Introduction to vectors via 3d printing

We’ve been using the F3 program to print various different mathematical 3d prints. As my older son explores the program more deeply, it is becoming a really fun way to teach him about vectors.

Last night I had him describe his first 3d print and how he made it. The concepts remain new to him and he has not understood all of the concepts correctly, but I hope talking through the ideas will help him learn:

After the discussion about the shape we went upstairs to look more carefully at the code this gave him an opportunity to talk a bit more about vectors. It also gave me an opportunity to see a bit more about what he understood:

I’m excited to use 3d printing and the F3 program to help him learn about vectors. I am an absolute beginner when it comes to the “signed vector field” idea that the F3 program uses (not to mention that I’m also an absolute beginner when it comes to programming), but I’m really excited about the learning opportunities here.

Taking about Kate Nowak’s shape

Saw this neat drawing from Kate Nowak the other night:

I was interested to see if we could make the shape from our Zometool set, and . . . .

The boys really enjoyed making the shape last night and both also made several comments about how interesting it was. This morning we talked about it a bit. Both kids focused on symmetry. I spent a bit more time with my older son exploring the different kinds of symmetry, but it was great to hear what both kids had to say. It really is an amazing shape!

Younger son first:

Older son next:

This was a really fun project. The shape didn’t take that long to build, which was lucky. It is always fun to be able to pull out the Zome set to explore something that we saw on Twitter ๐Ÿ™‚

Learning math by studying 3d printing

My son spent the last couple of months preparing for the AMC 10. Now that the test is behind him I’m going to spend some time with him studying 3d printing.

Today we looked at some simple code in the F3 program:

sphere.jpg

The details of the code don’t matter that munch – all the code is doing is testing whether or not a point is inside of a sphere by checking whether or not the distance from that point to the center is greater than or less than the radius.

Immediately two ideas come to mind:

(i) how do we compute distance in 3 dimensions?

(ii) is that distance measure unique.

So, after 1 minute of looking at code we went to the whiteboard ๐Ÿ™‚

Our previous 3d prints of the sphere and torus in different L^p metrics were still on the table, so I used those as props.

The first topic was distance in two dimensions:

The second topic was distance in three dimensions:

The last topic was how the L^p metrics vary as p varies – it was lucky we had the spheres handy ๐Ÿ™‚

Today’s conversation was actually a nice surprise – I think there’s going to be quite a lot of fun math review that comes from studying 3d printing more carefully.

Trying to understand the DeMarco and Lindsey 3d folded fractals

Quanta magazine’s article on 3d folded fractals from last month has really captured my imagination:

3-D Fractals Offer Clues to Complex Systems

Since reading the article I’ve been trying to understand the paper by Laura DeMarco and Kathryn Lindsey that inspired the story:

Convex shapes and harmonic caps on arXiv.org

Although I’m making progress digesting the paper, that progress is slow – who knew that trying to understand current research in a field you know nothing about would be so hard . . . ๐Ÿ™‚

One really nice thing in the paper that helped me get my bearings was figure 1.1:

screen-shot-2017-02-05-at-1-34-15-pm

This figure shows the curved “cap” which combines with a square to make a 3d shape. I tried to imagine what the shape formed by gluing the square and the curved shape would look like, but quickly reached the limits of my imagination.
Luckily, though, my wife was willing to help me sew a version.

It took two tries but eventually this shape emerged!

It is much flatter in reality than it was in my mind so seeing an actual version of the shape turned out to be really helpful.

I’m not sure what the next steps are for me. Either I have to get a better understanding of the Riemann mapping theorem (and I’ve already dug out my old complex analysis book for that) or maybe just play with some approximations and make some 3d prints like this one from Yoshiaki Araki that was part of a contest that Quanta Magazine had in their article:

The work trying to get a better understanding of these 3d shapes has been really fun. I’ll be really happy if I’m able to understand one or two more things from the DeMarco and Lindsey paper. It would be amazing to be able to make some (even very simple) shapes to show kids some new ideas from current math research.

Sharing advanced ideas in math with kids via 3d printing

Yesterday (after a few false starts!) I printed several different versions of the torus in different L^p metrics. Here they are next to spheres in the corresponding metric

The idea was inspired by an old project that was inspired by a Kelsey Houston-Edwards video

Sharing Kelsey Houston-Edwards’s Pi video with kids

Prior to the prints finishing I talked through some of the shapes as they appeared on the computer with my younger son:

Exploring different L^p versions of the torus

When the various torus prints were done I asked each of the boys to tell me what they thought about the shapes. I love how 3d printing allows you to share advanced ideas about math with kids so easily!

Here’s what my younger son had to say:

Here’s what my younger son had to say:

These are the kinds of math conversations that I’d like to have with kids.

Exploring different L^p versions of the torus

A few weeks ago we did a fun project on L^p spheres after watching Kelsey Houston-Edwards’s video on different ways of measuring distance:

sphere-shapes

Sharing Kelsey Houston-Edwards’s Pi video with kids

Playing around a little with our 3d printing software last night made me want to try a similar project with a torus in various L^p metrics. I made 5 different ones and set the printer to print them overnight . . . and the print failed. Boo ๐Ÿ˜ฆ

So, I’m re-printing them to use for a project this afternoon, but for now the project with my younger son just used the shapes on the computer.

Here’s what he thought about the usual torus and the torus in L^1

Next we moved on to looking at the torus in the L^3 and L^5 metrics:

Finally, we looked at some of the shapes when p was not an integer. We looked at p = 0.75, 1.5, and 1.05.

Using the computer program was a nice way to save the project after the print failed. I’m really hoping that the 2nd time is a charm with the print and we can explore the 3d printed shapes this afternoon!

4th and 5th grade “notice and wonder” Family Math night

I’m kicking around the idea of doing a “notice and wonder” project for 4th and 5th grade Family Math night next week. I’m expecting about 20 kids and families and I think I have enough prints to make it work, although this type of hour long activity is something that I’ve not tried before.

The idea would be for the kids and families to circle through 6 stations writing down what they noticed and wondered about the various shapes. I’d let that go for 30 min and they we could talk in a group about the shapes.

The 6 stations would be:

(1) Two shapes made with intersecting cylinders plus the shape from Quanta Magazine’s contest based on Laura DeMarco’s work.ย  The 3d print for the shape was designed by the contest winner, Yoshiaki Araki:

(2) Various forms of the trefoil knot made by Laura Taalman and by Henry Segerman:

(3) Penrose tiles and Gosper Islands

The tiles were designed by Laura Taalman and the Gosper Islands were designed my Dan Anderson:

(4) Incredible 3d shapes –

The Sierpinski pyramid was designed by Laura Taalman (and was our first ever 3d print!) as was the space filling curve.ย  The Gyroid and the evolving space filling curve were designed by Henry Segerman (I think – can’t remember for sure where the gyroid came from).

(5) 4 dimensional cubes:

The “balloon” version was designed by Henry Segerman and the red version is “Hypercube B” by Bathsheba Grossman.

I will also include the version of Hypercube B that we made from our Zometool set.ย  Even though it is not an exact replica of Grossman’s shape it is still pretty cool:

(6) Squircles and an really fun shadow-casting shape

The sphere was designed by Henry Segerman and is on the front cover of his book:

img_1677

The thin squircle came to us via Dave Richeson and Brenda Landis, and I’m not sure of the origin of the other one:

The Plastic Number

Last night I got an idea from Ian Stewart’s amazing book:

The idea was about a sequence of integers related to the “plastic” number.

I started the project by building a very simple shape out of our Zometool set and asking the kids what they thought about it. One lucky surprise was that my older son guessed the recurrence relation for the sequence we’d be studying!

Just listening to my introduction now I totally butchered the definition of the plastic number – sorry about that . . . .

Next we built several more triangles and studied the sequence more carefully. My older son confirmed the recurrence relation he saw previously and my younger son found a different one!

Also – sorry about the lighting.

Finally – we checked out the Wikipedia page for the Plastic Number and then explored the equations relating to the two recurrence relations that we found previously.

Sorry about the bumbling around in Mathematica – I needed the screw up trifecta ๐Ÿ™‚

This was a fun projects. I think it would be a neat one for kids learning learning about factoring, too, so they could see how the two equations we studied in the last video relate to each other.

Using 3d printing to share 4-dimensional spheres with kids

A few weeks back we did a project on 4-dimensional spheres intersecting a different sorts of 3d worlds:

What if Flatland wasn’t a plane!

Last night I got around to 3d printing some of the shapes from that project:

Today we talked through the idea of how objects from higher dimensions “look” as they pass through lower dimensional shapes. We started by talking about the idea from Flatland – a 3d sphere passing through a 2d plane. After that we moved on to talking about what the intersections would look like if the sphere was passing through a plane that was creased in to a “V” shape:

Next we moved on to talking about a 4d sphere intersecting the same sorts of objects – a flat 3d space and a “V” shaped one. To create the “V” shape, I just assumed that the 4th dimension – call it w – had a value equal to the absolute value of the x-coordinate.

Next we looked at the 3d printed shapes I made last night. These shapes show a few different stages of a 4-d sphere passing through the “V” shaped 3 dimensional space:

Finally, rather than looking at 4d sphere passing through a “V” shaped 3d space, we went and looked at the shapes made when a 4d sphere passes through a 3d space that is bent like a parabola. So, using my language from above, the 4th coordinate in the space, w, is set equal to x^2.

The shapes here are really cool and also pretty surprising.

Our second facets project

We did this about a week ago and I never got around to publishing it. Both kids are sick today so it seemed like a good day to revisit the old movies.

Our first projects with our Facet set is here:

Our Facets have arrived!

The kids have enjoyed making little creations with the Facets ever since.

Here’s how their 2nd projects looked – my younger son went first. It is so fun to hear a week later how many different topics we covered in the talk about the ring he made:

Here’s my older son’s work. He made an interesting shape that we tried to extend with the camera off. Unfortunately that shape was too heavy for the magnets and collapsed. We got a little lucky, though, and some interesting shapes survived the collapse and we turned them into new, fun shapes:

I’m really happy with the Facets and can’t wait to do more projects with them!