[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 
and 
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 
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 
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:
Mike's Math Page 