to see more videos in the collection! Hopefully some of the videos and projects above can help younger kids see some beautiful ideas in math, too.

# 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 🙂

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.

# A partial response to Sam Shah

[I apologize at the beginning – this post was written very quickly and probably doesn’t read that well, but I loved Shah’s post and wanted to show a few projects for kids on about 8 out of 10 of the topics he mentioned]

Saw this post from Sam Shah today:

Sam Shah’s Inspiration and Mathematics blog post

It is a great post – honestly, I can’t recommend it enough. He talks about being inspired by Ed Frenkel’s book Love and Math. I loved that book and even turned part of it into a short talk about $\sqrt{2}$ and $i$ with the boys:

Ed Frenkel, the square root of 2 and i

Frenkel’s interview with Numberphile is excellent as well (and probably the thing I’ve linked to the most on this blog!):

My post today is a response to this paragraph from Shah’s post:

But what do my kids learn about modern mathematics — from school or popular culture? Are there any weirdnesses or strangenesses that can capture their imagination? Yes! Godel’s incompleteness theorem. Space filling curves. Chaos theory. The fact that quintic and higher degree polynomials don’t have a general “simple” formula always works like the quadratic formula. Fractals. Higher dimensions. Non-euclidean space. Fermat’s Last Theorem. Levels of infinity. Heck, infinity itself! Mobius strips. The four color theorem. The Banach-Tarski paradox. Collatz conjecture (or any simply stated but unproven thing). Anything to do with number theory! Anything to do with the distribution of primes! But do they capture students’ imaginations? No… because they aren’t exposed to these things.

I only have about an hour to write today, but let me see what I can do . . .

In order:

(1) Godel . . . I wish I had an project, but I don’t. Not sure how I would explain this one to kids. My own introduction to the ideas was reading Godel, Esher, Bach in high school.

(2) Space Filling Curves

Our first project involving space filling curves is here (and hits several of Shah’s poinst):

Banach Tarski, Hilbert Curves, and Infinite Sets

Our second project with the boys involving space filling curves was inspired by a 3D Printing blog post from Laura Taalman:

Laura Taalman’s Peano Curve post

That post led to this conversation:

One other project that could lead to a discussion of space filling curves is studying the Gosper Curve:

Dan Anderson’s Gosper Curves

Oh, and I almost forgot, Evelyn Lamb wrote a wonderful piece on space filling curves just last week:

Evelyn Lamb’s piece on Space Filling Curves

(3) Chaos Theory

I know this isn’t exactly “chaos theory” but it remains one of my all time favorite projects with the boys:

Computer Math and the Chaos Game

The 30 seconds starting around 2:46 shows why:

I also believe that the basics of dynamical systems are accessible to kids. I have fond memories of playing around with the logistic equation in high school.

Steven Strogatz’s video lectures and dynamical systems for kids

Finally, James Gleick’s Chaos is a wonderful book and any high school kid interested in learning about chaos theory will love it.

(4) The idea that quintic equations cannot be solved in general

Explaining this idea to kids is my secret dream. But now that the secret is out, here are a few thoughts that I’ve had.

First, I’ve been reading several books to try and try and try to figure out a way to make the idea accessible. Here’s one of the books I’ve been reading:

I have a few others, too. The problem is that the subject is pretty advanced. However, some of the ideas from group theory involved in the proof that you can’t solve quintics in general are accessible to kids. For example, we did this project about cubes inscribed in a dodecahedron:

A 3D Geometry project for kids and adults inspired by Kip Thorne

The group theory idea hiding in this project is that the group $A_5$ has an element of order 5, and that’s one of the key ideas in the proof of why quintics can’t be solved in general. The action of element or order 5 shows that there are 5 cubes inside of a dodecahedron.

There’s actually another way – and it is incredible – to see that’s there’s a cube inside a dodecahedron:

Can you believe that a dodecahedron folds into a cube

One other bit of 5 fold symmetry shows up with inscribe Tetrahedrons:

Five Tetrahedrons in a Dodecahedron

So, not perfectly getting at Galois theory, but at least a start down the path . . .

(5) Fractals

I’ve already mentioned a few. Our Gosper curve projects have been extremely fun:

A Fun Fractal project – exploring the Gosper curve

Dan Anderson’s Gosper Curves

The Koch snowflake is always fun:

Using the Koch Snowflake to introduce fractals

The idea that the area is finite and the perimeter is infinite really bothered my younger son:

and just last week we used Matt Parker’s latest video to talk about the Menger sponge and a strange relationship it has with $\pi$

Using Matt Parker’s Menger Sponge video to talk Fractions with kids

(6) Higher Dimensions

Oh gosh . . . this is such an exciting topic for kids thatI’m not even sure where to start!

Here are all of our projects with the word “dimension”:

All of our projects on dimension

These include projects from tiling shapes in 2 dimensions as in the picture above:

Zome Tilings

Up to a fun series of projects about 4 dimensional shapes inspired by a Patrick Honner Pi day post:

A link which includes all of our projects inspired by Patrick Honner’s Pi day post

A few others worth mentioning:

Sharing 4D shapes with Kids

Using Hypernom to get kids talking about math

Carl Sagan on the 4th Dimension

Counting Geometric Properties in 4 and 6 dimensions

(7) Non-Euclidean Space

I don’t have a lot here, and what I haven isn’t necessarily right on the money. We did use the Gosper islands to explore a little bit about non-integer dimensions:

Integer and Non-Integer Dimensions

I also think that we’ve done the classic “angles on a sphere” problem, but I just can’t find it.

(8) Fermat’s Last Theorem

I don’t have a specific project for kids on Fermat’s last theorem, but this Numberphile interview with Ken Ribet about a piece of the puzzle used to prove the theorem is a must see. The video is a wonderful illustration of what research mathematics is like:

(9) Infinity!!

There are so many ways to capture the minds of kids talking about infinity. Here are just a few of the projects that we’ve done:

To infinity . . . and to the next infinity

Exploring infinity and other Surreal Numbers

Possibly my favorite bit of math involving infinity to talk about with kids is this Numberphile video:

I’ve also loved talking about this series with my kids – using the idea of “algebraic intimidation” from Jordan Ellenberg’s How not to be Wrong:

One of the projects I’ve done about that video is here:

Jordan Ellenberg’s “Algebraic Intimidation”

(10) Mobius strips

[note: I’m low on time and am just copying this piece about some fun math to do involving Mobius strips directly from a recent post]

This sequence of tweets inspired a really fun set of projects with my kids as well as some other kids from the neighborhood. You just need strips of paper, scissors, and tape.

Here’s what the initial set up looks like – piece of cake!

Here’s the project:

Cutting a double Mobius strip

This is a wonderful project for kids because the results are so surprising and so hard to see ahead of time even when you’ve already been surprised a few times!

When you finish the project you can watch Wind and Mr. Ug with the kids!

(11) The 4 Color theorem –

I actually can’t believe that I’ve never talked about this theorem with my kids . . . but I haven’t. Oh well.

(12) Banach-Tarski

We did a whole project about it 🙂

Banach Tarski, Hilbert Curves, and Infinite Sets

(13) Finally – the Collatz Conjecture

This is another super fun bit of math to share with kids. We’ve talked about it a bunch:

The Collatz Conjecture and John Conway’s Amusical Variation

Having the kids listen to the music at the end of that project is one of my favorite math moments that we’ve ever had:

and here’s a more standard approach to the Collatz Conjecture with kids:

# Finishing up our talk about Matt Parker’s Menger Sponger video

We had a fun time with Matt Parker’s Menger sponge video this week:

Using Matt Parker’s Menger Sponge video to talk about fractions with kids

2nd day of using Matt Parker’s Menger Sponge video with kids

Here’s the video that inspired this set of projects:

Unluckily, while I was writing up the 2nd project I noticed that we’d made a mistake discussing the 3d modified Menger sponge whose volume is $4 \pi / 3$. I published the project with the mistake anyway, and returned to correct the mistake last night.

The error in the last project related to counting the number of small cubes which were removed from the modified Menger sponge at each step. In the normal Menger sponge, you are removing 7 small cubes from each small cube at each step. In the 2nd step of the modified Menger sponge you don’t chop the side lengths into 3 pieces, though. Instead each side length is chopped into 5 pieces. That means that when you removed the small center cubes that you remove 13 cubes rather than 7 cubes. We started the project by discussing that point:

Now that we had the correct formula for how the volume changes at each step in the construction of the modified Menger sponge, we went to Mathematica to see what we could say about the volume at each step (and in the limit):

So, a fun couple of days with Parker’s video. I’m sort of kicking myself for the mistake in the 2nd project, but it was fun to revisit the project and see how the volume of the shape converges to $4 \pi / 3.$

# 2nd day of using Matt Parker’s Menger Sponge video

Disclaimer right at the beginning of this post – we made a mistake in the last video and I didn’t notice it until I was writing up our conversation. I decided to write up the project anyway because mistakes are a part of learning and talking about math.

So . . . yesterday we did a project inspired by Matt Parker’s Menger Sponge video. Here’s the project:

Using Matt Parker’s Menger Sponge video to talk about fractions with kids

and here’s Parker’s video:

Yesterday we studied the 2d version – the Sierpinski carpet – and today I wanted to talk about the 3d shape. Since the 2d shape was fairly complicated, I thought it would be worthwhile to review yesterday’s confirmation before diving into the new 3D shape:

After that review, we talked about the 3d version. For the first part of the 3d discussion, we talked about removing the middle “thirds” of each part of a cube. How many cubes do we remove? How big are they?

The 3d geometry ideas here are great for kids to talk through. The conversation about the shape is also a nice fraction conversation for kids.

The next part is where we started to make a mistake – I’ll get to that in a second.

Here we looked at the next step in the modified Menger Sponge process. We began to discuss what happens when you chop up our remaining cubes into 5 pieces (on each side) rather than 3 pieces. The kids were able to see this step fairly easily on the square, but it was a little more confusing on the cube. Luckily we had one of the Menger sponges to aid in the visualization.

The mistake we made was in counting the cubes that get removed when we chop the sides into 5 pieces. We forgot we were chopping each side into 5 pieces and counted as if we were chopping each side into three pieces.

When you chop the sides into thirds and “punch out” each middle third, you remove 3 small cubes with each “punch” and have to put back 2 because you punched out the middle cube 3 times.

When you chop the sides into fifths, though, you punch out 5 cubes with each punch, so you really are punching out 5 cubes with each punch and having to put back 2 at the end. So, rather than taking away 7 cubes, you take way 13. In general, chopping the sides into (2n + 1) pieces and removing the middles results in taking way 3*(2n+1) – 2 cubes. Unluckily we always said that we took away 7 😦

For the last part we extended our ideas to higher levels of the modified Menger sponge. Our mistake of taking away 7 cubes every time came along for the ride . . . We’ll have to correct this mistake in a new project:

So, a neat project even with the mistake. I really like using the geometric ideas from Parker’s video to talk a little geometry and talk a little fractions with kids.

# Using Matt Parker’s Menger Sponge video to talk fractions with kids

Saw Matt Parker’s latest video via an Evelyn Lamb tweet yesterday:

Here’s the video:

Watching the video last night I thought that there’d be a lots of different ways to use this one with kids. I chose to use it as a way to talk about fractions and scaling. Kids, I think, will be surprised by the result involving $\latex \pi$ but diving into that part is probably too much for kids. They can appreciate the result, though, and the discussion of fractions and scaling leads right up to the Wallis formula.

I started by asking the kids what they thought about Parker’s video. We talked about their thoughts as well as a few other fractal shapes that the knew:

Next we talked about Sierpinski’s carpet and walked through the calculations for the area. The boys saw the area essentially as a subtraction problem, so I spent a lot of time trying to help them understand how to see it as a multiplication problem:

I think that the boys still didn’t really see the area change from one step to the next as a product, so we spent a little more time talking about that idea in the modified Sierpinski’s carpet that Parker talks about in the video. The way the area changes from step to step is a great fraction problem for kids.

I’m sorry that we got a little bogged down in the calculations here, but I really wanted to be sure that the kids saw the relationship between the subtraction and multiplication approaches to calculating the area.

Finally, we looked at the complete calculation for the area of this modified Sierpinski carpet. The boys noticed a pattern in the products, which was cool, and we were able to transform our product into the famous Wallis product.

So, a fun project for kids. Parker’s video is great and serves as a great motivation for diving into the calculations. The calculations themselves are a great exercise in fractions for kids.

I’d like to try to work through the 3d version, too, so maybe we’ll do that tomorrow. My back of the envelope calculation tells me that the 4d version doesn’t have the same property of having the “volume” of the 4-d sphere (which is $\pi^2 / 2$, though I may not have done the calculations correctly the first time around and will revisit them later this weekend, too.