mentioned in science fiction novels or modern physics discussions

about oddly curved spaces. The name seems to connote some kind of

weird form of geometry that doesn't match with our usual assumptions

about the world. And indeed, non-euclidean geometry does make it

possible to construct bizarre figures that seem to defy our basic

assumptions. But what exactly does "non-euclidean

geometry" mean?

To start with, let's review what Euclidean geometry is.

Basically, this is the type of geometry taught in most high schools,

based on the works of the greek mathematician Euclid over two thousand

years ago. Central to this system of geometry are five postulates, or

basic assumptions about how the world works, which are used as the

basis for proving more complex theorems. Four of the postulates are

very simple: any two points determine a line, any line segment can be

extended indefinitely, any line segment can be the radius of a circle,

and all right angles are congruent. The fifth postulate, or "parallel

postulate", is a little more complicated to state: Given a line and a

point not on the line, exactly one line can be drawn through that

point that is parallel to the first line. Quite a mouthful compared

to the other postulates, don't you think? Because it is so much more

complex to state, many mathematicians over the years tried to find

ways to get rid of it entirely, and prove it based on the other

postulates.

If you draw a few pictures on a piece of paper, you'll soon

realize that the fifth postulate has to be true. Well, drawing

pictures on paper might not be the best way to figure this out, since

you'll soon see that it's hard to even draw something the supposedly

violates this postulate. For example, suppose

that instead of the parallel postulate, *no* line can be drawn through

an external point that is parallel to a given line. That would mean

that if you draw two lines forming a right angle, and then a third

line that also forms a right angle with the second, then the third and

first lines must intersect somewhere, forming a triangle with two

right angles! Otherwise, the third line would be parallel to the

first, violating our modified postulate. Due to absurdities like

this, for many years it was assumed that the fifth postulate must be

true, and the only open question was whether it should be a

postulate or a theorem.

But in the nineteenth century, mathematicians were coming to a

growing realization. While mathematics is often very useful for

describing the real world, ultimately, it is a system for deducing the

consequences of your basic assumptions, or postulates. So why not try

modifying the fifth postulate, deducing the consequences, and seeing

where that takes you? Mathematicians Janos Bolyoi and Nicolai

Lobachevsky independently pursued this idea in the 1820s and 1830s,

developing whole geometries based on modifications of the parallel

postulate. At first, it looked like they were just playing some silly

symbolic game, and it was several decades before their work was widely

accepted. But gradually their colleagues realized that these new

geometries were very usefully modelling properties of different types

of surfaces.

For example, let's look again at our 'absurd' example of a

triangle with two right angles. How could such a thing be possible?

On a flat plane, it really is absurd. But look at a globe of the

Earth. Draw lines, which are actually great circle segments on the

surface of a sphere, from the equator to the North Pole at the 0 and

90 degree meridians. Each of these lines forms a 90 degree angle with

the equator, yet they intersect at the pole-- and we really do have a

triangle with two right angles! And the angle at the pole is also 90

degrees, so there are actually three right angles in this triangle.

In other words, our modified geometry may not make sense when viewed

on a flat plane, but is an accurate description of the properties of

shapes on the surface of a sphere. And you can now amaze your friends

by drawing triangles with three right angles, though they may get mad

at you for defacing their globes.

The development of non-euclidean geometries became vital when

Albert Einstein began working on his general theory of relativity in

the early 20th century. In Einstein's models, the three-dimensional

space we live in is curved in the fourth dimension in regions where

matter is present-- so these odd forms of geometry are what actually

describe the real world, rather than Euclid's so-called "obvious"

models! Of course, whether in a small region of the surface of a

sphere, or in a small region of relativistic space, Euclid's

conclusions are an excellent approximation of reality for most

practical purposes. And it's still true that Euclid made an

immeasurable contribution by showing how interesting and useful

conclusions could be deduced from a simple set of basic postulates.

But the real world is a lot more complicated than Euclid thought.

And this has been your math mutation for today.

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