A Better Way To Teach Geometry Using 3D Models

3D Printing & Imaging Education
Make: Geometry
Joan Horvath and Rich Cameron

So many students struggle to learn math. For many, math is so abstract. It doesn’t connect to anything in their physical world. For some, it’s the textbooks and their illustrations that just don’t explain concepts in a way that students can grasp them. Good teachers struggle to help students overcome these problems.

I first heard of a teacher using 3D printing to teach math several years ago at a Maker Faire. He talked about how some students who look at a 2D drawing on a page have trouble visualizing it in their heads as a 3D object. Holding a 3D printed object in their hands made a big difference. It made the math real.

What if there are new and better ways to teach math using the tools found in a makerspace and engage students in learning subjects like geometry. The authors of our new book, Make: Geometry, are our guests on this episode of Make:cast and they developed a fully new way to teach geometry using 3D design tools and 3D printers.

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Patrick: I’m your host, Patrick Di Justo, the book editor at Make Community. Today, we’ll be talking about the book, Make: Geometry, available at the Maker Shed or on Amazon. The official publication date is August 10th, 2021. The authors are Joan Horvath and Rich Cameron, co-founders of the consulting firm Nonscriptum, based in Pasadena, California. Welcome to the podcast.

Joan: Hello.

Rich: Hi there.

Patrick: Hi! Now, the central thesis of this book Make: Geometry is that it is possible to attain an understanding of the mathematical subject geometry, not just by learning theorems and formulas (theora? Formulae?), but by actually manipulating 3D geometric shapes by hand.

It’s an exciting way to teach. And we want to talk more about the pedagogy of that, obviously, but before we get more deeply into the book, let’s let the listeners know a little bit more about the authors themselves. Where are you from? What’s your life been like to get you to this spot right now? So Joan, let’s give the audience a quick recap where you’re from, what you’ve done, how you got here, that sort of thing.

Joan: All right. So thanks for having us on here. So I like to say I’m a recovering rocket scientist. I’m originally from New York, New York city then went up to, to MIT to go to school. And then out here to California, to UCLA, worked at JPL as a rocket scientist for awhile, and then went out on my own on various entrepreneurial things.

Most recently, 3Dprinting. So Rich and I were at a 3D printing company together. Moved on to form this training company, Nonscriptum, about almost seven years ago.

Patrick: Can I ask what JPL missions you’ve worked on?

Joan: Oh, is there for a long time? So I did a lot of research stuff. I worked on a supercomputer that would be laughable now called a Hypercube, in the eighties.

Patrick: I remember that!

Joan: Yeah. And worked on Topex Poseidon, which has its great grandchild missions. JASON and a couple of others, ocean topography, Magellan to Venus, Cassini, Saturn, and, worked on very, very early Europa’s surface mission to barrel through the ice of Europa. This was a actually landing on the ice and burrowing through the ice and releasing a little submersible and all kinds of stuff.

So we spent a lot of time learning how people did robot submersibles and, you know, all that stuff always comes full circle. It’s been interesting watching people do some of that exploration now.

Patrick: And Rich, how about you? What is your background? How did you get to this?

Rich: So I’m an old RepRap guy. I was building robots for fun, needed something to make robot parts and got into 3D printing back in the days when you really had to build your own machine.

Patrick: Oh yeah. Oh yeah.

Rich: Yeah. And, you know, first machine I got, actually I was the first person. Probably a first, first or second person to actually build one of these and get it running.

And it was a new design that, that had some problems. There, there were some parts passing through each other, so I needed to make some modifications in the process. So once I finally got the machine working, I spent all my time, you know, working on improving these designs and open had the 3D printer bug ever since.

Patrick: Okay. And both of you formed Nonscriptum and you’ve written several books together. And now, the one we’re here to talk about now is Make: Geometry, a book about teaching geometry, using the actual 3D geometric shapes, like cubes and spheres, but also parabola and right triangles and mutable right triangles. And you told me that this method had a very interesting origin.

How did you, you get involved in, you know, the, the physical hands-on teaching of, geometry using 3D geometric shapes.

Joan: Well, I think it had two threads that kind of came together. You know, we, when we started the company, we did a lot of consulting for schools that wanted to learn how to use 3D printers. And it became clear that they needed some content to use with their 3D printers.

And so we have some other books about science projects out there already. But people also really wanted to have an end-to-end subject. And about, at the same time we had for quite a while, been involved with the blind community or visually impaired community. As you might imagine, 3D printing for learning mass can be a real game changer in that community.

Those came together and we were fortunate to receive, receive a grant. And so, we were able to, to start working on geometry and then after that we realized we could also do the book. And so that both sides benefit from that.

Patrick: [Now you’ve got a book. So who are you hoping to reach with Make: Geometry and this method of teaching geometry?

Joan: There’s a, there’s a concept called universal design, which is that if you make design accessible for folks such as the visually impaired, or perhaps other people who learn differently, that it will be better for everyone. Also, I think, anybody who learns better by making something, which I think is almost everywhere.

I have a degree from MIT, but I found that I learned a tremendous amount messing around with the models with the book as we were working on it. And, I know, Rich learns, learns a little bit differently than I do. So maybe he can talk about that too.

Rich: I’ve always kinda taught myself my own way to do things in school. You know, some teachers got along with it better than others. And a lot of it is, you know, spatial reasoning, and you know, making things and stuff. And so we tried to capture a lot of that in this.

Patrick: Sure. Now, just to make it clear, do you regard this as a textbook?

Joan: No, it’s not a textbook. It tries to address the things that, where it makes sense to build a 3D model. There are a lot of things where you can learn more by having a 3D model, but there are some things that are perfectly straightforward without it. We do have some classical constructions in this, which you would use a compass. So you don’t have to use a 3D printer for absolutely everything in the book.

But we tried to focus on the areas where we found that felt that that made a bigger difference.

Rich: And there are a lot of things where you could make a 3D model of something, but it really doesn’t add anything. A lot of attempts to create 3D models to teach subjects have been creating a 3D printed table of elements, periodic table.

So what good is that doing? I mean, what? For a blind person you could use that, maybe, but a Braille one’s probably going to be better, right?

Patrick: Exactly. You, you find you have the tool and suddenly, people think that, oh, we can solve so many problems with the printer and not necessarily every problem can be solved this way.

Rich: Yeah. So we’re trying to keep it to things where it actually makes sense to do this and where it adds something.

Patrick: I distinctly remember back when I did take geometry, I didn’t like creating things with compass and a straight edge. But now as far as — for whatever reason — I found that, the most exciting part of the book that you can just sit there with basically what amounts to sticks and a pencil and come up with all these relationships between things in the world.

Joan: I think one thing that’s I think one thing that’s interesting is that as you start to play with this after you haven’t played with it for a very long time, you see a lot of connections. And so we’ve tried to capture some of that as we talk about it. I think the other thing that has been interesting and I want to acknowledge our collaborators.

Thanks. Sue and Chancey fleet here. Among the visually impaired grant that people who are purely, when you think about making a model for somebody who is purely going to experience it by touch, you think about it a little differently. And you think about how the model can be narrative. And has to have a start and an end in a way to walk through it.

And I think that that’s a very powerful thing that I think we’ve learned from working with that community and the grant from the US Administration for Community Living, in Health and Human Services, they really have that powerful message that if you have a model that is narrative stamp, freestanding by itself, it really teaches in a different way.

Rich: Yeah, there’s some really interesting stuff you can do with the compass and straight edge. It didn’t seem all that interesting to me when I was in school either, but since then, you know, I’ve discovered that you can do a lot more with it. There’s this awesome, game called Euclidea that I just love. You start with a compass and a straight edge, and you have to create these different things using only those tools. And you can use those tools to create better tools that you can use in later stages of the game. But it’s all based on those two things.

Patrick: Yes. I’ve played that, that can easily be addicting.

Okay. So I, I just, I also want to make one thing. It’s not vital or absolutely mandatory to have a 3D printer right there in the room. Is it?

Rich: Yeah. You need it. Okay, go ahead.

Joan: Okay. Sorry, you don’t need one. Certainly you can do more with it, but we have tried where it made sense to also say here’s how you could use craft materials or paper or do something alternative.

And in some cases we include a PDF. So you can print something on paper. If you want to do it that way.

Rich: So we have a number of models that, 2D printable alternatives. For example, we have a chapter on nets, which are basically you unfold the, all the sides of a three-dimensional shape to create a 2D pattern.

And that’s often done with paper. We have some, some unique ways to make 3D printing useful for that to work even better. But, all the code that we wrote for that can also generate the 2D.

Patrick: Yes. I found that absolutely fascinating. To be able to go to your 3D printer, that’s sitting there at your desk or to go to the 3D printer that’s in the school library or the public library, or if there’s no 3D printer that you have easy access to, you can just sit there with cardboard and, and make these shapes yourself.

Rich: Or laser cutters are very fast and a great tool for some of these jobs when things can be made, 2D too.

Patrick: To get into the technical aspects of this book. And by the way, we’ll just keep saying the title of the book over and over again. It’s Make: Geometry. You provide the code to make the 3D shapes, but you also encourage experimentation with the code to make these 3D shapes and let’s get very technical.

What design language did you use to make these 3D shapes?

Rich: So everything is written in OpenSCAD, which is an open source CAD program that uses a sort of a C-like programming language to describe objects, so it has primitives like cube and sphere and circle and square. And you build up shapes, by adding and subtracting these things from one another, because it’s a programming link.

You can do all kinds of math in it. It’s got, you know, it’s got the math libraries built in and stuff. So there are, I believe we have some models in this one where I calculate a series of points in space and build up a polyhedron directly from it. But a lot of it is simply adding and subtracting cubes and spheres and triangles and whatnot.

So all the code is open source and it’s written in a parametric way. So there are variables at the top of each file. And just by modifying those variables, you can get lots of different output. You don’t need to necessarily understand all of what the code is doing. You can just edit that top section and get a lot of different variations on each one.

Patrick: The variables at the top of the program will say something like number of sides. And you can change that from four to make a square, to five to make a pentagon, to six to make a hexagon, and so on and so on and so on. So you really don’t need to know how to code the actual shapes. You have handled all of that. And the reader can just change a few parameters and get totally different outcome.

Joan: Yes, exactly. And another point to make here is that if you don’t have a 3D printer, you can still simulate in open SCAD, get a lot of what you can get out of the book by playing around open SCAD, which is free and open.

Patrick: Right. You can just download it. Don’t have to pay anything. Install it on your computer, run these programs. And if you cannot print out the shapes, you’ll still see them on your screen. You’ll still be able to manipulate them, change them just as we had said.

Okay. The basic shape of geometry is the triangle, and that’s the focus of Chapter Five of this book, the Triangle Bestiary, which, I love that title! Can you give us a brief rundown of what you do with triangles in Chapter Five and how that leads us into the rest of the book?

Joan: So, as you say, the triangles, as simple as shape that you can make, are just putting together three sides, but there is a whole list of categories of triangles.

And I’m not going to recite them here because I’m sure your readers will know them. But you know, one of the interesting things that gets a little meta here: Any 3D print surface is broken into triangles. And so you can ultimately make just about a shape or approximate, I should say, with, with triangle.

So we have some fun models, some of which go back to Pythagoras and other ancient Greeks. We quote ancient Greeks a lot, because we try to say, well, you know, suppose we were inventing this, how did they invent it? And try to get into the mindset of: when you’re learning it, you’re inventing it for yourself.

And so try to get inside their heads a little bit and say, you know, these people figured all this stuff out, from nothing and, and they did it for a reason. We, later in the book, let you use some very simple measurements, basically just taking the angle of the sun, either with the 3D print or with some string or with a protractor and whatever you’ve got lying around.

And you can actually figure out your latitude pretty easily by just finding out when the sun is highest and, and looking up a few things, and you can figure out your longitude with that plus a clock. And so it’s fun to think about that. You can get pretty close where you are. Which was making a couple of measurements outside around noontime.

Rich: And, one of my favorite models in the triangle section is we have a little model of a right triangle where the ends of the hypotenuse can slide in a little tray. And we have gradations along the sides where they meet. And so if you consider the length of the hypotenuse has to be one in some units, the number that you read off of the sides, one is the sine? One is the co-sine of the angle. That’s the hypotenuse. And of course the ratio of those two is the tangent. So it really makes this the concepts of these trigonomic these basic trigonometry functions, very concrete and yes, that’s something.

Patrick: The book is Make: Geometry, but there’s quite a bit of trigonometry that comes in under the radar, which is great.

I guess I’d say the middle third of Make: Geometry gets pretty heavily into three-dimensional geometry and especially the conic sections, which are basically very interesting curves you get, if you take a cone, like a traffic cone or an ice cream cone or whatever, and slice it at various angles. The resulting shapes you get are pretty important in geometry. Slice the cone parallel to its base and you get a circle, but slice it at different angles, increasingly oblique angles, and you wind up with some very interesting three-dimensional shapes. Now slicing three dimensional shapes is exactly what a 3D printer does. How difficult was it to design these sliced shapes for 3D printing​?

Rich: So the 3D printing does slice the shapes, but not in exactly the way we’re doing here, but it’s very easy with the primitives that OpenSCAD provides. You can make a cone and then you can take a large box to basically cut off the top of it and get one of these shapes.

And you can rotate that box to cut it at a different angle and get a, as you said, a circle, if you’re going through it level, yeah. But you get an ellipse if you cut it at an angle, and an increasingly eccentric ellipse, a very wide thin ellipse as that angle increases until the ellipse essentially becomes infinitely long, at which, at which point the equation becomes that of a parabola.

And, beyond that point, it becomes a hyperbola. So you can get all these different curves, just by cutting your cone, which is interesting.

Patrick: A little bit of knowledge of certain ellipses: You can find where you are on the planet earth when your phone is dead. When the GPS satellites don’t work, when all you have is a watch and being able to look at the sun with this knowledge, you, you cannot get lost.

The book ends with something that you call the geometry museum. What can you tell us about that? Because I think this, this last chapter is just fascinating.

Joan: To give a lot of applications. So as you say, you know, a lot of this stuff is very, very practical and we wanted to show some interesting applications. Because if you think about it, when people were making things, making cathedrals in the middle ages, they didn’t have much in the way of instrumentation. They had rope, they could make circles, they could make right angles and that’s kinda it.

So one of the things we do early on is we play around with making arches. And one of the objects on the cover that I particularly like is a Gothic window.  And a Gothic window is really pretty easy to construct in the mass construction sense. And so we have a little program that generates that, and we also looked at the difference between circular arches and Gothic arches, and Gothic arches are the ones that are kind of pointing on the top.

Patrick: We should explain the Gothic arches, they’re tall and skinny. They come to a point, they’re clearly not a hemisphere, but they sort of are related to hemisphere.

Joan: Yeah. So take, take a look at Notre Dame, for instance, for something much in the news. And then a lot of those are Gothic windows and Gothic arches, and it turns out that that’s a tremendously strong form. And, so we, we talk about arches and, and why various of these structures are strong. And then we have you make some and show that some fall apart and some will withstand about as much force as you want to put on them.

And that’s kind of fun to do and reasonably convincing. Yeah. And sort of a fun thing to do in a, in a classroom also. And then later on, we there’s a lot of frankly, just beautiful, forms in and we wanted to include those in there. There’s something that you wouldn’t normally hit in a geometry class.

Partly because you’re going to have a lot of required things to cover and it can be challenging. So we, we added a lot of those like Reuleaux figures. Reuleaux figures are three-dimensional objects that are a constant with everyone.  And they have some fun properties and they kind of look like a tetrahedron that’s been squashed or some other shapes and they, they roll and they do all kinds of fun things.

So, things that are fun to play with and amuse your friends. And when we do events, people don’t want to give them back. And so that’s what we want. We would like teachers to have things that the geometry students won’t want to give back.

Patrick: Right. The, the thing that astounds me about the Reuleaux triangles, they look like triangles with round corners, basically.  And you don’t think that there’s anything all that special about them.  Yet, it turns out that if you make a drill bit in the shape of a Reuleaux triangle and stick it on the end of your drill and hold it up against a piece of wood, you will actually drill a square hole using a triangular drill bit.

It’s one of the most mind blowing things you could ever think of and in the book, (and what’s the name of the book?  Make: Geometry!)  you talk about how that works.

Rich: Yeah, you can do that. It’s a little more complicated than that because the, the center of the drill bit doesn’t stay in the center of the hole.

So you need to need to run it through a little linkage or just let it wobble kind of it. Let your drill wobble, I suppose, but, yeah, so that you can cut a hole, a square hole with a Reuleaux triangle, as your drill bit. You can also, there are higher order Reuleaux shapes. There is a Reuleaux pentagon, that would drill a hexagonal hole.  Seven sides on a Reuleaux would get you eight sides on your hole that you drill, and so on.

Patrick: I’m detecting a pattern here.

Rich: Yep. One more side.

Patrick: Yes. Okay. Well, as I said, this whole thing, the book itself, but also the amazing things that you can do with just learning geometry, is just amazing.

Joan: Well, I think one of the other things that we found exciting about this is that kids really do love it.

You know, we’ve piloted this including online, to kids at a distance, and where they’ve had to cobble together whatever they happen to have lying around at home, to use the material. And it’s, it’s still, been something that they’ve found fun and found exciting.

And, and we’ve even had one blind kid that, that we had at a distance. And, and he had some fun with some of the models and it helped us out actually debugging some of them. And so I think it is something that will work in just about any environment, obviously. A couple of 3D printers in your classroom and you can, or at home for people who are perhaps going to try this at home, which we would encourage, you know, I think it is something that, that isn’t intimidating.

And I think it’s something that you can, you can grab and take. Yeah, start wherever you want. And as the constructivists say, you know, it has, I think a pretty, low floor, high ceiling and wide walls. And by which we mean that, you know, you can, you can do something pretty straightforwardly, but there’s a lot of room for growth. If you have a kid that really wants to run with it, or if you do for that.

Patrick: That hadn’t occurred to me with that really, really in Make: Geometry. It’s not intimidating, whether you’re a parent who has to teach this thing to your child, or whether you’re the student who learns. It’s done in a way that it is just not intimidating. I think that’s one of the greatest things about Make: Geometry.

So, thank you very much for being with us! We’ve been speaking with Joan Horvath and Rich Cameron, authors of the book Make: Geometry, available at the Makershed at Makershed.com, or on Amazon. The official publication date is August 10th, 2021. My name is Patrick Di Justo, the make book editor. Thank you for listening.

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DALE DOUGHERTY is the leading advocate of the Maker Movement. He founded Make: Magazine 2005, which first used the term “makers” to describe people who enjoyed “hands-on” work and play. He started Maker Faire in the San Francisco Bay Area in 2006, and this event has spread to nearly 200 locations in 40 countries, with over 1.5M attendees annually. He is President of Make:Community, which produces Make: and Maker Faire.

In 2011 Dougherty was honored at the White House as a “Champion of Change” through an initiative that honors Americans who are “doing extraordinary things in their communities to out-innovate, out-educate and out-build the rest of the world.” At the 2014 White House Maker Faire he was introduced by President Obama as an American innovator making significant contributions to the fields of education and business. He believes that the Maker Movement has the potential to transform the educational experience of students and introduce them to the practice of innovation through play and tinkering.

Dougherty is the author of “Free to Make: How the Maker Movement Is Changing our Jobs, Schools and Minds” with Adriane Conrad. He is co-author of "Maker City: A Practical Guide for Reinventing American Cities" with Peter Hirshberg and Marcia Kadanoff.

View more articles by Dale Dougherty