Tag Archives: surface

A Slice of Algebra

and a nice cup of tea. I always find that helps. Before we get down to business, you might want to put this delightful recording on. It’s always nice to have a bit of background music, and Strauss just seems to fit with Algebra somehow.

A broad definition of Algebra could be the study of equations and their solutions. This is perhaps the type of algebra we’re all familiar with from school. Here’s a typical problem

Find x\in \mathbb{R} given that x^2-2x+1=0

That was easy, of course. Let’s try another one

Find all x,y \in \mathbb{R} such that y-x^2 = 0

Perhaps you had to think for a moment before realising that this just defines a parabola in 2D space, pictured below.

These example illustrate that the solutions to equations can come in the form of points, or curves, and it’s not hard to see that solutions to equations in sufficiently many variables can define surfaces of any dimension you like. For example the equation z=0 defines a plane in 3D space.

So we can easily see that Algebra gives rise to geometrical structures of the type we discussed in the last post. It should now seem natural to study geometrical structures from an algebraic point of view. Voila – we have the motivation for Algebraic Geometry.

There’s nothing to restrict us to studying the solutions (often referred to as zeroes) of a single equation. In fact many interesting and useful geometric constructions arise as the simultaneous zeroes of several equations. Can you see two equations in (some or all of) the variables x,y,z whose simultaneous solutions give rise to the y-axis in 3D [2]?

The technical terminology for the collection of simultaneous zeroes of several equations is an algebraic set. It is the most fundamental object of study which we will focus on.

Here we reach a slight technical impasse. For what follows I’ll assume a familiarity with elementary abstract algebra as outlined on the Background page. This may be viewed as a technical toolkit for our forthcoming studies. I’ll also assume some very basic knowledge about Topology, though not much more than can be gleaned by a thorough reading of the Wikipedia page. If you’ve never come across abstract algebra before, now is the time to do some serious thinking! I can’t promise it’ll be easy, and it might take a couple of days to get your head around the concepts, but I promise you it’s worth it. I’ll be happy to answer any questions commented on the Background page, and will flesh out the currently sparse details in the near future.

Good luck!

[1] We only every consider polynomial equations, which are those of the form f(x_1,\dots,x_n)=0 where f(x_1,\dots,x_n) is a finite sum of nonnegative integer powers  of products of the variables x_1,\dots, x_n. Thus f(x,y)=x^2+y^2=0, the circle, is admissible for study but f(x,y)=x^y=0 is not. It turns out that not much is lost by restricting our study to polynomials only. In some sense any mathematically interesting curve can be approximated arbitrarily closely by the set of solutions to polynomial equations. (This entirely depends on your definition of mathematically interesting though)!

[2] The equations are of course x=0 and z=0. Geometrically this is true since the y-axis is the intersection of the two planes defined by x=0 and z=0.

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Algebraic Geometry – Sorry, What?

Okay it’s a bit of a mouthful. Let’s break it down a bit. You probably remember geometry from school. Drawing triangles and calculating angles. Maybe even a few circle theorems. Pretty arcane stuff, you probably agree. Turns out that this is just one tiny area of what mathematicians call Geometry.

Roughly speaking Geometry is the study of any kinds of curves, shapes and surfaces you can imagine. We naturally think of curves as “1-dimensional objects” you can draw on a “2-dimensional” piece of paper. Similarly we think of surfaces as “2 dimensional objects” that exist in “3-dimensional space”. A piece of paper is an example of a surface, as it the surface of a beach ball. We can also think of “3-dimensional objects” like a solid snooker ball. In general geometry answers questions about what properties these things have.

Now you may be thinking that this is all a bit pointless. After all we know quite a lot about how a beach ball behaves. There are two caveats however. Firstly the surface of a beach ball is a very symmetrical object. We want to be able to make conclusions about vastly asymmetrical surfaces, and possibly ones it’s hard to imagine. These kind of general observations are useful because then if someone asks us about a specific case, a beach ball with a ring donut stuck onto it for example, we can tell them it’s geometrical features with no extra work.

The second pertinent observation is that sometimes we want to know about geometry in more than “3-dimensions”. Hang about, that’s completely pointless, I hear you say. Fair point, but you are forgetting we live in a 4-dimensional universe – 3 space and 1 time dimension. And thanks to Einstein’s General Theory of Relativity we know that gravity bends space. So knowing about how geometry works in 4D is vital for sending men to the moon, or getting accurate GPS signals from satellites.

Now you might be starting to see that Geometry is quite broad, quite useful but also quite hard. After all there doesn’t seem to be much a triangle has in common with the surface of the Earth! Nevertheless we’ll see in the next post that using Algebra we can start to pin down some classes of curves and surfaces that do share some surprisingly strong properties.

As a challenge before the next post, make a Mobius Strip, pictured above, and count how many sides it has. Okay that’s easy, if a little odd if it’s the first time you’ve seen it. You can see that even an easily constructible surface can have some surprises. Try and imagine some other odd surfaces; if you think of anything good then comment it! One such is the Klein Bottle. Don’t worry about the technical terminology on the Wikipedia page, just have a look at the pictures. It’s a 2-dimensional surface that can only be “drawn” in 4-dimensions. Looks like we’ll have our work cut out!