Casual maths.

In the defence of 'imaginary' geometry:

Euclid wrote a book called "Elements". He wasn't the only guy to write a book on the topic of geometry with that title - it was kind of the thing to do at the time. But his "Elements" now owns the name, and has stood the test of time remarkably well. It was actually 13 books, written in 300BC and when I read Anne of Green Gables the other day (set in 1890ish) it was her geometry textbook. That's pretty impressive in itself and there is a lot of cool stuff in there. But really, we're mostly going to ignore it.

In Elements, Euclid wrote down his axioms of geometry, that is, those things from which he proved everything else. These axioms wouldn't generally be considered too controversial.

I quote (wikipedia, quoting Elements) that these are things you can do, things that exist in the world, facts, as it were:

1. "To draw a straight line from any point to any point."
2. "To produce [extend] a finite straight line continuously in a straight line."
3. "To describe a circle with any centre and distance [radius]."
4. "That all right angles are equal to one another."
5. The parallel postulate: "That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles."

For many, many years mathematicians fixated on the last of these, the so called parallel postulate. They discovered it was logically equivalent to the much simpler:

"If you have a line, and a disjoint point, there exists one line through that point which does not meet the original line"

They still felt that it wasn't as nice, necessary or obvious as the other axioms though, so they tried to prove it from the others. In great mathematical tradition many lives were spent trying to prove this, until one day it all came to a grinding halt.

Someone (well, actually a few someones) came up with the idea of non-Euclidean geometry. That is, they threw away the parallel postulate and replaced it with:

"If you have a line, and a disjoint point, there exist no lines through that point which do not meet the original line"(Elliptic)

or, even weirder in my mind,

"If you have a line, and a disjoint point, there exist infinitely many lines through that point which do not meet the original line"(Hyperbolic)

If you need some reasons why such things are important, geometrically, consider the surface of a sphere, google a mobius strip or talk with a physicist.

Til next time!