Tag 3D Printing

Sharing “developable surfaces” with kids thanks to a brilliant lecture from Heather Macbeth

[This is a redo of a blog post from January 2018 that somehow ended up 1/2 deleted. Not sure what I did to that old post, but I didn’t want to lose the ideas.]

In January 2018 I attended a terrific public lecture given by Heather Macbeth at MIT. The general topic was differential geometry, and the specific topic she discussed was “developable surfaces.”

Here’s an example from the talk:

I also printed a few examples and shared them with the boys the next day:

Here’s what my older son had to say about the shapes:

Here’s what my younger son thought:

These are really neat surfaces to explore. If you look at some of the mathematical ideas for “Developable surfaces” you’ll find that some of the surfaces are actually pretty easy to code, print, and share with kids!

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Sharing Annie Perkins’s Cairo pentagons with kids part 2

[sorry at the beginning that this post feels a little rushed. I wrote it during an archery class my son takes, but I forgot the power cord to my laptop and only had 20% battery at the start . . . . ]

Over the last week I saw two really neat videos from Annie Perkins on the Cairo pentagon tiling:

Yesterday I did a project with my older son on this shape of the pentagon. That project’s focus was on coordinate geometry:

Exploring Annie Perkins’s Cairo Pentagons with kids

Today I did a project with my younger son with 3d printed versions of the pentagons that I made today (after a few glorious fails . . . .). Sorry that the tiles don’t show up super well on camera when they are pushed together – I’d hoped that the white background with show through the gaps, but not so much 😦

Before starting I showed my son the two videos from Perkins and began the project by asking him to try to recreate the shapes he saw. He liked the tiling but ran into a little trouble trying to recreate it. It turns out that tiles also fit together in a way that doesn’t extend to a tiling of the plane. My son had a nice geometric explanation about why the shape he found wouldn’t extend to the full plane.

After running into a little difficulty in the last video, he started over with a new strategy. That new strategy involved putting the tiles together in groups of two and fitting those groups together. This method did lead to a tiling that he thought would extend to the full plane.

Definitely a fun project. You can see some links to other tiling projects we’ve done in yesterday’s project with my older son. Tiling is definitely a topic you can have a lot of fun with on a few different levels – from younger kids talking about the shapes they see, to older kids learning how to describe the equations of the boundary lines and coordinates of the points. Making the tiles is a fun 3d printing project, too.

Exploring Annie Perkins’s Cairo Pentagons with kids

I saw a great tweet from Annie Perkins a few days ago:

I thought it would be a fun idea to add to the list of our growing list of pentagon projects. At this point I’ve lost track of all of them, but they got started with this amazing tweet from Laura Taalman:

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

and the most recent project (I think!) is this one:

Evelyn Lamb’s pentagons are everything!

Oh, and obviously don’t forget pentagon cookies 🙂

Screen Shot 2016-07-17 at 9.46.03 AM

Tiling Pentagon Cookies

After seeing Perkins’s tweet I started down the path of making the Cairo tiling pentagons but super unluckily had a typo in my printing code. At least my cat made good use of the not-quite-Cairo pentagons:

So, while I wait for the correct pentagons to print, I thought I’d talk about the special shape of the Cairo tiles with my older son. One of the neat things about all of these pentagon projects is getting to talk about geometry with kids in sort of non-standard, non-textbook way. Tonight’s conversation was about coordinate geometry using the properties of the Cairo pentagon.

Here’s a pic from the Wikipedia page on the Cairo tiles:

Wikipedia’s page on the Cairo tiling pentagon

Cairo.jpg

To start the project I drew the shape on our board and asked my son to find the coordinates of the points. This is a bit of an open ended question because you have to know the lengths of the side so know the coordinates – I was happy that he noticed that problem (and, just to be 100% clear, I don’t know for sure if there are restrictions on the sides for the Cairo tiling – I’ll learn that when the new pentagons finish printing – ha ha).

Here’s how he started in on the problem:

For the second part of the project he had to make one more choice for a side length, and then he was able to find the coordinates of all of the corners of the pentagon.

One of the great (and happy) surprises with math and 3d printing is that you get neat opportunities to explore 2d geometry. Some of our old projects exploring 2d geometry with 3d printing are here:

Using 3d printing to help kids learn algebra and 2d geometry

I’m excited to play with the Cairo tiles when they finish printing tonight. Hopefully the 2nd time is a charm!

3d printing ideas to explore math with kids

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:

The Prince Rupert Problem

(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 nice little triangle puzzle

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”:

3d printing and 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!

Screen Shot 2016-07-17 at 9.46.03 AM

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 x^2 + y^2 was not the same as (x + y)^2

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:

Screen Shot 2018-01-02 at 11.52.13 AM.png

(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:

Dan Anderson’s 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:

My week with juggling roots

(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 L^p metrics:

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:

Revisiting an old James Tanton / James Key pyramid project

Building Paula Beardell Krieg’s cube

Playing with 3d printed versions of shapes theorized by Hermann Schwarz

Saw a neat tweet earlier today about 3d printing, math, and engineering:

I recognized some of the shapes in the article as ones that we’d played with before:

The grey shape displayed in the article is a “made thicker for 3d printing” version of the surface \cos(x) + \cos(y) + \cos(z) = 0. I thought it would be fun to print that shape today and use it for a little project with the kids tonight. Here’s the Mathematica code and what the print looks like in the Preform software:

8 hours later the print finished and I asked the boys to describe that shape plus the gyroid. It is always fascinating to hear what kids see in unusual shapes. My younger son went first:

Here’s what my older son had to say (and he’s starting to study trig, so we could go a tiny bit deeper into the math behind the shape I printed today):

Next we watched the video about the shapes made by Rice University:

After watching the video I asked the boys to talk about some of the things they learned:

Of course, mostly they didn’t want to talk about the shapes – they wanted to stand on them! So much for an 8 hour print and 45 min of trying to clean out the supports . . .

Here’s how the standing went:

Definitely a fun project and a fun way to show kids a current application of both theoretical math and 3d printing!

Using 3d printing to help explore a few ideas from introductory algebra

Last spring I was playing around with some different 3d printing ideas and found a fun way to explore a common algebra mistake:

Does (x + y)^2 = x^2 + y^2

comparing x^2 + y^2 and (x + y)^2 with 3d printing

Today I decided to revisit that project. We started by looking at the same idea from algebra:

Does x^2 + y^2 = (x + y)^2 ?

At first we talked about the two equations using ideas from algebra and arithmetic.

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Now I asked the boys for their geometric intuition and then showed them the 3d printed graphs of the two functions.

This part ran a little long while my younger son was stuck on a small but important point about the graph z = (x + y)^2 – I didn’t want to tell him the answer and it took a couple of minutes for him to work through the idea in his mind.

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Next I showed them 3d prints of x^3 + y^3 and (x + y)^3 and asked them to tell me which one was which. It is really neat to hear the reasoning that kids use to go from shapes to equations.

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For the last part of the project I asked the boys to come up with their own algebra “mistakes” for us to explore. My older son chose to compare the graphs of \sqrt{x^2 + y^2} and x + y.

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My younger son chose the two equations x^2 - y^2 and (x - y)^2. Changing the + to a – in our first set of equations turns out to have some pretty interesting geometric consequences – “it looks sort of like a saddle” was a fun comment.

One especially interesting idea here was exploring where x^2 - y^2 = 0. We used Mathematica’s ContourPlot[] function to explore those two lines because those lines weren’t immediately obvious on the saddle.

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I’m happy to have had the opportunity to revisit this old project. I think exploring simple algebraic expressions is a fun and sort of unexpected application of 3d printing.

Playing with sin() and cos()

My older son has just started the trigonometry this week. I know the topic can be a little dry at the beginning, so I wanted to show him more than just unit circle exercises.

Today we looked at a few fun curves that you can make just by playing around with trig functions.

I stared by showing some simple graphs and then we moved to some 3d shapes:

Next we took a cue from an old project inspired by Henry Segerman:

Playing with Shadows inspired by Henry Segerman

Hopefully both the shape and the shadows come through in the filming – I haven’t figured out how to shoot shadows very well yet.

Finally I let the kids play around with the Mathematica code for a bit to create their own shapes. They had a couple of pretty fun ideas.

There was one little issue that came up on my younger son’s plot, unfortunately. I didn’t have enough detail in the plot to multiply the range by 10. That’s why his picture fuzzed out quite a bit. I didn’t see the problem on the fly, though, and wasn’t able to fix it in real time.

I’m excited to help my son learn about trig. Hopefully a few projects like this one will help him see that that there’s more to trig than just triangles!

Building Paula Beardell Krieg’s cube

Yesterday we studied how to build the pieces of Paula Beardell Krieg’s dissected cube:

Paula

3d printing Paula Beardell Krieg’s dissected cube shapes

Today the shapes were done printing and I had the kids talk about them one more time:

After that short conversation I had each kid tweak the code that we used to make the shapes to make a new shape. Here’s what my younger son made:

Here’s what my older son made:

Definitely a fun couple of days with these shapes. Will probably revisit them again in a few months.

3D Printing Paula Beardell Krieg’s dissected cube shapes

I’ve been thinking about exposing the boys to math through 3d printing lately. Today I decided to explore making Paula Beardell Krieg’s cube shapes with them. Here’s the exploration the boys did back in March when we first got them:

Even though we’ve played a bit with these shapes before I still thought that thinking through these yellow and pink shapes would be a fun challenge. The project turned out to be a tiny bit harder than I thought it would be, but it still was a nice conversation.

We started by first looking at the three pyramids that can come together to make a cube and continued by looking at what happens when you slice those shapes in half.

In the last video the boys were thinking about trying to describe these shapes by describing the lines that formed the edges. At the beginning of this video I told them that this particular approach was going to be tough since they didn’t know how to write equations of lines in 3 dimensions.

So, I had them continue to search for properties of the shapes that they could describe.

The boys were still struggling to find some ideas about the shape that went beyond the lines on the boundary, but we kept looking.

My older son hit on the idea that the shape was made from “stacking squares on top of each other.” We spent the rest of the video exploring that idea.

Now that we had the idea about stacking squares we went to Mathematica to try to create the shape. It took a few steps to move from the ideas about the squares to generating the code for the shape. We didn’t get all the way there during this video, but we did figure out how to make a cube.

Unfortunately I had to end the video since the camera was about to run out of memory.

While I was getting the videos off the camera the boys worked on how to change the cube shape to the pyramid shape. It was a good challenge for them and they got it. We talked about that shape for a bit and then moved on to the challenge of creating the “pink” and “yellow” shapes that Paula Beardell Krieg created from paper.

We had a little bit of extra time today and it was fun to walk through this challenging problem. I think creating shapes to 3d print is a really fun way to motivate math with kids. Can’t wait to use the printed shapes in a project tomorrow!

Ten 3D Printing math projects to help students explore math

Yesterday I was able to watch the Global Math Project presentations (well, most of them) via the Facebook Live feed. Hopefully those videos will be preserved here:

The Global Math Project’s Facebook page

One tank that caught my eye was given by Henry Segerman. I’d guess that his work and Laura Taalman’s work account for at least 80% of what I know about exploring math through 3d printing.

As I write this post there are 96 prior posts with the “3D Printing” tag on my blog.  3D Printing is still pretty new, and I think many people around math are only starting to see its use in education. Segerman’s talk made me want to throw together a list of fun projects that we’ve done just in case anyone is looking for a starting point after seeing his talk.

Some of my original thoughts on exploring math through 3d printing can be found in this blog post from March 2014 which features two really neat videos from Brooklyn Tech and Laura Taalman:

Learning from 3D Printing

Here are some sample projects:

(1) James Tanton’s Geometry Problem and 3d printing

Since this blog post was inspired by a talk a James Tanton’s Global Math Project, it seems appropriate to kick it off with a project inspired by Tanton:

Here are some of the shapes we printed as we explored what the shape itself looked like:

Shapes

and here are the two projects that we’ve done exploring this problem

James Tanton’s geometry problem and 3d printing

Revisiting James Tanton’s Tetrahedron Problem

(2) Hard to highlight just one project that Segerman Inspired, so here’s the first of 2

One of the Segerman’s examples in yesterday’s talk was about bubbles. He showed a few complicated bubble examples but there are simple ones that are amazing, too. Here’s an example showing that the “bubble” formed by dipping a tetrahedron in soap is the same shape as a 4-dimensional shape:

Talking about Henry Segerman’s 5-cell with my 5th grader

(3) A second idea from Segerman – exploring shadows

One of Segerman’s most beautiful creations is on the cover of his book:

It is incredibly fun to have kids explore this shape:

Here’s the project we did after seeing Segerman give a talk last fall:

Playing with Sahdows inspired by Henry Segerman

Here’s a link to all of our project inspired by him:

All of our Henry Segerman-inspired projects

(4) There is also no way to limit Laura Taalman’s work to one example.

Here’s a project where we explored some of here 3d printed knots – one of which was featured in Segerman’s talk yesterday:

Playing with some 3d printed knots

(5) Here’s another project inspred by Taalman – tiling pentagons

Taalman’s 3d printed tiling pentagon designs are one of the most amazing pieces at the intersection of math and education:

We’ve used them for several projects including making cookies!

Screen Shot 2016-07-17 at 9.46.03 AM

Here’s that project

Learning about tiling pentagons from Laura Taalman and Evelyn Lamb

and here’s a link to all of our projects inspired by Laura Taalman:

All of our projects inspired by Laura Taalman

(6) Exploring connections between algebra and geometry

3d printing can come in handy for looking at math ideas that previously you could only study on paper or on the computer screen. For example, a common algebra mistake is to think that:

(x + y)^2 = x^2 + y^2

Here’s what these two surfaces look like:

3d prints

Here’s two projects exploring these algebra ideas with the boys:

Comparing x^2 + y^2 and (x + y)^2 with 3d printing/a>

Comparing Sqrt(x^2 + y^2) and Sqrt(x^2) + Sqrt(y^2) with 3d Printing

(7) 3D printing can also be surprisingly useful for studying 2d geometry

We’ve done a few neat projects in this area.

(i) Which triangle has larger area, a 5-5-6 triangle or a 5-5-8 one?

puzzle

A nice little triangle puzzle

A few follow ups to the triangle puzzle

https://mikesmathpage.wordpress.com/2017/03/01/a-nice-little-triangle-puzzle/

(ii) A neat geometry idea from Patrick Honner

Here’s how we used 3d printing to explore this triange:

Inequalities and Mr. Honner’s Triangles

(iii) A neat geometry problem shared by Tina Cardone

Here’s how I explored this problem with 3d printing

A Cool Geometry Problem Shared by Tina Cardone

which led to a fun and unexpected follow up:

A Follow Up to Our Tina Cardone Geometry Project

(8) 3d printing can be a fun way to review ideas from elementary geometry

In his talk yesterday Segerman mentioned a few prints that his undergraduate students created. As he showed this projects he talked about how the creation process really helps students understand and explore the underlying math.

In the project below, creating the shape of the tile helped me review and explore equations of lines with the boys:

Screen Shot 2017-06-27 at 10.17.26 AM.png

Sharing a Craig Kaplan post with kids

(9) 3d printing can also make abstract math / advanced problems accessible

A few months back I saw this problem shared by Alexander Bogomolny:

Nassim Taleb’s look at the problem on Mathematica made me think that the problem could be shared with kids:

Taleb1

Taleb2

A project for kids inspired by Nassim Taleb and Alexander Bogomolny

After getting some intuition from this problem we extended the problem to 4 dimensions using Taleb’s approach. The prints were really fun to play with and it is amazing to hear kids talk about these shapes that come from 4 dimensions:

Here’s that project:

Extending our Bogomolny / Taleb project from 3 to 4 dimensions

(10) Using 3d printing for calculus and beyond

I’m written a few posts and done a few projects about how to use 3d printing to explore some basic ideas from Calculus.

Circles

That collection of posts is here:

Posts about 3d printing and calculus

But 3d printing can help you see even more advanced ideas. Here’s a cube inside of a dodecahedron, for example:

and, of course, many (most!) of examples that Henry Segerman showed in his talk yesterday are perfect for showing how 3d printing can help everyone experience some advanced ideas in mathematics.

I’ll end with the project we did yesterday, which is a delightful example of how 3d printing can help you explore a math idea:

Revisiting the Volume of a Sphere with 3d printing