Tag 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 ๐Ÿ™‚ 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!

Exploring some fun 3d transformations

Today’s project with the boys was exploring some simple (to code!) transformations. The question was how would the shapes change under those transformations.

I started with introducing the idea in 2d. It isn’t necessarily the simplest idea, and I had no intention to go into any details. The basic question I wanted them to think about was this – would a straight line stay straight under this transformation?

Next we looked at a tetrahedron (actually two tetrahedrons) under some similar 3d transformations:

Now for the punch line – what do the same transformations do to an octahedron?

Finally, I wasn’t planning on doing this part, but to clarify some of the ideas from the first part of the project we went up to the computer to show them what the transformations did to a line in 2 dimensions:

So, I think this is a fun way for kids to explore some 3d shapes and also begin to understand a little bit about how algebra and geometry are related

Revisiting James Tanton’s Tetrahedron problem

A little over 2.5 years ago I saw this very neat question from James Tanton:

The question led to a really fun – and also one of our first – 3d printing projects:

James Tanton’s geometry problem and 3d printing

We’ve now got a few more years of 3d printing under our belts and a new program we are using – F3 by Reza Ali – is opening completely new 3d printing ideas for us.

Somewhat incredibly, F3 has a one line command that draws all of the points that are a fixed distance away from a cube. Here’s that beautiful shape:

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F3 Box.jpg

Seeing that command inspired me to revisit James Tanton’s old question. I wasn’t quite able to do it in one line (ha ha – my programming skills are measured in micro-Reza Alis . . . .), but I was still able to make the shape. Here’s how it looked on the screen:

After the boys got home from school we revisited the old project together and used both the old and the new 3d prints to help us describe the shape (sorry for the noise in the background – that’s a humidifier I forgot to turn off):

Maybe because it is one of our first projects ever(!), but I love this problem as an example of how 3d printing gives younger kids access to more complex problems.

Using 3d printing in the college classroom

I saw a couple of tweets from Steven Strogatz yesterday that got me thinking about how you might use 3d printing in the college classroom:

The last tweet, in particular, made me think that having the 3d print versions of the two shapes would be useful. Before I get too far in to this post, though, I had to throw this post together pretty quickly to be able to fit in an hour of shovelling prior to heading to work! Sorry if it isn’t the most well-written or well-argued post. The main takeaway I want is that I think there are many great uses for 3d printing in the college math classroom.

The topic of 3d printing and calculus is one that I’ve thought about briefly before – see these old posts:

3d Printing and Calculus Concepts for kids

Using 3d printing to explore some basic ideas from calculus

Here are the shapes from the first post linked above – I think they would help students understand ideas like Riemann sums and volume by slicing:

Here are the two 3d shapes from the second Strogatz tweet from yesterday. Unfortunately we lost power in the middle of the night before the print project was complete, but you’ll get the idea. One of the things that comes through immediately in the prints is the difference in size of the two shapes:

Finally, an important shape from advanced algebra – a cube inside of a dodecahedron. This shape appears (and plays an important role) in Mike Artin’s Algebra book:

Dodecahedron

I found it hard as a student to understand the shape solely from the picture. Holding the shape in my hand, though, makes it much easier to see what is going on (I have made the cube slightly larger to highlight it):

So, while I’m sure it is true that learning to draw some of these shapes by hand is useful, I also think that 3d printing can be an important tool to help students see, understand, and experience the same shapes.

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.

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!

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.