History of Geometry

Humans evolved as pursuit predators, so it makes sense that understanding the relationships between form, space, and physical objects is an essential part of the way our brains work. Geometry is such an intuitive human faculty, many people never give it a second thought, but the history of knowledge is full of profound insight lurking when the most intuitive assumptions are contemplated. The history of geometry is no different, and you can read more about what I mean here:
History of geometry

Literally, geometry means "Earth measurement" in Greek, but since Earth is complicated, geometry typically starts with more primitive thing like points, lines, and arcs. It's difficult to define exactly what a point, a line, or an arc are, but we seem to have a reasonable grasp of these things even though they're difficult to define by themselves. As one of my math-major friends put it:

"What is a point? Geometricians can't say, but the axioms describe how they behave, and how they interact with lines, also undefined. You might as well call them tables and chairs, or pips and peeps."

Since the time of Euclid geometry has been built on the process of deriving theorems from each other and from fundamental axioms. Theorems are the children of axioms and the rules of logic, so they're transparent and well-insulated from mystery, but the rules and axioms themselves are a bit mysterious. They don't really come from anywhere but assumptions about reality. To clarify, let's talk about parallel lines for a moment.

What does it mean for two lines to be parallel? Intuitively you might guess that two lines are parallel if they point the same direction, that if this is the case the lines never meet (or, more rigorously, meet at infinity), and that there ought to be a way to prove this from some more basic information about the construction of reality. Euclid and a small army of mathematicians who came after him for two millennia tried to do that and failed every time, having to accept that "parallel lines meet at infinity" is an axiom.

Since axioms aren't proved from anything, it's reasonable to ask what happens if you change them. What if parallel lines meet at a finite distance, or diverge forever at infinity? That seems nonsensical if you think of space as an infinite homogeneous grid, like a sheet of graph paper in every direction, but what if space curves and puckers like a hammock on a lazy afternoon?

In the 1800s geometers began seriously looking at what would happen in such a world. Surprisingly, such a world is counterintuitive but conceivable, and geometry where parallel lines bend toward or away from each other seemed to be fertile to new, consistent theorems. It was interesting, but many thought there was a touch of esotericness to these theories. Charles Lutwidge Dodgson went so far as to write a scathing critique of what he saw as a debasement of mathematical rigor among those working on non-Euclidian geometry. All this critiquing came to a screeching halt when the theories of Albert Einstein and the experimental confirmation that shortly followed showed unambiguously that we live in a world where straight lines ain't necessarily straight.

The history of geometry is a rich subject, and there's much more to it than knowledge of parallel lines. There's more than I could possibly hope to cover as a layman in one blog post, so here's one last story, as told by someone indispensible in any discussion of math, Vi Hart. Pythagoras, one of the giants of pre-Socratic philosophy and founders of geometry, may totally have murdered a dude over the square root of 2:

You might say that's an... irrational thing to do.