[0:00:02] So one of our members is a radiologist, and he cat scanned a broccoli for us, and then we printed it. So this is, there was a real broccoli, and then unfortunately we were keeping it in the freezer so that when we printed a full-size version of this someday, we'd be able to put them side by side, but somebody threw it out. So I was really irritated about that. And then we have lots of other cool 3D printed things. So this is a 3D printed vacuum cleaner. The motor actually is burnt out, so it doesn't work right now. But there's a propeller inside, and then there's a filter, and it goes and catches stuff.
[0:00:31] And we have some math visualizations like this in particular one, which is a diffusion one. So this is the diffusion equation with the initial conditions of the Heaviside function. And basically you can think of it as having a metal rod that's warm on one side and cold on the other. And over time, of course, entropy increases, heat diffuses, so it smooths out.
[0:00:54] And it's also an example of going and doing a multicolored print by swapping the filament. And then there's a buffer in the extruder, so it smooths out a little, and you get this nice color gradient. And we have other math visualizations, like this is a minimal surface Minkowski space. I actually don't know much math about Minkowski space. I was just asked to print that for somebody. But I went and I made some molds based off it. So this is dental plaster, and this is polyester casting resin.
[0:01:22] And actually, speaking of polyester casting resin, I've been doing experiments with making lenses. So this is a really bad one. But I got better ones. And actually, there's an article in the Open Hardware Journal about that. And what I was doing was I was going and using saran wrap to smooth over it so that you can compensate for the imperfections in the 3D print and use some oil as well. And then I was doing...
[0:01:45] This is a slice of a four-dimensional generalization of the Mandelbrot set. So you can think of all the Julia sets as being perpendicular to the Mandelbrot set, so they form a four-dimensional object. But you go and you slice it, and this is the real Julia sets stacked on top of each other.
[0:02:01] Yeah, and this is the real component of cosine on the complex plane. So your usual cosine is down here, and it's really neat because cosine can be thought of as two complex exponentials cancelling out their imaginary components. So one spins this way, and one spins this way, and so on. But when you add an imaginary component, the radius of one increases exponentially, and the other one falls exponentially. And so you get these, well, you get hyperbolic cosine along here. And if we could show the imaginary component, you would see that there is an imaginary component.
[0:02:30] We're also spinning through the cracks here. So this is a wave with the initial conditions of a so-called three-fingered wave. So imagine you have an infinite guitar string you pulled up to be a triangle. And when you let go, it goes and shoots off just two waves in different directions because it's an infinite guitar string. But this is why it's very important what happens at the boundaries if we have a finite one. So these are two different.
[0:02:58] Well, the same wave. They both have the initial conditions of a half cosine wave right here. But what happens is they go and they reflect off the edges. Well, they interact with the edges in different ways. So this is so-called Dirichlet boundary conditions, which basically means they're fixed. Imagine a guitar string where you have the edges fixed at two sides and they can't vibrate. So it goes and you get your usual notion of a wave where it goes up and down and up and down.
[0:03:22] Whereas this one is so-called Neumann boundary conditions, which just means it's more like a cup where you have water and it drops in and it reflects off the sides, right? Just like a wave, if you were waving like a bathtub, it's going to go and reflect off the sides. You can see here, it's not the greatest print, but you can see how it sort of goes and bounces off the sides. And this is really great. It's a really great opportunity actually to use this to teach math, because you can talk about, it's a partial differential equation, and if somebody knows calculus, you can talk about how the Dirichlet boundary conditions are actually just fixing the...
[0:03:51] But the Neumann boundary conditions are fixing the derivatives. Or if somebody doesn't know that, you can talk about how a partial differential equation, we can do algebra, essentially, on things that have unknown values that aren't numbers, but are functions themselves. Because that's what differential equations is about. So that's really neat to me. And there's lots of other neat things here. I'm writing a CAD program, implicit CAD. And this was the first object actually printed making with it.
[0:04:21] Printed, made with it. So you can see that the top is a rounded union of a bunch of circles. So it's not just a union. We go and we round the interfaces. And there's a really interesting connection between the math that I'm...