*Euclid dammit
In: 6
Euclid’s first through fourth postulates are called neutral geometry. Think of doing some geometry but without any comment on parallel lines. It’s neutral!
A postulate is just a rule or statement that can be used to make new rules.
Parallel lines are lines that never touch. Kind of like lines on a highway.
Euclid had a fifth postulate to talk about parallel lines. The postulate says that if you have a line and a point not on that line, then there is only one unique line that can be drawn through this point and never touches the other line.
People throughout history disagreed on the necessity of this fifth postulate. Many had tried, and all had failed to show that you only needed the first four postulates (or rules) to make the fifth postulate. In the process, a lot was learned about this type of geometry. The big takeaways are the first four postulates cannot make the fifth postulate true, and Euclid’s fifth is not neutral on parallel lines. This Euclidean geometry is probably what you and many others use day-to-day.
Another takeaway was that if you tweak Euclid’s fifth, you get another type of geometry. This one says that if you have a line and a point not on that line, then there are infinitely many lines that intersect that point that don’t touch that other line. This isn’t a neutral take on parallel lines. This is hyperbolic geometry. Hyper- meaning “many” and -bolic meaning “bundle” I suppose. Wanna see a cool example of this? [Check out MC Escher!](https://en.m.wikipedia.org/wiki/Circle_Limit_III)
Latest Answers