Geometry: I should have paid attention to it during high school!

 I have been doing some reading and researching on some stuff that I hated in high school and one of them was geometry. Anything that had figures bored me, but now I am realizing that these stuffs are really cool.

Euclid.

Anyway, Geometry is one subject that never got through my thick head. I read some stuff about Euclid, the father of geometry. Euclid lived around 300 BC. He was famous for his book "The Elements" which was a book about geometry. The geometrical system described in the Elements was long known simply as geometry, and was considered to be the only geometry possible, that is until it was challenged indepently during the 19th century by three mathematicians Janos Bulyai, Nikolai Lobachecsky, and Carl Friedrich Gauss. It's amazing that the Elements went unchallenged for at least 2300 years--a testament to its influence.

Here are five Euclidean postulates or axioms:

• "To draw a straight line from any point to any point."

infinite points: infinite lines

                                            

• "To produce [extend] a finite straight line continuously in a straight line."

to make the line/s longer, just extend; remember, lines are infinite

• "To describe a circle with any centre and distance [radius]."

• "That all right angles are equal to one another."  

• 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."

Two parralel lines are intersected by the lines producing right angles. According to Euclid, if the angles  are equal to 90 degrees, the lines if produced indefinitely would not meet. Common sense tells us that this is a self evident truth.

Here we see twoo parallelel lines with angle A less than 90 degrees and angle C more than 90 degrees. According to Euclid's fifth postulate line 1 and line 2 will inevitably meet while line 3 and 4 will move farther away from each other. Well, this self evident too.

Euclid's postulates are simple enough to understand conceptually, except the math, maybe. But postulate number 5 was challenged and proven to be false during the 19th century hence a new geometry, a non-Euclidean geometry, was born. 

Anyway, Euclid's geometry works only on flat surfaces. The parallel postulate does notr work on curve surfaces.

On curve surface, the angle produced parallel lines and an intersecting line  are 90 degrees but the  parallel lines move away from each other. They do not stay parallel.

On a sphere, parallel lines meet.

The amazing things about Euclidean geometry is that it took 2,300 years before it was challenged. Many who challenged it was afraid to publish their work, Gauss for example, because they were afraid of the repercussions. Even the great Philosopher Kant himself considered Euclid's geometry (and Newtonian mechanics) as a priori knowledge--truth apart from experience. 


Of course, Eucledian geometry  is still valid and still works but only on flat surface.  

Geometry is interesting, sans the math, and I wondered why I was not interested with it during my high school days.