I’m doing some work in Michael Serra’s Patty Paper Geometry book with my younger son over school break.

Yesterday’s project was about finding angle bisectors and perpendicular bisectors using paper folding. Today he wanted to extend that work by finding incenters and circumcenters of triangles.

He did the folding to find the circumcenter of a triangle first – here’s that work:

Next up was finding the center of the circumscribed circle. He had one misconception about the inscribed circle – that the circle touches the circle at the feet of the angle bisectors. We talked about that point for a while at the end of the video:

I really do love how simple ideas in folding allow kids to explore and discuss fairly advanced ideas in geometry!

I had my copy of Michael Serra’s Patty Paper Geometry out on my desk because of a geometry discussion on Twitter.

It is vacation week this week and I asked my younger son if he wanted to keep working in his algebra book this week or do a little vacation project. He saw the book and said he wanted to do some patty paper projects this week. Yay!

Today he picked two –

(1) Making a perpendicular bisector via folding:

(2) Making an angle bisector via folding:

I love the idea of exploring geometry through folding. As you can see in the above videos, it allows kids to experience ideas in geometry naturally. It also gives plenty of opportunities to talk through other (all, I assume!) important ideas along the way, too!

Though it is hardly a book for kids, one problem from the first section jumped out as ones kids could actually work through, so I shared it with the boys today.

Here’s my younger son (in 7th grade) talking through the problem – he struggles a bit, but eventually finds the right idea to get to the end of the problem:

Next my older son (in 9th grade) talked through the problem. He was able to work through the problem without too much difficulty, so I asked him to explain his solution in more detail at the end:

I’m really excited to go through this book on my own over the next few months – hopefully will find a few more problems here and there to pull out and share with kids.

I saw an interesting tweet from Jordan Ellenberg earlier this week – here’s writing a new book on geometry and was asking for suggestions for neat geometry ideas people have seen:

I've got tons of material, probably too much already, but still — if something cool & geometric crosses your eye this year, tell me!

Having spent close to 10 years now searching for fun math ideas to share with kids, the tweet from Ellenberg was a good motivation to catalog some of them.

The first thing I did was ask my kids what their favorite geometric project was – my younger son answer was about tiling pentagons and my older son mentioned Platonic solids.

So, with those ideas to start things off, below a list of some of the neat ideas related to geometry that we’ve played with in the last few years.

(1) Tiling (and non-tiling!) Pentagons

After some new results about tiling pentagons came out a few years back, math professor and 3d printing super grand master Laura Taalman made some 3d printed models available and we had an enormous about of fun playing with them. Several projects are below, and even more information is in Patrick Honner’s article about tiling pentagons in Quanta Magazine

I was really happy to hear my older son bring up the idea of Platonic solids. We’ve done more than 100 projects with our Zometool set – one of the most amazing was putting all of the Platonic solids together in one shape. Other projects were inspired by the GIF above and a Matt Parker video:

The idea of approaching geometry through folding hadn’t really ever been on my radar. This video featuring Katie Steckles opened a new world to me (also see the Patty Paper Geometry book below):

The image above was inspired by a tweet I saw from the artist Ann-Marie Ison. It shows an incredible connection between geometry and number theory and you can play more with that connection with this Martin Holtham Desmos program:

I’ve come across several amazing – and I’d say fairly non-standard – books related to geometry in the last few years. Pics of those books plus a sample project from each of them are below:

I'm half trying to write a 2018 year in review for my math blog but other things keep getting in the way. So, in case I never write up anything, my favorite math book that I got in 2018 was Martin Weissmann's An Illustrated Theory of Numbers. It is absolutely incredible. pic.twitter.com/ghGfwYnEJy

The connections between geometry and topology have been some of the most eye-opening projects that we’ve done. The James Tanton project at the bottom of the list below is one of the most amazing math projects that I’ve seen.

The exercises for K-12 students from Moon Duchin’s Geometry and Gerrymandering conference are an absolutely terrific example to go through with kids. I’ve also used some ideas from Katherine Johnson’s NASA technical papers and a computer program about black holes to share interesting applications of geometry with kids:

(10) A few miscellaneous topics of interest to math professors that made for really fun geometri-realted projects for kids.

I didn’t really know how to classify these projects, so consider this last section “other”. The Larry Guth “no rectangles” project below is a super fun activity to do with a group of kids (of any age!). When I played around with the problem with a group of 3rd graders, I actually couldn’t end the session when the parents came to pick up the kids – the kids wouldn’t stop working on the problem!

This week learned about the book Experiencing Geometry by David Henderson and Diana Taimina. Unfortunately I learned about the book through people sharing news about David Henderson’s death. But despite the terrible circumstances, the book was captivating.

When I learned that Cornell professor David Henderson had died I bought a copy of the geometry book he wrote with @DainaTaimina . The book came today and I’ve spent the last few hours enjoying it. What an absolute treasure – can’t wait to use some of these ideas with kids. pic.twitter.com/aZXRY9K8QC

This morning I picked an idea from the book to share with the boys. The idea is from chapter 16 and is about drawing a circle through three points in a plane chosen at random.

Here’s the introduction to the problem. My younger son struggled a bit in the beginning to remember the ideas, but they did come to him eventually. That little struggle made me happy that we were looking at these geometric ideas today:

After we’d talked through some of the introductory ideas, I had the boys talk about their thoughts on the geometry in a bit more detail. I was especially happy that my younger son was able to sketch a proof that the perpendicular bisector was equidistant from the two endpoints of a line segment:

I had the boys work through the constructions off camera and then explain what they did. My older son approached the problem through folding:

My younger son worked for about 15 min on his construction – he works in a way that is so much more detailed than me! Here’s his work and his explanation which includes a nice discussion of why the center of the circle is outside of the triangle he drew:

I saw an amazing tweet from Craig Kaplan this week:

Tactile is a modern reimplementation of my PhD code for manipulating isohedral tilings, now available as open-source libraries in C++ and Javascript. I've also created a few fun web-based apps for playing with tilings. https://t.co/vnt8hLIPbl

Ever since seeing it I’ve been excited to share the program with the boys and hear what they had to say. Today was that day 🙂

So, this morning I asked the boys to take 15 to 20 min each to play with the program and pick 3 tiling patterns that they found interesting. Here’s what they had to say about what they found.

My older son went first. The main idea that caught his eye was the surprise of distorted versions of the original shapes continuing to tile the plane:

My younger son went second. I’m not sure if it was the main idea, but definitely one idea that caught his attention is that a skeleton of the original tiling pattern seemed to stay in the tiling pattern no matter how the original shapes were distorted:

Definitely a neat program for kids to play around with and a really fun way for kids to experience a bit of computer math!

My older son is working on a different math project this morning, so once again my younger son was working along. While cleaning up a little bit yesterday we found our old collection of “facets” – so I asked my son to build something for the Family Math project today.

He built a really neat shape:

We have done two previous projects with the facets (including making a big circle 🙂