So what's Euclidean mean anyway?
Ever wonder just exactly what "Euclidean" means?
Well in honor of the last day of school, I'll tell you.
Euclid was a Greek mathematician born somewhere around 320ish BC. He's famous for a lot of things including proving that there are in fact an infinite number of prime numbers and devising the Euclidean algorithm which finds greatest the common divisor of any two numbers. He also composed the text "Elements", which was studied extensively in Western Europe by scholars for centuries.
It's this part of his work that interests me the most. Euclid is basically to geometry what Chuck Berry is to rock and roll.
Today we take Euclid's work for granted, but it's some of the most essential every day "common sense" type math. For centuries Euclid was
pretty much it as far as geometry goes. It wasn't until 1899 when
David Hilbert turned Euclid's work into a logically complete system that
geometry really advanced very much.
Are you a carpenter or a cabinet maker? If you know what a 3-4-5 triangle is you can thank Euclid. He proved the Pythagorean Theorem.
Euclid is most famous for these 5 postulates of geometry. A postulate is a self evident statement that's assumed to be true without formal proof.
1. A straight line segment can be drawn joining any two points.
Or between any two points, there is a line. This may seem obvious but it cannot actually be proven.
2. Any straight line segment can be extended indefinitely in a straight
Or in other words lines extend in both directions forever.
3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
Basically a circle of any radius > 0 can exist.
4. All right angles are congruent.
This too seems obvious but cannot be objectively proven. It must be assumed.
5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.
This postulate is equivalent to what is known as the parallel postulate. My favorite version of this is that given a line and a point
not on that line, there is exactly one line through the aforementioned point which is parrallel to the given line.
It has been proven that this postulate cannot be proven either. It too must be assumed.
And that's what fascinates me. It seems obvious that this is true, but we can't prove it. Working on the assumption it's not true, other mathematicians such as Gauss have found geometry systems where the first
4 postulates hold but the fifth one does not.
This leads one to speculate if the universe we live in is Euclidean or not. Every mathematician muses over this at some point, realizing the question itself is pointless.
Personally, I think it's simply more worthwhile to see the Euclidean universe than the non Euclidean one.
We all know the world is round and space is curved. The concept of parrallelism on the Earth's surface or in space itself therefore seems questionable at best.
But we treat the world around us as if it were flat, and in that system Euclidean geometry makes sense and others don't. The universe itself may or may not be Euclidean, but you try to build a house without operating on the assumption all right angles are congruent or that straight boards are (more or less) straight. You can't do it. You can't build a house without demonstrating Euclidean geometry.
Thus, while one can build a logical set of truths around the first four postulates, if one ignores the fifth postulate, you get a complete set of geometry that just doesn't describe the world most people live in very well.
Reminds me of certain political philosophies...