The article is interesting all by itself, and the mathematical question Dan is asking was the subject of our project this morning.

First I asked the boys to read the article – here’s what they thought:

I was happy that the idea about the cone having the least surface area for a given volume came up when the boys were summarizing the article. We now moved on to investigating that question.

We first looked at a cube:

The calculations for the cube were pretty easy. Now we moved on to a slightly more complicated shape -> half of a sphere.

Working through the various volume and surface area formulas is a nice introductory algebra exercise for kids:

Now we moved on to looking at cones. Looking carefully at cones is quite a bit more complicated than looking at cubes or spheres. So, first we played with the formulas and reduced the surface area formula to one variable. We got that formula at the end of this movie:

The formula we found in the last video was a bit complicated, so we moved to Mathematica for a bit of help. The graph of the surface area for different values of radius of the cone is a shape that the boys haven’t seen before.

It was fun to talk about how this shape could be helpful in studying the question that Dan asked in the tweet.

It was also fun for me to hear how they thought about ways to zoom in on the minimum.

Definitely a fun project – would be especially good for a calculus class, I think.

A tweet last week from John Baez made for a really fun week of playing around. I’ve written several blog posts about it already. Here’s the summary to date, I guess:

(3) A video from a comment on one of the posts from Allen Knutson that helped me understand what was going on a bit better:

So, with that as background, what follows are some final (for now at least) thoughts on what I learned this week. One thing for sure is that I got to see some absolutely beautiful math:

Playing with the "juggling roots" and printing some of the structures has been the most fun I've had with math in ages. #math#mathchatpic.twitter.com/4cL583Ix2y

For this blog post I’m going to focus on the 5th degree polynomial . I picked this polynomial because it is an example (from Mike Artin’s Algebra book) of a polynomial with roots that cannot be solved.

So, what do all these posts about “juggling roots” mean anyway?

Hopefully a picture will be worth 1,000 words:

What we are going to do with our polynomial is vary the coefficients and see how the roots change. In particular, all of my examples below vary one coefficient in a circle in the complex plane. So, as the picture above indicates, we’ll look at all of the polynomials where moves around a circle with radius 8 centered at 10 + 0 I in the complex plane. So, one of our polynomials will be , another will be , another will be , and so on.

The question is this -> how do the roots of these polynomials change as we move around the circle? You would certainly expect that you’ll get the same roots at the start of the trip around the circle and at the end – after all, you’ve got the same polynomial! There’s a fun little surprise, though. Here’s the video for this specific example showing two loops around the circle:

The surprise is that even though you get the same roots by looping around the circle, with only one loop around the circle two of the roots seem to have switched places!

Here’s another example I found yesterday and used for a 3d print. Again for this one I’m varying the “2” coefficient. In this case the circle has a radius of 102:

When I viewed this video today, I realized that it wasn’t clear if 3 or 4 roots were changing places in one loop around the circle. It is 4 – here is a zoom in on the part that is tricky to see:

Next up is changing the “-16” in the x coefficient in our polynomial. Here the loop in the complex plane is a circle of radius 26:

Finally, there’s nothing special about the coefficients that are 0, so I decided to see what would happen when I vary the coefficient of the term that is initially 0. In this case I’m looping around a circle in the complex plane with radius 20 and passing through the point 0 + 0i:

So – some things I learned over this week:

(1) That the roots of a polynomial can somehow switch places with each other as you vary the values of the coefficients in a loop is incredible to me.

(2) The idea of thinking of these pictures as slices of a 3-dimensional space (which I saw on John Baez’s blog) led to some of the most visually striking 3d prints that I’ve ever made. The math here is truly beautiful.

(3) I finally have a way to give high school students a peek at a quite surprising fact in math -> 5th degree polynomials have no general solution.

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 đź™‚

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!

These inversions are such a fun area of geometry and one that kids hardly ever get to see, so I was really excited to play around with Dan’s program. Today I used it for a fun project with the boys.

We started with a one dimensional introduction to inversions:

The next thing I did was have the boys try out an inversion for a simple 2d shape – they took about 5 min to draw out how they thought the process would work. Here are their drawings and their explanations:

Next I introduced the boys to Dan’s program – there were lots of surprises!

At the end of the last video my younger son noticed that pentagons always seem to invert to be pentagons. That led to a fun discussion about lines, angles, and eventually circles.

So, another really fun program from Dan Anderson and a great opportunity to talk through some really interesting math with kids!

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:

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:

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:

One thing that gets me still: the dissonance between my belief that anyone can do math, and the recognition that most people won't want to.

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?â€ť

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:

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!

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.

I decided to introduce Dan’s shapes to the boys by putting them on our whiteboard and seeing what the boys would do.

My younger son went first and chose to slightly surprising shape to try to make out of the Gosper Island pieces:

Next he wanted to see if he could make a larger rectangle using the big pieces that Dan sent us. It would not have occurred to me to try to make a rectangle out of these pieces – it is fun to see the ideas that kids have!

Now I had my older son look at the pieces. He started building and made some holes. He had a little trouble seeing that the holes were the same shape as the other Gosper island pieces – he saw that they were hexagon-like, but they were so small that seeing that they had the same shape as our pieces was difficult.

At the end of the last movie my son wondered if you could make the same kind of holes using starting with the big pieces. He ended up building a slightly different shape than the one in the previous video, but the holes in this shape were indeed exactly the size of the small Gosper island pieces:

So a fun project just playing around with the pieces. It is pretty amazing to me how well the three pieces all fit together – these laser cuttings are a great way to explore fractals and fractal dimension.

(and there are *many* more pictures if you look at Dan’s twitter feed +/- a few days from this picture)

One of Mandelbrot’s finance books was on my mind, too:

So, when my son said that he wanted to do a geometry project this morning, I suppose that I was basically primed to think of something relating to fractals. The idea that I chose was a dimension.

I had the boys make a few simple shapes from our Zometool set and then we talked about what they thought about dimension:

In the last video we talked about 1 dimensional objects, but it wasn’t at all clear where the “1” was. Next we talked about 2 dimensional objects. That helped my younger son see the the “1”. Yay!

Having talked about the 1st and 2nd dimension, the third dimension was a piece of cake – well . . . other than a tiny bit of confusion about area and volume.

Now we got to the punch line of the project – shapes that don’t scale with integer powers. We used our Gosper Island pieces that we printed from Thingiverse:

These shapes have the unexpected property that when you scale lengths by a factor of 3, the area scales by a factor of 7. This idea was quite a surprise to my younger son – “I did not know that there were half dimensions.”

We have looked at this shape previously. It was almost a year ago, though, so I’m not all that surprised that it wasn’t fresh in their mind:

I also wish that the shapes fit together just a bit better than they do so that the scaling by a factor of 7 was clearer. Still, though, I thin that this is a fun project and a neat way to introduce kids to the idea that not everything is a perfect shape from Euclidean geometry.

So, tonight we revisited the idea of rotation to try to make things a little clearer. First up was an equilateral triangle. In the less abstract setting of the dining room table, the kids were able to talk through the rotational ideas a little more easily:

Next we looked at a square. After the discussion my older son gave about the equilateral triangle, my younger son was able to give a nice description of what was going on with the square:

The last shape we looked at was a regular pentagon. My older son thought that the rotation angle + the interior angle of each shape (or at least a regular polygon) would add up to 180 degrees. I asked the kids to figure out why that was true for the pentagon:

Finally, as a special little treat / challenge, I showed the boys a strange situation where you have to rotate something 720 degrees around the center to get back where you started. This surprising rotational trick is something that I learned back in my abstract algebra class in college from Mike Artin.

Sorry the video quality isn’t so great, but it was fun talking through these patterns with the kids. We started with the square pattern:

Then we moved on to the triangle pattern. It was surprisingly difficult for the kids to understand how to describe the rotations, but eventually they figured it out. I think I’ll revisit a bit more about rotations for a Family Math Project this weekend.

The patterns in the Numberphile video make a great project to talk through with kids. I can’t wait to try a few more ideas from their video.

I had a 7 hour round trip drive yesterday and spent a little time thinking about the math ideas that really grabbed me in high school. Three really stuck out in my mind:

(A) The Extended law of sines:

We learned in our trigonometry class that for a triangle with sides A, B, and C, and corresponding angles a, b, and c that:

= =

But it turns out that these ratios are equal to 2R where R is the radius of the circumscribed circle. I learned this idea from the wonderful book Geometry Revisited by Coxeter:

In fact, the first math movie I uploaded to youtube was about this idea:

This identity made me think that there was a lot more going on in geometry that met the eye. One neat particularly neat thing that the identity shows is that the area of a triangle with side lengths A, B, and C is equal . Beautiful!

(B) 1 + 1/4 + 1/9 + . . . . =

Mr. Waterman used the idea that the coefficients of a polynomial were symmetric functions of the roots to prove this sum. It blew me away. (yes, this is a non-rigorous proof, but it is what captured my attention)

In general, for a polynomial of degree n, , sum of the reciprocals of the roots is given by

We know that Sin(x) = Factoring out an we are left with a polynomial whose roots are namely:

making the substitution u = x^2, we see that the polynomial

are

By the “sum of the reciprocals of the roots” formula above, we see that

Multiplying both sides by gives us the result.

This result showed me that there was more going on with the integers than I realized! How could they be connected to ? A few years later I’d see this identity in a complex analysis class and see that and were connected in a strange way, too!

(3) A formula for the Fibonacci numbers

I think it was my sophomore year in high school when a former student, Anita Barnes, came back to lecture to Mr. Waterman’s Enrichment Math class. Her talk showed a way to find closed form solutions for simple recurrence relations like the one for the Fibonacci numbers:

The idea seemed incredibly simple – for the Fibonacci numbers just assume the solution took the form and solve for x. Solving the recurrence relation for the Fibonacci numbers was reduced to solving the quadratic equation From there it was not hard at all to show that the Fibonacci numbers were connected to the Golden ratio. If we let , then

That just blew me away – there was a simple formula for the Fibonacci numbers (and any simple recurrence relation). You could calculate the 100th Fibonacci number by just knowing the first 2 plus the recurrence relation. I think this was the first idea from advanced math that totally blew my mind.