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More math with bubbles

Bubbles were just in the air this week!

and last night flipping through Henry Segerman’s math and 3d printing book I found these bubble project ideas:

So I printed two of Segerman’s shapes overnight and tried out a new bubble project this morning.

I started with some simple shapes from our old bubble projects – what happens when you dip a cube frame in bubbles?

The next shape we tried was a tetrahedron frame:

Now we moved on to two of Segerman’s shapes. These shapes are new to the boys and they have not previously seen what bubbles will form when the shapes are dipped in bubble solution.

If you enjoy listening to kids talk about math ideas, their guesses and descriptions of the shape are really fun:

The second shape from Segerman we tried was the two connected circles. We actually got (I think) a different shape than I’d seen in Segerman’s video above which was fun, and the boys were pretty surprised by how many different bubble shapes this wire frame produced:

Definitely a fun project. I tried a bubble project for “Family Math night” with 2nd graders at my younger son’s elementary school last year. Kids definitely love seeing the shapes (and popping the bubbles).

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Dave Richeson’s Knotted bubbles project

Saw this tweet from Dave Richeson last week which basically “had me at hello”:

here’s the video in cast the twitter link doesn’t work:

We’ve done a few bubble projects in the past, so the boys were already familiar with the basic concept:

Zometool and Minimal Surfaces

Trying out 4 dimensional bubbles

More Zome Bubbles

Anyway, I ran out to home depot and got some wire and we made some knots. I had each of the boys make a trefoil knot and then make a random knot of their own choosing. In retrospect I wish I’d spent maybe just 5 minutes explaining some of the ideas in Richeson’s blog post – oh well, the excitement got the better of me 🙂

Here’s my older son playing with his trefoil knot and making a Mobius strip bubble. I love the “hey, I actually think I got it” moment:

Here’s him playing with the knot me made – in retrospect I’d argue for a knot that was slightly less complicated:

Next up was my younger son. First up was the trefoil knot and we got another great moment “I think this might be a Mobius strip” !!

Finally we made his own knot and explored. Again, I’d probably ask for a less complicated knot if I was doing this again:

So, that so much to Dave Richeson for posting his old project – this is an incredible project, and an especially great one for kids. The appearance of the Mobius strip is really quite an amazing little math miracle!

More zome bubbles

I asked my older son to choose a project for today and he wanted to dip more zome shapes in bubbles. We’ve done a few of these project previously:

Zometool and Minimal Surfaces

Trying out 4 dimensional bubbles

My older son went first – his two shapes were a dodecahedron and an icosahedron:

My younger son made two neat shapes including a non-planar loop, or, as he said “a squigly decagon”:

Trying out 4 dimensional bubbles

At the end of the project with my younger son this morning he remembered that we’d see some of the 4 dimensional shapes we were looking at in our Zome Bubble project. He went on to wonder if we dipped our 4 dimensional shapes in the bubble solution would we get a 5 dimensional shape. Well – We had to try that!

First, though, we looked at what happened when you dipped a cube and tetrahedron in bubble solution:

Next we tried the 4 dimensional shapes – what happens when you dip the zome versions of the 5-cell and the Hypercube into the bubble solution?

Ahead of the dipping, my younger son had this thought:

“I think we are going to see a 5 dimensional shape”

Here’s what happened:

I’m really loving just playing around with the 4 dimensional shapes with the boys. Soon we’ll move on to looking at the 4d version of Patrick Honner’s Pi Day project – can’t wait for that!

Finally, here’s the project from this morning that led to my younger son wondering about bubbles:

Sharing 4d shapes with kids

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:

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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

Playing with 3d printed knots from Mathematica

Yesterday I learned that Mathematica has a wide variety of knots that you can 3d print. We’ve done a few knot projects in the past. Here are 3 of them:

Playing with some 3d printed knots

Dave Richeson’s knotted bubbles project

Exploring Colin Adams’s “Why Knot?”

I thought that actually being able to hold the printed versions of so many different knots in your hand was going to be a game changer for knot projects, though. So, I printed a few as test cases and had the boys look at them.

My older son went first:

My younger son went next – he had a couple of things to say, but wanted to point out some of the knots in Colin Adams’s book, so we cut this video a little short so that we could go find the book:

After finding the book we were trying to match one of the printed knots with the knot in the book that he had wanted to print. The knot he wanted to print had 8 crossings and the one that we thought matched it turned out to have 7. Whoops – we had the wrong knot 🙂 A good accidental lesson that comparing two knots isn’t super easy!

I’m really looking forward to trying more projects with these prints. There are a little over 30 different knots with 8 or fewer crossings. It’ll probably take a week to print them all, but that’ll be a fun collection to have for future knot projects!

K-1 Family Night 2017

Here’s my plan for the K-1 Family Math nights for this year:

For the first project I’m going to borrow Lior Patcher’s idea about the 4-color from this incredible blog post:

Unsolved Problems with the Common Core

I have 30 min for the 3 projects with the kindergartners and an hour with the 1st graders, so I think for the younger kids we’ll just do one coloring sheet. The sheets I’m going to use come from an old post from Richard Green discussing a really neat result about tiling octagons. The result is a pretty deep result from geometry, but with the side benefit of hopefully producing images that young kids will enjoy seeing and coloring with 4 colors. I learned about Green’s post from Patrick Honner about 2 years ago:

Our project using Green’s post is here:

Using a Richard Green Google+ post to talk about geometry with my son

Here are the two images octagon tilings I’ll use in the project.

image2-copy

image1

Next we’ll move on to making bubble shapes with our Zometool set. As I write this, at least, I think I’ll dip the shapes in the bubbles myself. I’m worried that letting a room full of younger kids loose on a container full of bubble solution will end up with bubble solution everywhere. Also the zome parts are small and I won’t be able to supervise all of the kids on my own. Anyway, we’ve done a few zome bubble projects and my kids and the neighborhood kids have really enjoyed them. The shapes are really incredible to see and trying to guess what the bubble shapes will look like is a fun challenge for kids. Here are a couple of our old zome bubble projects:

Zometool and Minimal Surfaces

Trying out 4 dimensional bubbles

More Zome Bubbles

Zometool

Finally, with the kindergartners I’m going to do the paper folding project that we did for our original Family Math. I did the same project with the kindergartners last year and it went reasonably well (assuming that you set your expectations on the “me dealing with 30 6 year olds” setting!). I’ve got the first graders in a week, so I’ve got a bit of time to think through a replacement project to avoid duplicating last year’s work. Here’s the project as we did it in 2011:

Our math year in review

I’m sick and not working today but instead have been sort of day dreaming about all of the math the boys and I worked on this year. The sheer amount of absolutely great ideas that people are sharing via blogs, twitter, or otherwise makes it incredibly easy to find fun projects. Here are some of my memories from 2016.

Again, though, I’m sorry that this likely reads so poorly. Despite being so sick it was fun to write this and think back through the year.

(1) Sharing math that I saw from professional mathematicians

Two of my favorites in this category came from Bjorn Poonen, Eugenia Cheng and Laura Taalman:

The project we did after seeing Poonen’s problem about n-dimensional spheres was the most viewed math project for kids on my blog this year:

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Bjorn Poonen’s n-dimensional sphere problem with kids

A neat video from Eugenia Cheng inspired me to revisit an old post from Laura Taalman and do a project on “tiling pentagon cookies”

Screen Shot 2016-07-17 at 9.46.03 AM

Tiling Pentagon Cookies

At the risk of failing to mention lots of people in my current Dayquil-induced state, I’m also incredibly grateful to these professional mathematians who have inspired tons of our projects with the math they’ve shared:

Jim Propp

Steven Strogatz

Evelyn Lamb

Dave Richeson

and one of the neatest things that happened to me all year was when Joel David Hamkins created an amazing “fold and punch” activity based on an activity that I found in some old material from “Family Math Night” at my younger son’s school:

img_0699

Math for nine year olds: fold, punch and cut for symmetry

(2) James Tanton

I have to give him his own category because the work he is doing to share math with kids (and everyone, really) is astonishing.

This project on cutting Mobius strips that I saw in his book “Solve This” is one of the most incredible math projects I’ve ever seen:

cutting

An absolutely mind-blowing project from James Tanton

Here are all of our project inspired by Tanton – you can basically pick one at random and have an amazing math conversation with kids (though if you don’t want to pick at random the candy dividing one is really cool!):

Project inspired by James Tanton

(3) Laura Taalman and Henry Segerman’s work in 3d printing

img_1576

The work that Taalman and Segerman are doing with math and 3d printing is stunning. I mentioned one of projects inspired by Taalman above – there are dozens’s more:

Projects inspired by Laura Taalman

Segerman published a new book about math and 3d printing this year.  I was incredibly lucky to be able to bring the boys to a talk he gave about his work:

segerman

Projects inspired by Henry Segerman

(4) Sharing new and / or popular math ideas with kids.

Erica Klarreich and Natalie Wolchover are doing amazing science (and especially math!) journalism work at Quanta Magazine. Oh to have had writing like theirs around when I was a kid!

I’m so happy to see pieces like Klarreich’s article on Maryna Viazovska’s sphere packing result

In fact, when I asked my older son what his favorite math memory was from 2016 he said it was learning about sphere packing. Yay!

Wolchover’s article about hyperuniform distributions blew me away and led to a really fun project with the boys:

Screen Shot 2016-08-20 at 8.42.03 AM

Using a Natalie Wolchover article to talk about hyperuniform distributions with kids

I don’t think it is possible to overstate the importance of Klarreich’s and Wolchover’s writing. They are going to influence a generation of young mathematicians and physicists.

Another fun math-related item that got a lot of attention this year was Sugihara’s “ambiguous cylinder”:

squircle

We really had fun playing with this shape and I want to give a special thanks to Dave Richeson and Brenda Landis for sharing a 3d print of the shape.

Playing with Sugihara’s “ambiguous cylinder”

(5) The sphere packing problem reminded me of the new PBS Infinite Series work that Kelsey Houston-Edwards is doing.

Holy cow are these videos amazing! Here’s just one example:

Wwe are one behind because of the holidays, but each of Houston-Edwards’s videos has inspired a really fun project. Her videos are great tools to use to share math with kids.

Projects inspired by Kelsey Houston-Edwards

(6) Three projects from twitter that completely blew my mind:

(i) Can you believe that a dodecahedron folds into a cube?

There are actually a couple of projects that Simon Gregg’s tweet inspired. The main picture is this one (which always has weird embedding problems, so sorry it isn’t aligned correctly):

dodecahedron fold

Prepping for this project to make sure that we could do it with our Zometool set was really fun, too:

A neat post from Simon Gregg

(ii) A zipper Mobius strip from Mathsjams

Much like the James Tanton “cutting a Mobius strip” project above, the idea is to try to guess what the shape is going to look like when you unzip it!

(iii) Ann-Marie Ison’s math art

This was a great project with the boys and I also used it for a talk to a high school math camp at Williams. If you play with the Desmos program below your mind will be blow, too 🙂

(7) We played with more math-related art, too:

Paula Beardel-Kreig’s “Puff Boxes” were incredibly fun:

Playing with Paula Beardell-Krieg’s Puff Boxes

And Henry Segerman’s 3D Printing book introduced me to the work of Bathsheba Grossman, and we explored several of her creations:

Our projects with Bathsheba Grossman’s work

(8) Our Zometool work

I know there are lots of ways to spend money on math-related games, books, and toys in general. Building up a good Zometool set is my #1 recommendation. The opportunities to play and learn and study are endless!

Here are all of our Zometool projects:

Our projects with our Zometool set

I particularly recommend the bubble projects and Nesting Platonic Solids

(9) Dan Anderson’s Gosper Curves

When I asked my younger son what his favorite project from this year was he said that it was playing with the Gosper Curves. He really likes fractals!

We got a nice surprise in April when Dan Anderson sent us some laser-cut versions of the Gosper island shape:

Playing with Dan Anderson’s Gosper Curves

We did a few more Gosper-related project (including a Zome one) which are here:

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Our Gosper projects

And, of course, we did a million projects inspired by ideas that Dan shares on twitter:

Our Projects Inspired by Dan Anderson

(10) Patrick Honner’s Pi Day exercise

On March 14th Patrick Honner shared a fun little “Pi Day” exercise:

This terrific project inspired me to try it out in 4 dimensions. That led to a fun multi-day project with my older son as we search for which 4-dimensional platonic solid was the most spherical (according to Honner’s definition).

This project combined ideas from geometry, Zometool, and 3D printing.

Screen Shot 2016-03-14 at 6.48.08 PM

Here’s a collection of the projects:

Patrick Honner’s Pi Day Exercise in 4 dimensions

Honner’s “pi day” exercise is a perfect example of why I love all of the sharing of math ideas that people are doing these days. Not in a million years would I have come up with an idea like that – luckily he did, though, and it turned out to be a really fun way to explore more than just 3d objects!

It really was a great year in math for us. Can’t wait to see what 2017 brings.

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

Last night we printed a shape from Henry Segerman’s new 3d printing book Visualizing Mathematics with 3D Printing:. We’ve done many project based on Segerman’s work and even were lucky enough to be able to attend his talk at MIT earlier this fall:

segerman

The shape we printed last night is Henry’s 3d representation of the 5-cell – a 4 dimensional “platonic solid” ( You can read more about the shape here: The Wikipedia page for the 5-cell)

If you search “Segerman” in the blog you’ll find more than 10 projects we’ve done based on his work!

I started off the project today by asking my younger son for some thoughts on the 5-cell:

One interesting thing that he remembered is that he’d seen the shape previously in some of our bubble projects, so we brought out the bubble solution to make the shape out of bubbles. It was really interesting to hear how he viewed the two shapes differently.

Sorry for the absolutely awful camera work in this video – you’d think I’d have gotten the hang of this after 4,000 videos . . . . .

Taking kids through John Baez’s post about the Gyroid

[sorry for no editing on this one – had some computer problems that ate up way too much time. I finished typing with 2 minutes to spare before rushing out the door.]

I saw this neat tweet from John Baez earlier in the week:

You should be able to click through to Baez’s blog post from the tweet, but just in case that isn’t working, here’s the link:

The Butterfly, the Gyroid and the Neutrino by John Baez

I spent the rest of the week sort of day dreaming about how to share some of the ideas in the post with kids. Last night the day dreaming ended and I printed a gyroid that I found on Thingiverse:

The specific gyroid that I printed is here:

Alan Schoen’s Gyroid on Thingiverse by jamesosaurus

This project connects with several of our prior projects on 3d printing (particularly the recent ones inspired by Henry Segerman’s new book) as well as projects on minimal surfaces. Though the list below is hardly complete, here are a few of those projects:

Zometool and Minimal Surfaces

Trying out 4 dimensional bubbles

More Zome Bubbles

Playing with shadows Inspired by Henry Segerman

Playing with more of Henry Segerman’s 3d Prints

Henry Segerman’s flat torus

Using Hypernom to get kids talking about math

So, with that introduction – here’s what we did today.

First we revisited the zome bubbles to remind the kids about minimial surfaces – it is always fun to hear kids describe these complicated shapes:

Next we looked at the Gyroid that I printed last night. This shape is much more complicated than the zome bubbles and the kids sort of had a hard time finding the words to describe it – but we had a similar shape (and I don’t remember why or where it came from) that helped the kids get their bearings:

So, after playing with the blue shape for a bit and seeing some of the symmetry that this shape had (yay!) we returned to the Gyroid. The boys still struggled to see the symmetry in the gyroid (which is really hard to see!) but we made some progress in seeing that not all of the holes were the same:

Finally, we turned to Baez’s article to see the incredibly surprising connection with butterflies and physics. There’s also a fun connection with some of the work we’ve done with Bathsheba Grossman’s work and Henry Segerman’s 3D printing book:

So, a fun project. I love how 3d printing helps open up advanced ideas in math to kids. After we finished the boys kept reading Baez’s article to find the connection with neutrinos – it is really gratifying to see how engaged they were by today’s project!