What is geometry? Circles and triangles—right? Not quite! A geometry is an axiomatic system which contains the following undefined terms: points, lines, and the relation incidence, that is, “a point is incident to a line” or “a point is on a line.”
If that still doesn’t help much, just think about points, lines, and a set of consistent axioms (or, statements that are assumed to be true) to play with, and voila, you’ve got a geometry. There are three axioms, called the Parallel Axioms or Parallel Postulates. Many geometries have one of the following axioms: Elliptic Parallel Postulate where all lines intersect. Euclidean Parallel Postulate where any point not on a specific line is on exactly one line parallel to the specified line. Hyperbolic Parallel Postulate where any point not on a specific line is on more than one line parallel to the specified line.
Given a single point, we are all used to there being one line through this point that is parallel to any other specific line. This is because we all studied Euclidean geometry in high school. But if we had all studied hyperbolic geometry in high school, we would be used to having multiple lines through this single point that are all parallel to any specific line. Why does this matter? Have you ever wondered what the shape of the universe is? A cosmologist at Princeton University, David Spergel, claims, “The shape of the universe tells us about its past and its future.”
That seems like a pretty big deal! So, what is the shape of the universe then? While any shape is technically possible, there are really just three basic shapes it could take: a flat universe, a spherical universe or a hyperbolic universe.
Each of these correspond to the one of the Parallel Postulates. A flat universe would exhibit the Euclidean Parallel Postulate. As discussed, spherical geometry exhibits the Elliptic Parallel Postulate. And (big surprise) a hyperbolic universe would have the Hyperbolic Parallel Postulate. Geometry matters! Just by studying spherical geometry, we’ve been able to understand our world better. Imagine if we could discover what the geometry of the universe is. We would then be able to learn whether or not the universe will eventually collapse or if it will keep expanding forever. However, if the discussion of the shape of the universe is making your head explode, don’t be alarmed. There are many everyday uses of geometry. If you like art, engineering, nature, sports or pretty much anything else, you can find a use for geometry in your life.
In addition to this, there are some good puns with geometry, which I would argue is sufficient reason to study the subject. What did the triangle say to the circle? You’re pointless!