The shape sort of stuck in my mind and last night I finally got around to making two shapes inspired by Phelp’s shape. My shapes are not the same as his – one of my ideas for this project was to see if the boys could see that the shapes were not the same.

So, we started today’s project by looking at the two shapes I printed overnight. As always, it is really fun to hear kids talk about shapes that they’ve never encountered before.

Next we looked at Phelp’s tweet. The idea here was to see if the boys could see the difference between this shape and the shapes that I’d printed:

Finally, we went up to the computer so that the boys could see how I made the shapes. Other than some simple trig that the boys have not seen before, the math used to make these shapes is something that kids can understand. We define a pentagon region by 5 lines and then we vary the size of that region.

I’m not expecting the boys to understand every piece of the discussion here. Rather, my hope is that they are able to see that creating the shapes we played with today is not all that complicated and also really fun!

This was a really fun project – thanks to Steve Phelps for the tweet that inspired our work.

— Alexander Bogomolny (@CutTheKnotMath) June 27, 2017

and the mathematical point that caught my eye was the question -> which positive integers are close to being integer multiples of ?

One possible approach to this question uses the idea of “continued fractions.” I learned about continued fractions from my high school math teacher, Mr. Waterman, who taught them using C. D. Olds’s book.

So, today I stared off by talking about irrational numbers and reviewing a simple proof that the square root of 2 is irrational:

Next we talked about why integer multiples of irrational numbers can never be integers. This I think is an obviously step for adults, but it took the kids a second to see the idea:

Now we moved on to talk about continued fractions. I’m not trying to go into any depth here, but rather just introduce the idea. I use my high school teacher’s procedure: split, flip, and rat ðŸ™‚

We work through a simple example with and also see that the first couple of fractions we see are good approximations to .

With that background work we went up to use Mathematica to explore different aspects of continued fractions quickly. One thing we did, in particular, was use the fractions we found to find multiples of that were nearly integers.

Finally, we wrapped up by using continued fractions to find good approximations to , and a few other numbers.

Definitely a fun project, and one that makes me especially happy because of the connection to Mr. Waterman. Hopefully the boys will want to play around with this idea a bit more tomorrow.

It is a nice compliment to some of the 4th dimensional projects we’ve been doing. Here’s what my younger son thought after playing around with it for a bit:

Here’s what my older son thought after playing with it for 10 min:

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.

So, I think the path that the animation took to our eyes was from twocubes to curiosamathematica to John Baez to us. Sorry if I do not have the sources and credit correct, but I will make corrections if someone alerts me to an error.

I’d never made any sort of animation before, but since the pictures looked like they came from Mathematica I started to play around a little bit last night to see what I could do. In doing so I learned about Mathematica’s “Animate” and “Manipulate” functions and made some progress, though the animations that I made were not nearly as good as the ones from the above posts. This Stackexchange post was helpful to me in improving the quality of my animations, but still mine aren’t in the same league as the original ones:

Anyway, with that introduction, I thought it would be really fun to share these animations with kids and do a tiny bit of background explanation. I stared this morning by just showing the boys some of the pictures and asking them to describe what they were seeing:

Next I showed them one of the animations that I made and asked them to see if they could see some similarities with any of the previous animations:

Next we went down to the living room to talk about roots of equations. My older son knows a little bit about quadratic equations, but only a little bit. I didn’t want this part of the conversation to be the main point, but I did want them to get a tiny peek at the math behind the animations we were looking at today:

Finally, we went back up to the computer to look at some of the animations for quadratic and cubic equations. My maybe too open-ended task for them here was to compare the animations of the roots of quadratic and cubic equations to the animations of the roots of the quintic equations.

I’ve always wanted to be able to share some of the basic ideas from Galois theory with kids. I’ve never seen anything like these animations previously. They make for a neat starting point, I think, since kids are able to talk about the pictures. I would **love** to know what a research mathematician sees in the pictures. In particular, is there something in the pictures that gives a clue about why the roots of 5th degree polynomials are going to be more difficult to study than 2nd, 3rd, or 4th degree ones?

I’m falling way behind on Kelsey Houston-Edwards’s video series, sadly. Her “How to Break Crytography” video is so freaking amazing that it needed to be first in line in my effort to catch up!

So, this morning I watched the video with the boys. We stopped the video a few times to either work through some of the math, or simply to just have me explain it a bit. Overall, though, I think this video is not just accessible to kids, but is something that they will find absolutely fascinating.

Here’s what my kids took away from it:

Next we went upstairs to write some Mathematica code to step through the process that Houston-Edwards described in her video. In this video we (slightly clumsily) step through the code and check a few small examples:

When I turned the camera off after the last video my younger son asked a really interesting question -> Why don’t we just use Mathematica’s “FactorInteger[]” function?

We talked about that for a bit in this video and then tried to use Shor’s algorithm to find the factors of a number that was the product of two 4 digit primes.

So, we had the camera off for a little over a minute after the last video, but the good news is that Mathematica did, indeed, finish the calculation. It was a nice (and somewhat accidental) example of how quickly this algorithm runs into trouble.

The cool thing, though, is that it did work ðŸ™‚

Definitely a fun project, though it does require a bit more computer power than most of our other projects. I’m happy to be catching up a little on Kelsey Houston-Edwards’s video series – it really is one of the best math-related things on the internet!

Before diving in to the project, I’d really recommend thinking about the question – even just for a few seconds – just to see what your intuition tells you.

We started the project by looking at the tweet and trying to make sure that the boys understood the question. The question itself was harder for them to understand than I expected. One reason was that they weren’t used to thinking about ages in terms of days.

Next we went to Mathematica and wrote a little program using the “PrimePi” function which tells you the number of primes less than or equal to a number.

We played around a little bit. Their initial instinct was to zoom in on a specific number like 30 years old. There were some fun surprises since the number of primes between two numbers bounces around a bit. They also had some really interesting ideas about prime numbers.

Eventually they decided to check a range of ages.

At the end of the last video we decided to check a range of ages, and we did that with a “For” loop. Once we did that we found a couple of really fun surprises ðŸ™‚

Running the program over night, the largest age that I found was 179,676 years old! I doubt that’s the highest number, though, and I love that the boys thought that there might be infinitely many solutions to this problem.

I saw this problem today when it was re-tweeted by Matt Enlow:

I wonder if this is enough information to figure out the dimensions of a square. If not, I wonder what squares are possible. pic.twitter.com/oZry1z9U5C

It is a little advanced for my younger son, but I still thought it would be fun to turn into a mini project tonight with the boys.

We started by talking through the problem and taking a guess at what we thought the answer was -> Is there enough information to determine the side length of the square?

Although we didn’t really make any progress towards a solution in this initial discussion, I really like the ideas that we talked about. Specifically, I liked how much thought my older son put into how to label the diagram.

In this part of the project we began to discuss how to solve the problem. We found two equations, but had 3 variables. My older son began to think that we weren’t going to find a solution.

In trying to simplify one of our equations my younger son made a common algebra mistake. I spent most of the video slowly showing him how to tell that the algebra he thought was right was actually off.

At the end of the last problem we found an equation that seemed to be a step in the right direction of finding a solution to the problem. In this part of the project we explored that equation.

At the beginning my older son was really confused. I think he’s used to seeing problems where there is always a solution – the open endedness of this problem seemed to leave him puzzled.

We did get our sea legs back, though, exploring a few specific cases. The happy accident was that the two solutions we found to the problem gave us the same perimeter for the square – was a unique solution hiding here?

To wrap up the project we went up to the computer to look at our equation using Mathematica. We’d covered the important mathematical ideas already, but finding some of the exact solutions was going to be a chore and certainly finding the maximum perimeter wasn’t going to be in reach.

Nonetheless, there were a few fun surprises to be found ðŸ™‚

Based on the blog post, his talk at MoMath must have been incredible!

I decided to try out one of his explorations with the boys tonight. We did the first few parts by hand and the last part using Mathematica and the code from Wolfram’s blog post.

The process we studied works as follows:

(1) Pick an integer to start with and pick a number to multiply by in step (3),

(2) Cycle the digits of the number to the left. A few examples will make the process clear:

123 goes to 231
402 goes to 024, or simply 24
111 would stay 111

(3) Multiply the new number by and then add 1.

(4) Return to step (2) with the new number.
The video below shows how our exploration began. Our initial integer was 12 and we multiplied by 1 at each step (so, starting easy, though I picked 12 at random so I really didn’t know what was going to happen):

Now we moved to a slightly more complicated example -> the same process as in the first part but we’ll be working in binary rather than in base 10.

We started with the number 6 (110 in binary) and multiplied by 2 at each step. Once again we found a fun surprise:

To get one more round of practice in before moving upstairs to the computer we looked at the same situation as in part 2, but this time starting with 1 and looking at several cases – multiplying by 1, by 2, and by 3:

Finally, we went to the computer to explore the process in many different situations. We used code from Wolfram’s blog post to recreate the work from the MoMath talk:

What I *love* about this project is that the exploration works really well with kids on the whiteboard and on the computer. The whiteboard exploration gave us a great opportunity for a little practice with arithmetic, with binary, and with algorithms. We also saw some really fun surprises!

The computer exploration is obviously fantastic, too. I’m so grateful that Stephen Wolfram shared the ideas from his talk!

And there’s our fun Zometool Snowman, too, where the icosidodecahedron is the head:

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The name suggests that it is made from a combination of an icosahedron and a dodecahedron – but how?

Ahead of the project I made a few shapes:

(i) the icosidodecahedron
(ii) a dodecahedron with an icosahedron removed
(iii) an icosahedron with a dodecahedron removed

Here’s what the boys thought of those shapes:

Next we went upstairs to play with some code in the F3 program. Looking at the video now I see that I forgot to publish it hi def – sorry about that. I hope our explanation of the code is good enough if the code is too fuzzy to read:

Definitely a fun little project – it is so fun to be able to play with these shapes on the computer and then hold them in your hand!