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Updated: 2016-05-12T20:06:53+02:00

To Infinity… And Beyond!

2013-01-28T00:00:00+01:00

Exploring the outer limits “It is known that there are an infinite number of worlds, simply because there is an infinite amount of space for them to be in. However, not every one of them is inhabited. Therefore, there must be a finite number of inhabited worlds. Any finite number divided by infinity is as near to nothing as makes no odds, so the average population of all the planets in the universe can be said to be zero. From this it follows that the population of the whole universe is also zero, and that any people you may meet from time to time are merely the products of a deranged imagination.” – The Restaurant at the End of the Universe, Douglas Adams If there's one thing mathematicians have a love-hate relationship with, it has to be infinity. It's the ultimate tease: it beckons us to come closer, but never allows us anywhere near it. No matter how far we travel to impress it, infinity remains disinterested, equally distant from everything: infinitely far! $$0 < 1 < 2 < 3 < … < \infty$$ Yet infinity is not just desirable, it is absolutely necessary. All over mathematics, we find problems for which no finite amount of steps will help resolve them. Without infinity, we wouldn't have real numbers, for starters. That's a problem: our circles aren't round anymore (no $π$ and $\tau$) and our exponentials stop growing right (no $e$). We can throw out all of our triangles too: most of their sides have exploded. A steel railroad bridge with a 1200 ton counter-weight.Completed in 1910. Source: Library of Congress. We like infinity because it helps avoid all that. In fact even when things are not infinite, we often prefer to pretend they are—we do geometry in infinitely big planes, because then we don't have to care about where the edges are. Now, suppose we want to analyze a steel beam, because we're trying to figure out if our proposed bridge will stay up. If we want to model reality accurately, that means simulating each individual particle, every atom in the beam. Each has its own place and pushes and pulls on others nearby. But even just $40$ grams of pure iron contains $4.31 \cdot 10^{23}$ atoms. That's an inordinate amount of things to keep track of for just 1 teaspoon of iron. class='mathbox square autosize' src='/files/infinity-and-beyond/mb-1-austenite.html' /> The crystal structure of 32 iron atoms in the hot austenite phase.If your steel looks like this, your bridge is on fire. class='mathbox square autosize' src='/files/infinity-and-beyond/mb-2-cube.html' /> A chunk of solid, mathematical iron. Instead, we pretend the steel is solid throughout. Rather than being composed of atoms with gaps in between, it's made of some unknown, filled in material with a certain density, expressed e.g. as grams per cubic centimetre. Given any shape, we can determine its volume, and hence its total mass, and go from there. That's much simpler than counting and keeping track of individual atoms, right? Unfortunately, that's not quite true. The Shortest Disappearing Trick Ever Like all choices in mathematics, this one has consequences we cannot avoid. Our beam's density is mass per volume. Individual points in space have zero volume. That would mean that at any given point inside the beam, the amount of mass there is $0$. How can a beam that is entirely composed of nothing be solid and have a non-zero mass? class='mathbox fit' height='250' src='/files/infinity-and-beyond/mb-3-empty-cube.html' />Bam! No more iron anywhere. While Douglas Adams was being deliberately obtuse, there's a kernel of truth there, which is a genuine paradox: what exactly is the mass of every atom in our situation? To make our beam solid and continuous, we had to shrink every atom down to an infinitely small point. To compensate, we had to create infinitely many of them. Dividing the finite mass of the beam between an infinite amount of atoms should result in $0$ mass per atom. Yet all these masses still have to add up to the total mass of the beam. This suggests \$ 0 + 0 + [...]

Making Worlds 3 - That's no Moon...

2009-11-05T00:00:00+01:00

It's been over two months since the last installment in this series. Oops. Unfortunately, while trying to get to the next stage of this project, I ran into some walls. My main problem is that I'm not just creating worlds, but also learning to work with the Ogre engine and modern graphics hardware in particular. This presents some interesting challenges: between my own code and the pixels on the screen, there are no less than three levels of indirection. First, there's Ogre, a complex piece of C++ code that provides me with high-level graphics tools (i.e. objects in space). Ogre talks to OpenGL, which abstracts away low-level graphics operations (i.e. commands necessary to draw a single frame). The OpenGL calls are handed off to the graphics driver, which translates them into operations on the actual hardware (processing vertices and pixels in GPU memory). Given this long dependency chain, it's no surprise that when something goes wrong, it can be hard to pinpoint exactly where the problem lies. In my case, an oversight and misunderstanding of an Ogre feature lead to several days of wasted time and a lot of frustration that made me put aside the project for a while. With that said, back to the planets... Normal mapping Last time, I ended with a bumpy surface, carved by applying brushes to the surface. The geometry was there, but the surface was still just solid white. To make it more visually interesting, I'm going to apply light shading. The most basic information you need for shading a surface is the surface normal. This is the vector that points straight away from the surface at a particular point. For flat surfaces, the normal is the same everywhere. For curved surfaces, the normal varies continuously across the surface. Typical materials reflect the most light when the surface normal points straight at the light source. By comparing the surface normal with the direction of incoming light (using the vector dot product), you can get a good measure of how bright the surface should be under illumination: Lighting a surface using its normals. To use normals for lighting, I have two options. The first is to do this on a geometry basis, assigning a normal to every triangle in the planet mesh. This is straightforward, but ties the quality of the shading to the level of detail in the geometry. A second, better way is to use a normal map. You stretch an image over the surface, as you would for applying textures, but instead of color, each pixel in the image represents a normal vector in 3D. Each pixel's channels (red, green, blue) are used to describe the vector's X, Y and Z values. When lighting the surface, the normal for a particular point is found by looking it up in the normal map. The benefit of this approach is that you can stretch a high resolution normal map over low resolution geometry, often with almost no visual difference. Lighting a low-resolution surface using high-resolution normals. Here's the technique applied to a real model: (Source - Creative Commons Share-alike Attribution) Normal mapping helps keep performance up and memory usage down. Finding Normals So how do you generate such a normal map, or even a single normal at a single point? There are many ways, but the basic principle is usually the same. First you calculate two different vectors which are tangent to the surface at the point in question. Then you use the cross product to find a vector perpendicular to the two. This third vector is unique and will be the surface normal. For triangles, you can pick any two triangle edges as vectors. In my case, the surface is described by a heightmap on a sphere, which makes things a bit trickier and requires some math. I asked my friend Djun Kim, Ph.D. and teacher of mathematics at UBC for help and he recommended Calculus on Manifolds by Michael Spivak. This deceptively small and thin book covers all the basics of calculus in a dense and compact way, and quickly became my new favorite reading material. Differential Geometry In th[...]