History of Geometry

Some of you may have heard of Euclid. As in Euclid's Elements which set forth the axioms of Euclidean geometry which you should all have learned in high school. You know parallel lines never meet and in fact are equidistant, meaning two parallel lines are always the same distance apart. You know the angles of a triangle total 180 degrees. You know you can draw a square, which is a 4-sided polygon with right (90 degree) angles. You know a triangle has unbounded area, meaning you can always draw a triangle as big as you want. And since you know all this you are probably feeling very clever now. Enough of that.

You probably don't know that none of these statements can be proved on its own. One of them has to be assumed. Euclid assumed one, called the parallel postulate. For the next 2,000 years, mathematicians assumed they knew better than Euclid and could prove the statements without assuming any of them. After 2,000 years of failure, it occurred to someone that maybe Euclid was right after all. Why did it take so long? Well, if we don't assume the parallel postulate we open the door for some weird geometries. We all like Euclidean geometry because we can draw a picture and see what happens. There's no mystery. The next geometry people got happy with was spherical geometry. Spherical geometry is what you were doing when you learned in high school that the shortest distance between any two points on the globe isn't a straight line; it's a great circle on the surface of the planet. (Feeling clever again now?) As Dr. Maher said today in class:

Why were they happy doing spherical geometry? Well, they could draw a picture and this cheered them up!




In fact, drawing pictures cheers everyone up. As Zhenyi, another topology student, said:

That's why I chose topology. I like to draw beautiful pictures.

There you have it, folks. Mathematicians like topology (top picture) for the same reasons that 5-yr-olds like crayons and people with questionable taste like certain selections from the Museum of Modern Art (bottom picture).