Bjorn Poonen’s n-dimensional sphere problem with kids

[ sorry to be a little rushed (and unedited) on this write up – I had to be out the door by 10:00 am and wanted to get this post finished before I left]

Yesterday I wrote about a problem I overheard Bjorn Poonen discussing earlier in the week:

A strange problem I overheard Bjorn Poonen discussing

That blog post has the full problem – which is super fun to think about. This post is an attempt to talk about the problem with kids. There are some neat geometric ideas in the problem and also lots of surprises.

So, here’s the first part of a two part set up for the problem – looking at a fairly straightforward situation in 2 dimensions/. The question here is how can we compute the radius of the small circle in the center of our picture?

This question led to a great discussion about symmetry and geometry.

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The next question we tackled was the 3-dimensional version of the problem. The 2-d set up problem (and discussion) really helped my younger son understand the ideas required to solve the 3-d problem. The kids were able to see the generalization to n-dimensions pretty quickly!

At the end of this video we lay out the three questions in Poonen’s problem:

 

Next we looked at the solution to the first problem – does the inner n-dimensional sphere ever get bigger than the n-dimensional spheres inscribed in the n-dimensional cubes?

Working through this question turns out to be a really nice inequality / algebra exercise for kids.

 

The 2nd of Poonen’s questions is this – does the inner sphere ever get so big that it actually extends beyond the n-dimensional cube?

This question is a hint that some really strange things happen in higher dimensions. How could the inner sphere leave the confines of the cube?

It turns out that the solution involves looking at an inequality that is quite similar to the one we looked at in the first part of the problem:

 

The final part of the problem is a little tricky for kids, so I just presented the volume formula for even dimensional spheres and used that as a starting point.

From there we did a little algebra and arithmetic to come up with a fairly simple inequality that would tell us if our n-dimensional sphere had volume greater than the 2x2x2 . . . x2 n-dimensional cube:

 

Finally, we wrapped up the project by putting our inequaility into Mathematica and seeing what happened. The discussion about hunting around to find the first dimension when the sphere was bigger than the cube was really fun!

One thing to be careful about watching this video (that I was not as careful as I should have been making it!) is that our “k” value is equal to half the dimension. Since we are looking at even dimensions, we are calling the dimension “2k.”

 

So, a fun project that was full of surprises. Who would have thought that there was an interesting fact about the 1,206th dimension 🙂

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A strange problem I overheard Bjorn Poonen discussing

I was sitting in a lounge in the MIT math department on Tuesday evening reading and waiting to meet a friend when Bjorn Poonen walked in and began to discuss a strange problem with a student. I hadn’t really realized how strange the problem was until I started thinking about it just now. Here’s how the problem goes:

Start in two dimensions. Suppose that you have a 2×2 square and that you chop it into 4 1×1 squares. Inside each of those 4 squares draw the inscribed circle, and then draw the circle at the center of the square that is tangent to each of those 4 circles. The situation looks like this picture:

 

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Now extend the idea to 3 dimensions. So, start with a 2x2x2 cube, chop it into 8 cubes, inscribe spheres in each of those cubes, and finally draw the sphere in the middle of the cube that is tangent to each of those 8 spheres. I don’t know how much it helps, but here’s basically half of the picture with our Zometool set and some tennis balls:

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Here’s the problem:

(A) As you extend this idea into higher and higher dimensions, is there a dimension in which the sphere in the middle is actually larger than each of spheres inscribed in the 1x1x1 . . . x1 n-dimensional boxes?

(B) Is there a dimension in which the sphere in the middle actually extends outside of the 2x2x2 . . . x2 n-dimensional box?

(C) Is there a dimension in which the n-dimensional volume of the center sphere is larger than the 2x2x2 . . . x2 n-dimensional box?

Does this math course exist?

I’ve spent the last few days thinking about how students can learn about math that is normally outside of the school (both k-12 and college) curriculum.

The topic has been on my mind for a while, actually – pretty much since seeing this Ed Frenkel interview several years ago:

Frenkel’s talk has inspired several of my blog posts.

I wrote this one after seeing a project that Dan Anderson did with his students:

A list Ed Frenkel will love

Then, after seeing Lior Pachter write about how some unsolved problems in math fit nicely into the Common Core:

Lior Pachter’s “Unsolved Problems with the Common Core

I sort of combined Pachter’s idea and my thoughts about Frenkel’s interview into several different posts in the last couple of years:

Sharing math from Mathematicians with the Common Core

10 pretty easy to implement math activities for kids

A partial response to Sam Shah

This week I ran across two new ideas that got me thinking about sharing math, (and not just with kids). The first (I saw thinks to a SheckyR comment on a recent post) is this interview with Keith Devlin:

Keith Devlin’s interview: On learning and what it means to be human

This quote right at the beginning (around 3:40 into the interview) really struck me:

“If the last experience with mathematics is what you learned – certainly up to the middle level of high school – and to a large extent to the end of high school . . . you’ve basically never seen mathematics.”

Then I saw this tweet from TJ Hitchman:

I think the Hitchman and Devlin ideas are connected – if all you are seeing as a student is the math that is part of the normal school math programs (which, at least where I live, seem to be driven by what’s on the state tests) it would be pretty hard for anyone at all to get excited about math.

So, how do we, as Frenkel asks, get students to “realize that mathematics is this incredible archipelago of knowledge?”

A new idea crossed my mind this morning – and it isn’t that well thought out, but . . . .

One of the most influential-after-college classes that I took in college was a year-long physics course called “Junior Lab.” The idea in Junior Lab is that over the course of each semester you’ll do 6 (I think) famous experiments in physics (out of maybe 20 total choices). The website for the course is here:

Junior Lab’s website

After you do the experiments you present the results to your instructor as if you were the one doing the original experiment. As I wrote half-jokingly to my old lab partner, this is the most scary room on campus!

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You, of course, learn about the experiments, but there are so many lessons beyond that. The class teaches you about the breadth of physics, about experiments not working the way they are supposed to (!!), about presenting and defending results, and about writing papers.

It seems like the Junior lab format would be a great format for showing students math that isn’t typically part of a k-12 or college curriculum. It is a few steps beyond what Dan Anderson did with his “My Favorite” project, but, I think, would give students a totally different perspective on math.

It would be about as far away from a “learn this fact / take this test” type of math class as you can get. The students would have a wonderful opportunity to learn about many different areas of math and math research, and, as I mentioned above, the lessons from this class would reach far beyond the math.

In any case, I was wondering if there is a course like this anywhere. I hope there is because I’d like to think through the idea a little more carefully.

A nice algebra / geometry problem with lines

My son is working his way through Art of Problem Solving’s Introduction to Algebra book and was stuck on one of the challenge problems in the chapter about lines (chapter 8). We talked through the problem this morning.

Here’s the problem and his solution. He was able to find one of the points that satisfied the conditions of the problem, but he was surprised to learn that there was a second point:

Having learned about the second solution, I asked him to try to find it:

After he found the second point I wanted to show him that there was a little more to this problem that you might think. Here’s how we can get to a solution by using a little algebra and the Pythagorean theorem:

Rather than crunching through all of this algebra, I plugged the equation into Wolfram Alpha. He was pretty surprised to see what the general solution looked like 🙂

So, a fun problem and an pretty challenging one, too. I was happy to see that you could extend the “surprising” fact that there were actually two solutions not just one to a new “surprising” fact that there were actually infinitely many!

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:

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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.

Math Interviews I want to remember

Making a list that I’ll update from time to time so that I don’t lose track of these interviews. Happy to take more suggestions!

(1) Julie Rehmeyer on Wild about Math:

Julie Rehmeyer’s “Inspired by Math” interview

What’s always stuck with me from this interview is the story that begins around 31:30 and in particular the part beginning around 34:40 about proving that 0 + 0 = 0. It is a beautiful lesson about learning math.

(2) Nalini Joshi on the Australian TV show “The Weekly”

This is a stunning interview.

(3) Ken Ribet interviewd by Numberphile

Anyone wondering what research math is like – and especially anyone looking to go to graduate school in math so watch this video:

(4) The story about Paul Erdos on Relatively Prime

Moving in a different way that the Nalini Joshi interview, but I was captivated for the entire hour of this interview. Ron Graham’s stories are just incredible:

Relatively Prime’s story about Paul Erdos

 

(5) Ed Frenkel interviewed by Numberphile

(6)  Steven Strogatz interviewed by Kara Miller

Playing with Colin Adams’s “Why Knot?”

I met Colin Adams when I was giving a lecture at Williams college a few weeks ago.  He showed me some of the work he’s done on knots and it looked like there would be some fun projects for the boys hiding in that work.   I ordered his book Why Knot? right after our conversation and it arrived yesterday.

This morning instead of the review work that they’ve been doing in algebra and number theory I had the boys play around with the book. No help from me today – just open the book and go.  

Here are their thoughts.

First – initial reactions and a simple knot:

Second – an attempt to construct a slightly more complicated knot. This task was harder and, in fact, we didn’t quite complete it. Something to work on later today, though:

So, a good start, though this is probably a project that needs a bit more adult help than I provided this morning. I’m looking forward to exploring this book with them over the next few days.