Tag 3d geometry

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:

 

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.

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 🙂

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 🙂

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.

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.

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!

Paula Beardell Krieg’s intersecting squares

Saw a fun sequence of tweets from Paula Berdell Krieg last week:

Then . . . we received an envelope!

Next we compared the shape to the 3d prints it was based on:

After this I showed the boys the tweet above that shows how to unfold the shape into a cube and they were able to recreate the procedure:

Definitely a fun shape to explore – thanks to Paula for sending it to us! It really is amazing how much geometry you can explore just by folding paper 🙂

Our Facets have arrived

About two years ago we supported the Kickstarter campaign for Facets. The program had quite a few road blocks thrown in its way, but Ron Worley stuck with it and has nearly completed the project. Good for him.

Our package arrived last night and we did a quick project. Here’s the “unboxing”:

After opening the box I had each of the kids build a shape. Here’s what they had to say – younger son first:

Older son next:

I’m excited to do some more projects with the Facets – it was really cool to see how quickly the boys could create interesting shapes from 3d geometry. The Facets look like they’ll be a really fun tool to use to explore geometry with the boys.

Exploring 3 intersecting cylinders with 3d printing

Calculating the volume of 3 intersecting cylinders is a classic calculus problem.  The 3 cylinder problem caught my attention a few years ago when Patrick Honner shared this video about the 3d printing lab at his high school:

I wrote about my reaction to the video here:

Learning from 3D Printing

Today we used our 3d printer and the F3 program to explore the intersection of three cylinders. Here’s what they boys had to say when they saw the setup on the F3 program – my older son went first:

Here’s what my younger son had to say:

After the shape finished printing I had the boys talk about their thoughts when they had the shape in front of them. Here’s my older son’s thoughts:

And next my younger son:

Even though this is probably a better Calculus example, I loved being able to share the shape with the boys. It is fun to hear kids talk and wonder about fun shapes like this one.

Using 3d printing to talk symmetry with kids

We’ve done a lot of projects relating to platonic solids and dodecahedrons in particular. A really neat fact about dodecahedrons is that you can use the verticies to put 5 cubes inside!

It isn’t just a mathematical “fun fact” either – the symmetry groups involved play roles in important mathematical theorems.

For today’s project I wanted to explore one cube in a dodecahedron and look at the relationship between the rotations of the cube and the rotations of the dodecahedron.

We started by looking at the dodecahedron by itself:

Next we moved to looking at the cube in the dodecahedron and studied what rotating the dodecahedron did to the cube:

Finally we looked at some 3d printed models that we made to see if these models helped us explore the rotations a bit more:

I was a little disappointed that I made the 3d printed models a bit too small, but I still like how this project went. I’m going to try again with some slightly larger models with my older son.

Using 3d printing to explore some basic ideas from calculus.

A few days ago we did a short project with some shapes that the boys and I had made using the F3 program:

Using 3d printing to talk math with kids

My older son made a “twisted octahedron” based on one of the examples that comes with the program. When we talked about the shape he wondered how you would calculate the volume of the shape. Today I made some slicing models that you might see in a calculus class to help him see the answer to that question.

I’m still 3 steps below a novice at using F3 but am learning a bit more every day. The program I wrote to make the slicing models is pretty easy (and pretty short) and didn’t take that long to figure out. The code is below – I show it not because it will make sense to everyone, but rather to show how easy it is to make really cool shapes with the F3 program. The code is a very slight modification of F3’s twisted tetrahedron example that caught my son’s eye the other day, and running this code allowed me to make the three sliced examples in the second video below.

slicing-program

So, for our project tonight we first revisited the twisted octahedron and talked about the volume:

Next we looked at the sliced models I made. Unfortunately I made the models way too small. By luck they showed up ok on camera (though sorry for going off screen a few times) and the boys were able to see how the smaller slices converged to the shape we had originally:

I’m not teaching the boys calculus – don’t worry! It still is really neat to be able to show them some of the ideas from calculus using 3d printed models. Can’t wait to play with more shapes this week!

Sharing Kelsey Houston-Edwards’s Pi video with kids

We really enjoyed watching Kelsey Houston-Edwards’s latest PBS Infinite Series video last night:

We’ve played around with a few ideas about altering Pi a few times previously. One was with Vi Hart’s “Pi = 4” video in this project:

A fun fractal project – exploring the Gosper curve

A second time was a super fun series of projects on 4-dimensional shapes inspired by a tweet from Patrick Honner. That complete series is here:

https://mikesmathpage.wordpress.com/2016/03/24/patrick-honners-pi-day-exercise-in-4d-part-5-the-120-and-600-cells/

Patrick Honner’s Pi day exercise in 4d part 5: The 120 and 600 cells

Today we explored the 2-dimensional idea in Kelsey Houston-Edwards’s video in 3-dimensions using 3d printing.

First, though, I asked the boys what they thought about the latest PBS Infinite series video:

Next we took a look at 6 different “spheres” in 3-dimensions which were defined using the different way of measuring distance Kelsey Houston-Edwards introduced in her video. It is really fun to hear kids talking about these shapes, and even more fun to be able to actually hold these shapes in your hand!

The last thing we did was look at the 6 spheres all together. My younger son noticed how much bigger the shapes got as you moved from the L-0.75 norm to the L-4 norm. My older son noticed that some of the shadows of the shapes looked pretty similar even though the shapes didn’t look the same at all:

This was a really fun project to prepare. It is really fun to show kids ideas from advanced math that the wouldn’t likely see in school. It is also really fun to hold these strange shapes in your hand! Lately we’ve been using the F3 program to help us make objects to print – it was a really lucky coincidence to see the new PBS Infinite Series video *after* I learned to use the new program. It took less than 5 minutes to make the .stl files for the 6 shapes 🙂