Over the winter break I began to think about collecting some of our 3d printing projects into to one post to highlight various different ways that 3d printing can be used to help kids explore math.
The post got a little long, but if you are interesting in thinking about 3d printing and math, hopefully there are ideas in here that either catch your eye.
(1) Archimedes’s proof relating the volume of a sphere, a cone, and a cylinder
I asked my younger son to pick his favorite 3d printing exercise – here’s what he picked:
Our project with this shape is here:
The Volume of a Sphere via Archimedes
(2) Playing with a Rhombic Dodecahedron
My older son’s favorite project involved the rhmobic dodecahedron:
We’ve actually done a bunch of projects – both 3d printing and Zometool projects – with the rhombic dodecahedron. Here’s a link to all (or probably most) of them:
Our projects with the rhombic dodecahedron
(3) Sharing a Craig Kaplan post about tiling (or non-tiling) shapes
Here’s a fun example of how you can use 3d printing to explore 2d geometry:
Sharing a Craig Kaplan post with kids
Another project where we used ideas from algebra and geometry to make tiles is here:
Using Evelyn Lamb’s tiling pentagons to talk about lines and shapes with kids
(4) The Prince Rupert Cube problem
This is probably my favorite 3d printing project that we’ve done on our own. I didn’t do a specific project with the boys using the shape because it is really fragile (in fact, I have 3 other broken ones . . . ).
The problem is -> can you cut a hole in a cube large enough so that you can pass another cube of the same size through the first cube?
An old project where we talk about the problem (without 3d printing) is here:
(5) Playing with mathematical puzzles
Here are two fun mathematical puzzles we found on Thingiverse. There are lots of fun mathematical games you can find to play with:
One other incredible game is Iwahiro’s “Apparently Impossible Cube”:
The “Apparently Impossible Cube” on Thingiverse
The boys had really enjoyed trying to solve Iwahiro’s puzzle (which may be more difficult to get apart than it is to put together!).
(6) The Gyroid and other minimal surfaces
3d printing allows you to explore some incredible shapes. For instance:
Taking kids through John Baez’s post about the gyroid
Playing with 3d printed versions of shapes theorized by Hermann Schwarz
(7) Some simple examples for a calculus class
3d Printing and Calculus concepts for kids
Another calculus-related project is here, and it includes a great video from Brooklyn Tech that helped show me the possibilities 3d printing had for helping kids explore math:
Sharing a shape from Calculus with kids
(8) “Seeing” geometric probability
Working through an Alexander Bogomolny probabilty problem with kids
(9) Some amazing shapes – the “rattleback”
Here’s a really fun shape to play with – the rattleback. It wants to rotate one way, but not the other way. There’s very little indication when you look at it that it would have such an odd property:
(10) James Tanton’s tetrahedron problem
This one has a special place in my heart because it was one of the first times we used 3d printing to solve a “new to us” problem. I loved how these shapes came together. The problem involved understanding the locus of points that were 1 unit away from a tetrahedron:
James Tanton’s geometry problem and 3d printing
Revisiting James Tanton’s Tetrahedron Problem
(11) Exploring plane geometry
Some projects where we’ve used these ideas are here:
A few follow ups to the triangle puzzle
Paula Beardell Krieg’s 72 degree question
Another idea from plane geometry that we explored with 3d printing came from Patrick Honner:
Inequalities and Mr. Honner’s triangles
(12) Exploring 4d geometry:
We’ve done a bunch of projects related to the 4th dimension that have been aided by 3d printing. Most of this work has been inspired in one way or another by Henry Segerman. Here are a few examples:
Using 3d printing to share 4-dimensional shapes with kids
Things to Print and Do in the 4th Dimension
(13) Rollers
This tweet from Steven Strogatz inspired us to makes some “rollers”:
(14) Exploring a fun shape -> a surface with 2 local maximums
John Cook shared a shape with a surprising property last year:
John Cook’s neat surface example
(15) Exploring knots with 3d printing
3d printed knots were a great aid to us exploring the basics of knot theory.
Playing with some 3d printed knots
An intro knot activity for kids
(16) Tiling pentagons
Another one of my all time favorites projects came from Laura Taalman. Right after the discovery of a 15th type of pentagon that tiles the plane, Taalman created 3d print models of all 15 of the pentagons so that anyone could explore this new discovery:
We’ve used Taalman’s pentagons for several projects including making cookies!
Here’s that project
Learning about tiling pentagons from Laura Taalman and Evelyn Lamb
(17) Exploring some algebraic expressions
This was an idea that I started playing with on a whim. Turned out that 3d printing some surfaces was a great way to show that 
Comparing x^2 + y^2 and (x + y)^2 with 3d printing
Comparing sqrt(x^2 + y^2) and Sqrt(x^2) + Sqrt(y^2)
(18) Playing with trig functions
Playing with the algebraic expressions above game me the idea to introduce some concepts from trigonometry through 3d printing:
The last shape in the above video really blew me away – it the following description:
(19) Exploring fractals
A fun fractal project – exploring the Gosper curve
A fun follow up to that project came when Dan Anderson sent us some laser cut Gosper curves:
(20) The “squircle”
This is one of the most amazing illusions that you’ll ever see 🙂
(21) Laura Taalman’s hinged polyhedra nets
Laura Taalman created 3d prints for explorying Platonic solids:
Laura Taalman’s “Customizable hinged polyhedra” on her Makerhome blog
(22) Braid groups and polynomial roots
This tweet from John Baez led to a really fun week exploring roots of polynomials:
At the end of the week we created some 3d printed models showing examples of how the roots move:
(23) A neat shape shared by Steven Strogatz
This tweet from Steven Strogatz led to us playing with a really interesting shape:
An amazing shape shared by Steven Strogatz
(24) Playing with shapes made by Henry Segerman
Henry Segerman’s book is a must!
Playing with more of Henry Segerman’s 3d prints
and more of our projects inspired by Segerman are here:
Playing with shadows – inspired by Henry Segerman
(25) Exploring the Permutohedron
A comment on the blog post below from Cornell math professor Allan Knutson introduced us to a new shape:
A morning with the icosidodecahderon thanks to F3
Here’s the permutohedron:
A fun shape for kids to explore -> the permutohedron
(26) Exploring L^p metrics
This video from Kelsey Houston-Edwards led to a really fun series of projects exploring the 
Here are some of our 3d printed shapes and why my younger son thought about them:
The full series of projects is here:
Sharing advanced ideas in math with kids via 3d printing
(27) Dissecting a cube into 3 and 6 pieces
It all started with this tweet 🙂
Then our friend Paula Beardell Krieg showed some fun extensions of the idea through paper folding:
Mike's Math Page 