Today I had my son read through the post and then we discussed the ideas. His initial thoughts are in the video below – he understood most of the post and also had a couple of good questions:
After we talked through the post we went to Mathematica to take a look at some of the example “reflections” which preserve the 1st and 2nd derivatives:
One of my son’s questions in the first video was why the blog post was using functions like f(x/2) and f(x/4) to make reflections. I’d mentioned that these were essentially arbitrary choices. Below we saw what would happen if we used f(x/3) instead:
We finished up by going back to Mathematica to see what these new “reflections” would look like:
A few days ago we did a project using Mathologer’s amazing video on Fermat’s “two squares” theorem. At the end of the project the boys were wondering about why so many of he numbers we found that could be written as the sum of two squares in several different ways were multiples of 5. I was wondering the same thing and spent two days playing around and trying to learn more these sorts of numbers. Even after searching the positive integers up to 3,000,000, all of the numbers I found that could be written as the sum of two positive squares in exactly 7 ways were multiples of 5. What was so special about 5?
Overnight I got some great twitter advice on the subject form Stephen Morris and Alex Kontorovich. Their ideas helped me understand a bit more about what was going on. Tonight I explored some of the basic ideas with the boys. I know next to nothing about the number theory here, but am completly amazed by the never ending patterns that are hiding inside of the integers!
We started today’s project by looking at all of the positive integers less than 1,000,000 that can be written as the sum of 2 positive squares in exactly 7 ways. Here’s what they noticed:
At the end of the last video my younger son thought that it might be useful to factor all of the numbers on our list. We did that off camera and then the boys looked for patterns in the numbers and factors. Finding patterns in the factored numbers was more challenging than I expected, but they were able to make some progress.
Based on what we noticed we took some guesses at numbers that were not multiples of 5 that could be written as a sum of two positive squares in exactly 7 ways.
Finally, we used the Wolfram Alpha code that Stephen Morris showed us to check if the numbers we guessed really could be written as the sum of two positive squares in exactly 7 ways.
This project was incredibly fun. It shows how computers (and Twitter!) can really help kids explore some pretty advanced ideas. I’m really interested to see how we might be able to explore a few more related ideas in the next week.
Yesterday we watched this fantastic video by Mathologer on Fermat’s Two Squares theorem:
I’m hoping that we can do a couple of different projects based on the video. Today we talked about some ideas from the video and wrote a project to explore sums of squares.
We began by discussing some of the ideas in the video that the boys found interesting, and then talked through a few of the proof ideas from the vdeo:
Next we checked out one of the algebraic identities that came up in the Mathologer video. I thought that checking these identities would be good algebra practice.
Finally, we went to Mathematica to write a program to look at sums of squares. There was one slightly tricky part of the program that took a minute or two for the boys to explain. I thought that they had understood this concept while we were writing the program, so it turned out to be sort of lucky that we went back to talk about it.
Anyway, here’s the program that we wrote and a little bit of play with the numbers. The surprise was that it seemed like almost all of the numbers that could be written as sums of squares in lots of ways were multiples of 5. We might explore this idea a bit more in a later project.
Fermat’s Two Squares is really fun to explore with kids – and Mathologer’s video is a great way to show kids an proof that they can understand. I’m excited to explore the ideas a bit more with the boys later this week.
This week my son had two pretty neat homework problems – one from his math class at school and one from Art of Problem Solving’s Precalculus book. I thought it would be a nice and easy project today to go back and review these two problems.
The first one was a geometry problem from his school math homework:
The second is a problem about complex numbers from Art of Problem Solving’s Precalculus book:
My younger son has been learning about complex numbers in Art of Problem Solving’s Precalculus book lately, and today I thought it would be fun for him to see a little bit about what complex functions “look” like.
We started with a quick review of what he knows about complex numbers and then we talked a bit about why graphing a function of a complex variable is difficult:
Next we talked about graphing a specific function ->
Now we moved to Juan Carlos’s plotter. Here we discussed and then where varies between -1 and 1.
We had a couple of slips of the hand trying to type in between the tripod in this and the next video – that’s why you’ll see a few jumps in the vids.
Finally, we wrapped up by having my son graph a few functions that he created. It was fun to see what he thought about those functions:
Last week I saw this really neat tweet from Tom Ruen:
The projections are closely related to quasicrystals as well. The Penrose tilings come from projections of a 5-cube (which stack infinitely in 5D like cubes in 3D). This shows all the rhombic dissections of a decagon.https://t.co/hmk3Vt0fHnhttps://t.co/knff5ywvn9pic.twitter.com/PRClCio2Hu
Yesterday my younger son and I talked through the decagons after building them from our Zometool set. Today we talked about the projection of the 5d cube.
Here are his initial thoughts:
My son was interested in comparing this 5d cube shape to a shape that we’d built previously. So we got that shape and continued the comparison. We also talked a bit about where else the number 5 appeared in the 5d cube and in our shape:
I’m so happy to have seen the conversation that Nalini Joshi got started on Twitter last week. We’ve had two super fun projects so far inspired by it!
Earlier in the week I saw a really neat twitter thread that had this post:
The projections are closely related to quasicrystals as well. The Penrose tilings come from projections of a 5-cube (which stack infinitely in 5D like cubes in 3D). This shows all the rhombic dissections of a decagon.https://t.co/hmk3Vt0fHnhttps://t.co/knff5ywvn9pic.twitter.com/PRClCio2Hu
where 
varies between -1 and 1. 
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