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:

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

Using Laura Taalman’s 3D Printed Pentagons to talk math with kids

Evelyn Lamb’s Pentagons are Everything

Sharing Annie Perkins’s Cairo Pentagons with Kids

Sharing a Craig Kaplan post on “non-tiling” Pentagons with kids

(2) Platonic Solids

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:

Nesting Platonic Solids from our Zometool set

Can you believe that a dodecahedron folds into a cube?

Using Matt Parker’s Platonic Solid video with kids

(3) Folding / Cutting

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):

Math for Nine Year Olds -> Joel David Hamkins’s take on Fold and Punch

A fun folding exercise for kids from Paula Beardell Krieg

(4) Geometry and Art

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:

Our project inspired by Ison’s work and a few other art-related projects are below.

Extending our project with Ann-Marie Ison’s Art

Using Joel David Hamkins’s Perspective Drawing posts with kids

Paula Beardell Krieg’s Puff Boxes

An Amazing Visual math project for kids I learned from Jessica Rosenkrantz

**(5) Fun geometry ideas I’ve learned from math teachers**

I’ve learned about so many fun projects from ideas that teachers have shared on twitter – a few of the geometry-related projects are below:

Playing with Varignon’s Theorem thanks to Patrick Honner

Fawn Nguyen’s incredible Euclidean Algorithm project

Geometric Tilings inspired by Annie Perkins

Using Martin Holtham’s Inversion program (inspired by Dan Anderson) with kids

(6) Books

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:

Sharing an idea from Experiencing Geometry with kids

Playing with Patty Paper Geometry

Using Ernest Irving Freese’s “Geometric Transformations” with kids

(7) Number Theory

Using Mathologer’s Triangular Squares Video with kids

Writing an Integer as a Sum of Squares

Connecting Arithmetic and Geometry

A really neat problem that Gauss solved

**(8) Topology**

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.

Using the Infinite Galaxy Puzzle from Nervous System to talk Topology with kids

Dave Richeson’s Knotted Bubbles Project

A Zipper Mobius Strip from Mathjams

An Absolutely Mind Blowing Project from James Tanton

**(9) A few real world applications**

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:

Sharing some ideas about math and gerrymandering with kids

Using an idea from one of Katherine Johnson’s NASA Technical Papers to introduce polar coordinates

(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!

Has the Perfect Cuboid Problem been solved?

Larry Guth’s “No Rectangles” problem

A strange problem I overheard Bjorn Poonen discusing

Having kids play with Swarmalators