Tag 3D Printing

My talk at the 2018 Williams College math camp

[had to write this in a hurry before the family headed off for a vacation – sorry that this post is likely a little sloppy]

Yesterday I gave a talk at a math camp for high school students at Williams College. The camp is run by Williams College math professor Allison Pacelli and has about 20 student.

The topic for my talk was the hypercube. In the 90 min talk, I hoped to share some amazing ideas I learned from Kelsey Houston-Edwards and Federico Ardila and then just see where things went.

A short list of background material for the talk (in roughly the order in the talk is):

(1) A discussion of how to count vertices, edges, faces, and etc in cubes of various dimensions

This is a project I did with my kids a few years ago, and I think helps break the ice a little bit for students who are (rightfully!) confused about what the 4th dimension might even mean:

Counting geometric properties in 4 and 6 dimensionsf

(2) With that introduction I had the students build regular cubes out of the Zometool set I brought. Then I gave them some yellow struts and asked them to construct what they thought a hypercube might look like. From the prior discussion they knew how many points and lines to expect.

To my super happy surprise, the students built two different representations. I had my boys talk about the two different representations this morning. Funny enough, they had difference preferences for which was the “best” representation:

Here’s what my older son had to say:

Here’s what my younger son had to say:

At the end of this section of my talk I showed the students “Hypercube B” from Bathsheba Grossman (as well as my Zometool version):

(3) Now we moved on to looking at cubes in a different way -> standing on a corner rather than laying flat

I learned about this amazing way to view a cube from this amazing video from Kelsey Houston-Edwards. One of the many bits of incredible math in this video is the connection between Pascal’s triangle and cubes.

Here are the two projects I did with my kids a after seeing Houston-Edwards’s video:

Kelsey Houston-Edwards’s hypercube video is incredible

One more look at the hypercube

After challenging the kids to think about what the “slices” of the 3- and 4-dimensional cubes standing on their corners would be, I showed them the 3D printed versions I prepared for the talk:

Here are the 2d slices of the 3d cube:

Here are the 3d slices of the 4d cube:

(4) Finally, we looked at the connection between cubes and combinatorics

I learned about this connection from this amazing video from Numberphile and Federico Ardila:

Here is the project I did with my older son after seeing Ardila’s video:

Federico Ardila’s Combinatorics and Higher Dimensions video is incredible!

I walked the students through how the vertices of a square correspond to the subsets of a 2-element set and then asked them to show how the vertices of a cube correspond to the subsets of a 3-element set.

There were a lot of oohs and ahhs as the students saw the elements of Pascal’s triangle emerge again.

Then I asked the students to find the correspondence between the 4-d cubes they’d made and subsets of a 4-elements set. I was incredibly happy to hear three different explanations from the students about how this correspondence worked – I actually wish these explanations were on video because I think Ardila would have absolutely loved to hear them.

(5) One last note

If you find all these properties of 4-D cubes as neat as I do, Jim Propp has a fantastic essay about 4 dimensional cubes:

Jim Propp’s essay Time and Tesseracts

By lucky coincidence, this essay was published as I was trying to think about how to structure my talk and was the final little push I needed to put all the ideas together.

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3d printing totally changed my approach to talking about trig with my son

For the last two weeks we’ve been playing with this book:

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Our most recent project involved one of the pentagon dissections. My son wrote the code to make the shapes on his own. We use the RegionPlot3D[] function in Mathematica. To make the various pieces, he has to write down equations of the lines that define the boundary of the shape. Writing down those equations is a fantastic exercise in algebra, geometry, and trig for kids.

Here’s his description of the shapes and how he made the pentagons:

Next we moved on to talking about one of the complicated shapes where the method he used to define the pentagon doesn’t work so well. I wish I would have filmed his thought process when he was playing with the code for this shape. He was really surprised when things didn’t work the first time, but he did a great job thinking through what he needed to do to make the shape correctly.

Here is his description of the process followed by his attempt to make the original shape (which he’d not seen in two days . . . )

I’m so happy that he’s been interested in making these tiles. I’ve honestly never seen him so engaged in a math project. The original intention of this project was just for trig review, but now I think creating these shapes is a great way to use 3d printing to introduce basic ideas from trig to students.

Playing with the nonagon tiles

Two of our recent project have involved studying a tiling of a nonagon from the book “Ernest Irving Freese’s Geometric Transformations”

Those two projects are linked here:

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

nonagon tiles

After school yesterday I had each of the boys make a pattern with the nonagon tiles and then build the two patterns that were in the book. The videos below show there work. My younger son went first:

Here’s what my older son had to say:

This project was super fun from start to finish. Hearing the thoughts from the boys after seeing the pattern initially was really fun. Building and printing the blocks was a nice geometry / trig lesson. Then having the boys play around with them made for a really satisfying end to the project. I hope to do more like this in the near future.

Nonagon tiles

Last week we did a fun project using a pattern we say in “Ernest Irving Freese’s Geometric Transformations” by Greg N. Frederickson:

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

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I thought it would be fun to make some of the tiles – especially since my older son is studying trig right now. The tiles finished printing overnight:

Last night my son and I talked about how you could make these tiles, with a focus on the trig and algebra required to define the shapes.

Here’s the introduction to the topic:

Now we talked about how to define the kite shape in the tiling. This involves talking about 40 and 50 degree angles:

Finally, we talked through the last part – finding the final point is pretty challenging. Turns out, though, that we don’t have to find the coordinates of the point because we can write down the equation of the top line pretty easily:

I’ve been happily surprised that 3d printing is a fun way to help kids explore 2d geometry. I’m excited to have my son try to make some other tiles from the book on his own for our next project.

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!

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 🙂

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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!

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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:

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(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.