Yesterday my younger son and I explored the perimeter of the Koch snowflake. He was surprised that the perimeter was infinite, but he also thought that the area was finite. Today we calculated the area.
I started off by asking my son for a plan for how we should approach the problem:
In the last video we calculated the additional area that was added in each step of the process of creating the Koch snowflake. Here we look carefully at those amounts and take a guess at the pattern.
Now that we had the pattern, we added up all of the areas. I really enjoyed seeing my son’s approach to adding up the infinite series. I think this calculation is a great one to show to kids to sneak in a little arithmetic practice as well as introducing some simple ideas about sums of infinite series.
This morning I asked my son to flip through Martin Gardner’s The Colossal Book of Mathematics and pick out a chapter he thought would be interesting to talk through. One of the chapters had a discussion of the Koch snowflake that caught his eye.
Here’s that chapter and why he thought it was interesting:
For today we decided to explore the perimeter to see if we could figure out why it was infinite. For starters we tried to calculate the perimeter of the first four iterations. He had a little trouble with the 4th, but I think that trouble shows why this is such a great arithmetic exercise for kids:
Next we went back to look more carefully at the 4th step to make sure that we had the right number. With this review we found the correct perimeter.
For the last step we found the pattern for how the perimeter changed at each step. This was, unfortunately, slightly rushed as we were about to run out of memory in the camera. But still, I thought my son gave a nice explanation of why the perimeter eventually went to infinity.
I saw a really neat tweet (and subsequent twitter thread) yesterday:
I thought that Dave’s tweet would make for a great project, so we took a close look at it this morning. We started by looking at the tweet and then I asked my son what his definition of dimension was:
Next we worked through a few of the introductory examples in Dave’s tweet – a point, a line, a square, and a cube:
Next we moved on to the fractals. Here we also need to talk about logarithms, so we stuck to two of the examples – the Sierpinski Triangle and the Koch Curve:
Finally, we looked at the most complicated example – the Sierpinski Carpet. At the end of this video we recap and got my son’s thoughts on non-integer dimensions
This was a terrific project. Thanks to Dave for sharing the idea!
A few weeks ago Grant Sanderson published this amazing video about fractal dimension:
I’ve had it in my mind to share this video with the boys, but the discussion of logarithms sort of scared me off. Last week, though, at the 4th and 5th grade Family Math night the Gosper curve fractals were super popular. That made me think that kids would find the idea of fractal dimension to be pretty interesting.
Here are the Gosper curves that Dan Anderson made for us:
We’ve actually studied the Gosper curve several times before, so instead of just linking one project, here are all of them 🙂
A collection of our projects on the Gosper curve
So, today we started by watching Sanderson’s video. Here’s what the boys had to say about it:
At one point in the video Sanderson makes a comment that fractals have non-integer dimensions. I may have misunderstood his point, but I didn’t want to leave the boys with the idea that this statement was always true. So, we looked at a fractal with dimension exactly equal to 2:
Next we looked at the boundary of the Gosper island. I wanted to show that this boundary had a property that was a little bit strange. I introduced the idea with a square and a triangle to set the stage, them we moved to the fractals:
Finally – to clear up one possible bit of confusion, I looked at a non-fractal. For this shape we can see that the perimeter scaled by 3 and the area scaled by 7. Why is this situation different that what we saw with the Gosper Island?
Definitely a fun topic and I think Sanderson’s video makes the topic accessible to kids even if they don’t understand logarithms. I’m excited to find other fractal shapes to talk about now, too!
Quanta magazine’s article on 3d folded fractals from last month has really captured my imagination:
3-D Fractals Offer Clues to Complex Systems
Since reading the article I’ve been trying to understand the paper by Laura DeMarco and Kathryn Lindsey that inspired the story:
Convex shapes and harmonic caps on arXiv.org
Although I’m making progress digesting the paper, that progress is slow – who knew that trying to understand current research in a field you know nothing about would be so hard . . . 🙂
One really nice thing in the paper that helped me get my bearings was figure 1.1:
This figure shows the curved “cap” which combines with a square to make a 3d shape. I tried to imagine what the shape formed by gluing the square and the curved shape would look like, but quickly reached the limits of my imagination.
Luckily, though, my wife was willing to help me sew a version.
It took two tries but eventually this shape emerged!
It is much flatter in reality than it was in my mind so seeing an actual version of the shape turned out to be really helpful.
I’m not sure what the next steps are for me. Either I have to get a better understanding of the Riemann mapping theorem (and I’ve already dug out my old complex analysis book for that) or maybe just play with some approximations and make some 3d prints like this one from Yoshiaki Araki that was part of a contest that Quanta Magazine had in their article:
The work trying to get a better understanding of these 3d shapes has been really fun. I’ll be really happy if I’m able to understand one or two more things from the DeMarco and Lindsey paper. It would be amazing to be able to make some (even very simple) shapes to show kids some new ideas from current math research.
Saw this tweet from Holly Krieger this morning:
and, oh bother, the embedded tweet isn’t coming through. Here it is:
I thought it would be fun to talk about some sort of similar shape and stumbled on this neat design by eduardoviruena on Thingiverse:
eduardoviruena’s “Sierpinski cubes” on Thingiverse
I sent the print to the printer as I was running out the door. Unfortunately in my haste I made the print much smaller than I intended. Oh well . . . we still got to have a fun conversation.
Here’s what he had to say about the eduardoviruena’s “Sierpinski cubes” – I was really interested in his description of how to make this shape:
Because the print was so small I wanted to see if he had any other thoughts when he saw a larger version on the screen:
I’m really excited to do more 3d printing projects like this one – hopefully giving kids shapes like this to play with will help them see a fun and exciting side of math 🙂
I’m starting to think about what to do for the Family Math nights at my younger son’s school this year. During the day today I 3d printed 2 of Laura Taalman’s Peano Curves to see if that might somehow make a fun project for a group of 4th and 5th graders.
Taalman’s blog post about the curve is here:
Laura Taalman’s 3d-printed Peano curve blog post
The plan tonight was to have each kid talk about the curve (they’ve seen it before) and see what they thought was interesting. My older son went first:
Then my younger son:
For the last part of the project we took the curve off the base and stretched it out (almost) into a line:
I think there’s a fun project here – these take a long time to make, but I think with a week or two of printing prep that there’s a good 45 minute project for 4th and 5th graders in here somewhere.
This project is the 2nd of two projects on the Koch snowflake. The reason for the projects was that my younger son wondered how the Koch snowflake could have an infinite perimeter but a finite area.
The first project (about the perimeter) is here:
Exploring the perimeter of the Koch snowflake
Our approach to studying the area was similar to the approach for studying the perimeter. Essentially we looked at the steps in the construction of the Koch Snowflake and then looked for a pattern. Here are the initial thoughts from the kids about the area:
The first step in studying the area was to look at the total area of the first few iterations of the Koch Snowflake.
I decided to avoid the complexity of geometry triangle formulas and just talked about scaling. My younger son also came up with a really nice argument for
Now that we’ve seen a first few cases, can we find the pattern?
The amount of area that we add each time has a fairly simple pattern – it is just multiplication by 4 and division by 9. The only time that doesn’t happen is in the first step.
Can we connect the numbers with the geometry?
Now that we’ve seen and understood the pattern, how can we figure out the sum? I love that the boys saw that the main sum we were looking at here was less than 2.
I didn’t want to derive the geometric sum formula, so I just gave it to them. We can talk about it another time. That formula seems to be the easiest way to find the exact value of the sum, though.
Finally we wrapped up and discussed the process we used to study the area and perimeter. I don’t really believe that my younger son now understands every detail of what we talked about, but I hope that he’s a little bit less confused about the area and perimeter of the Koch snowflake.
I think the math here is something that all kids would find interesting.
Last week we have a fun talk about the boys “math biographies”:
Math Biographies for my kids
When I asked my younger son to tell me about a math idea that he’s see but that he doesn’t believe to be true, he brought up the area and perimeter of the Koch snowflake. The perimeter is infinite while the area is finite, and he does not believe that these two facts can go together.
Today I thought it would be fun to talk about the perimeter of the Koch snowflake – no need to tackle both ideas at once. Here’s the introduction to the Koch snowflake and some thoughts from my younger son on what he finds confusing about the shape:
After that introduction we began to tackle the problem of finding the perimeter. We began by looking at the first couple of iterations in the construction of the snowflake to try to find a pattern. At this point in the project the boys didn’t quite see the pattern:
As a way to help the boys see the pattern in the perimeter, I asked my younger son to calculate the perimeter of the 4th iteration. My older son had been doing most of the calculating up to this point, and I hoped that my younger son working though the details here would shed a bit more light on what was going on as you move from one step to the next.
The counting project we reference at the end of this video is here:
John Golden’s visual pattern problem
Finally, we looked at how we could use math to describe the pattern that we found in the last video. We also discuss what it means mathematically for the perimeter to be infinite.
We need fairly precise language to describe the situation here, so this part of the project also gives the kids a nice way to learn the language of math.
I asked the boys what they wanted to talk about today and got a fun response – the Koch Snowflake with squares!
Luckily we still had the Zome set out from the “tribones” project from last week – so making the first couple of iterations wasn’t that hard.
Before we started, though, I asked the boys what they thought the shape would look like:
While they were building I searched for something on line that would let us play with this particular fractal. I found these two Wolfram Demonstration projects:
Creat Alternative Koch Snowflakes by Tammo Jan Dijkema
Square Koch Fractal Curves by Robert Dickau
We’ll explore these to demonstrations below.
Here’s what the boys had to say about the first three iterations of the square Koch Snowflake. A fun thing that happened here was that during the discussion the boys found a small mistake in the construction of their level 3 curve:
Next we moved to the computer to explore the Wolfram Demonstration projects. First up was Robert Dickau’s “Square Koch Fractal Curves.” Sorry that this video (and probably the next one, too) is so fuzzy – it looked ok in the view finder. Oh well.
Finally, we explored Tammo Jan Dijkema’s “Create Alternative Koch Snowflakes” demonstration project. This project allows you to alter the fractal. The boys had a lot of fun playing with this project. Again, sorry for the fuzz in the video.