Introduction to Simplices.
November 11, 2010
There are a number of ways in topology to make shapes. We can use the euclidean plane and make them out of equations. For example, we can make this torus:
by plotting the equation in and .
Another way we could think of the torus is taking one circle centered at some non-origin point on the -axis and rotating it about the -axis, say. You could think of this as taking one of those bubble wands, holding it at arms length, and spinning around in a circle. If the bubble didn’t pop, you’d make a torus bubble around you. Wouldn’t that be cool?
There’s another way we could do it, though. Think about peeling an orange,
We’re going to peel this a special way. First, we’re going to cut little almost-triangular pieces out (see the picture above, the right-hand picture) and then we’re going to peel these pieces off perfectly. We’re that good at peeling oranges. Either way, now we have four triangular pieces of orange. That’s breaking up something which is a sphere into a set of four almost-triangles.
Also, perhaps less excitingly, given four triangular pieces, we can construct the following type of thing:
called a tetrahedron. Now, instead of orange peels, pretend these pieces are rubber and we’ve melted all of these together nicely. We can do the following thing: let’s stick a bike pump in and blow it up. What’s going to happen? It’s going to turn into a sphere. Think about pumping up a basketball: it doesn’t start out as a perfect sphere, but once you blow enough air into it, it becomes a nicely shaped sphere.
We can do the same thing to a lot of other shapes: we can cut them into triangles or things that look like triangles, take them apart, reassemble them, or even add and subtract pieces of them, so long as the resulting structure is sufficiently nice. Think about geometry for a second. Given a figure, say, an octagon, is there any way to break this up into triangles? Of course there is! There are many ways. Here’s one of them:
What about a circle? Can we make that into triangles? Well, no, not really, but in topology, we don’t really care what the shape looks like exactly, so long as we can continuously deform it. So, let’s deform a circle into something nice. Specifically, if we just squeeze the sides in, and make pointy edges, we can make the circle into a square:
The point of all of this is to say: given any sufficiently nice figure, we can make it out of pieces which look like triangles or higher-dimensional versions of triangles.
Let’s actually do some mathematics and define what a simplex is exactly.
A simplex is can be defined in abstract space or in euclidean space. We’ll do it concretely in euclidean space just to show that there’s no funny business going on here, and that we can actually legitimately define such things.
Definition: An n-simplex is set of ordered pairs in defined by
This might sound sort of confusing, but what we’re doing is taking all of the possible points such that the sum of the points is exactly 1.
For example, what is the 0-simplex (zero-simplex)? By definition, this is every point in such that the sum of that point is 1. Well, the only point that satisfies this is the point , so a 0-simplex is just a point.
Less trivially, what is a 1-simplex? It is every point in such that and . So, what is this? This is every point on the line above the x-axis and to the right of the y-axis. This is exactly the red line segment below. Notice that any point on that red line has coordinates that add up to 1.
Even less trivially, what is a 2-simplex? This is, by definition, all points in such that and .
Well, what is this? It’s much less obvious, but we can plot it. It will look something like this:
So, to sum things up so far, up to rearranging coordinates and translations and scaling and whatnot, we have the following cheat-sheet.
(Note: Another way to think about simplices is abstractly. Given some sufficiently nice space, an -simplex can be created if we have points such they don’t all lie on the same -dimensional hyperplane. Once we have these points, we take the convex hull, which is sort of like “connecting the dots and filling in the inside.” The resulting “filled in” space is our simplex. It is useful to think about what this means for -simplices: take 3 points not all of them are in a -dimensional plane (aka, a line) and what do we have? If they’re not all colinear, they make a triangle. Connect the dots and fill in the middle, and you get a triangle. This is exactly what we’d expect! Try to make a 3-simplex, a 1-simplex, and a 0-simplex this way!)