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!

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