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Month April 2016

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 🙂

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.

A conversation about primes with a 10 year old

My younger son is doing a little review work in Art of Problem Solving’s Introduction to Number Theory book.  This book is a wonderful way for kids to learn all sorts of interesting properties of numbers.  Today he told me that he was looking at a section on “primes and composites.”

So, we talked and the conversation led to a discussion about why there were infinitely many primes.

He remembered the “add one” piece of the proof that there are infinitely many primes, but what that “add one” bit was a key idea wasn’t something that he understood. So we talked some more:

I love how he found his way back to the proof and realized why the “add one” part was important. It was also pretty cool to see that he wasn’t bothered by the fact that the infinite list he was making might not have all of the primes – it was still infinite!

Fun morning 🙂

Tadashi Tokieda’s “World from a sheet of paper” lecture

Yesterday we saw an incredible public lecture from Tadashi Tokieda. He showed an hours worth of amazing mathematical ideas that come from paper folding.

This is actually our second project inspired by Tokieda. The first came from his “freaky dot patterns” video with Numberphile:

Numberphile’s Freaky Dot Pattern Video

We’ve also done a project on some of the paper cutting activities in this other Tokieda video, though we saw the idea in a different place:

Here’s that project:

Cutting a double Mobius strip

If you are interested in seeing a longer presentation from Tokieda, he has also given a public lecture at the Museum of Mathematics:

Tokieda’s public presentations are absolutely incredible!

We started our project today by exploring an idea at the beginning of the MoMath lecture – I wanted to show the boys something that they’d not seen previously. The exploration here is the noise that a coffee mug makes when you strike it with a spoon at various different locations (and sorry that my hand was blocking a lot of the shot here):

Next we looked at one of the surprising paper folding patterns you can see without doing any careful folding. It is fascinating to see folding patterns arising as naturally as this one does:

For the last project today we looked at the demonstration that Tokieda used to start his lecture yesterday – passing a large circle through a small hole. It seems as though the task is impossible, but some clever folding makes it work. They boys had a bit of a hard time explaining how this one worked – but that’s fine – it is hard to believe that it works at all!

So, a fun project today following yesterday’s fascinating lecture. It is so great to see lectures like Tokieda’s that bring amazing math to everyone – from kids to tenured MIT math professors!

My respose to my challenge

Earlier today I wrote about problem #! from the 2016 European Girls’ Mathematical Olympiad:

A Challenge / Plea to Math Folks

I thought the problem was wonderful and “live blogs” of different people solving the problem would help people (kids especially) learn about different aspects of mathematical thinking.

Here’s the problem:

 

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And here’s a sketch of the process I went through to solve it. First, though, here’s what my scattered notes looked like by the time I got to the end:

Step 1: Total confusion after reading the problem. You just have a seemingly random set of numbers . . . how can you show that the max of one set of expressions is even related to the min of another set of expressions? No idea what to do.

Step 2:  Oh wait, I remember the arithmetic mean / geometric mean inequality – that’s what’s written down in the middle of my note paper.  You’ll see the scratched out inequality when I first wrote it down, because I remembered the inequality backwards.

This inequality comes from the simple idea that the quantity (a - b)^2   has to be greater than or equal to zero.  The idea is that any real numbers squared has to be greater than or equal to zero.  When you expend the algebraic expression you get  a^2 - 2ab + b^2 \geq 0 which simplifies to a^2 + b^2 \geq 2ab.

That’s what I’ve got written down in the middle of the page.   Feeling good at this point in the problem solving process!

Step 3:  Wait a minute, the inequality in the problem goes the other way.  It says less than or equal to not greater than or equal to.  What??  This problem seems to want me to prove the exact opposite of the arithmetic mean / geometric mean inequality.  But this inequality is so true that it has its own Wikipedia page:

The Arithmetic Mean / Geometric Mean inequality on Wikipedia

So proving the opposite is probably not going to be the right strategy!

Ummmm . . . . . Not feeling good anymore . . . .

Step 4:  Well . . .  x_{i}^2 + x_{x+1}^2 sort of looks like a geometric expression.  It relates to the distance between the point (x_i,x_{i+1}) and the point (0,0) in the plane – maybe there is a geometric angle to the problem.   What would the term 2x_j x_{j+1} mean in this setting?   It would be an area, or maybe twice the area of a rectangle, but how would I compare that to a length?   Hmmm . . .

No idea . . . .

Step 5:  Time for an example – you’ll see that I wrote down the numbers 1, 3, and 5 in the upper left hand corner of the paper and in the upper middle right of the paper I wrote down the values for the two sets of expressions in the problem:

 

The min of x_{i}^2 + x_{x+1}^2 was 10 and the max of 2x_j x_{j+1} was 30, so the inequality we are trying to prove seems to be true for the numbers 1, 3, 5.  Yay, I guess, but what now?

Step 6:  I started wondering about why the problem wanted you to use an odd number of numbers, so I tried to see if I could find a set of 4 numbers that didn’t work.  Maybe that would help me see something.  Just to keep the numbers as simple as possible I tried 1, 2, 1, 2 and found that the min / max inequality in the problem didn’t work.  x_{i}^2 + x_{i+1}^2 was always 5 and 2x_j x_{j+1} was always 4.

Interesting.

Step 7:  You’ll see that I’ve got a < b < c written down in the lower left hand part of my paper.   Maybe what was special about my 1, 3, 5 example is that the numbers were increasing.   When you have three increasing numbers like this a^2 + b^2 will be less than 2bc because 2bc is greater than 2b^2 and 2b^2 is greater than a^2 + b^2 since b is greater than a.  That’s what this part is saying:

That’s it for the notes in the lower left hand corner.  A nearly identical argument would show that three decreasing numbers would produce a solution to the problem, too.

By the way, once you find that one of the 2 x_{i} x_{i+1} expressions is greater than one of the x_{j}^2 + x_{j+1}^2 expressions, you’ve solved the problem because the max of the 2 x_{i} x_{i+1} expressions has to be greater than or equal to the one you’ve found and the min of the x_{j}^2 + x_{j+1}^2 expressions has to be less than or equal to the one you’ve found.  That means the max of the 2 x_{i} x_{i+1} terms will be larger than the min of the x_{j}^2 + x_{j+1}^2 terms.

So, yay, we can solve the problem if we have three increasing (or decreasing) numbers in  a row, but why does there have to be three numbers like this in a row?  It seems as if the numbers are chosen totally at random.  Hmmmmmmm . . . .

Step 8:  The triangle on the right hand side of my notes – it wasn’t just a doodle!

The numbers x_1 to x_n aren’t really arranged in a row.  The problem points out that the last comparison involves x_{n+1} which is equal to x_1.   The numbers are really going around in a circle – or a triangle as I’ve drawn.

As you move from one number to the next you will either increase or decrease (because if two are ever equal the inequality in the problem is an equality and the problem is solved immediately).  You are making n up or down movements as you move around the circle of numbers, but since n is odd you have an odd number of total steps around the circle, and  the number of times you move up in value as you step cannot be equal to the number times you move down in value.

Aha.

You can imagine going around the triangle and having 2 up comparisons and 1 down.  The two ups have to be next to each other because there are only three sides, so any arrangement of the two ups and one down has to have the two ups next to each other.  As soon as you have two ups in a row you know you have three increasing numbers.

The general case isn’t that much harder.  Assume that you have more ups than downs and you want to try to spread out the ups and downs as much as possible so that there are never two of the same type  in a row.  Put, for example,  an up in the first position, a down in the second, and etc.  The ups will always be in odd positions and the downs will be in even positions.  If you are really lucky and can spread them out as much as possible you’ll have an up in the last position (since that position is odd) with no ups or downs next to each other.

But WAIT, since the numbers are going around in a circle the last position is really next to the first position.  oops – even if you are lucky enough to be able spread the ups and downs out as much as possible you end up with two next to each other.  So . . . you are forced to have either two ups or two downs together (which ever type of comparison there are more of)  when you put an odd number of numbers around a circle!

That’s it – that’s the main idea in the problem!

I know this write up isn’t perfect and certainly not perfect enough for something that a contestant would need to write up during the contest, but I wasn’t going for perfection here.  I just wanted to communicate my thought process as I worked my way through the problem.

This was a fun problem to think through and I hope that some ideas about how people trained in mathematics think about problems like this one came through in my write up.