Hard Tweets Explained: Non-Euclidean Space

And here we have another example of me beating a twitter joke format horse to death. You can suffer through other examples of this in regard to Confections and Movies.

In this case, we are playing with the same pun about an opening line vs a geometric line that I used with Benjamin Button. Close your eyes, think back, and remember geometry from school: the shortest path between two points is a straight line; parallel lines never intersect; if you have a line and a point, there is only one line through that point that doesn’t intersect the line.

All that stuff, like so much of what you learned in school, is not exactly true.

There is a convention, when teaching, to fail to mention the context. For example, when they teach you Newtonian physics, they fail to mention that most of what you are learning does not apply at very high speed, or at very small scale. When you learn about music, they fail to mention that the convention of 12 notes is a Western construct, and there are other music systems that divide the frequencies of sounds in an octave differently. And when you learn about geometry, they fail to mention that you are learning about Euclidean geometry, but there are other geometries with different rules.

It’s a pet peeve of mine. It’s fine to teach one “truth,” but make it clear that this truth is contextual. There are other “truths” that seem to contradict this one, in other contexts.

So let’s focus on geometry. You learned about Euclidean geometry. That’s the geometry that Euclid defined about 2,300 years ago. He started with a small set of rules, and then derived a whole bunch of other rules from it. One of those rules says that if you have a line (think a road), and a point (think a sign next to the road), then there is only one line that goes through that point that doesn’t intersect the line (such as the sidewalk: it goes through the sign, and it never intersects the road). That line is parallel to the one it doesn’t intersect. Seems intuitive, and obviously correct, right?

OK, so suppose that I tell you that there is a different set of rules where every line through that point is going to intersect that  first line. That parallel lines do, in fact, intersect. That lines, in fact, always intersect themselves. And suppose, further, that you’ve worked with a geometric system like this your whole life. This other set of rules are different from Eulcid’s rules, so we call them Non-Euclidean. And your very existence seems to be in a Non-Euclidean space. As is mine. As is everyone else’s on the planet.

It’s like one of those Sphinx puzzles. You exist in a place where all lines intersect themselves. Where are you?

Answer: On a globe. Shoot a line in any direction, and eventually it comes around and hits you in the ass.

The surface of a sphere is a Non-Euclidean space. Every line through a point intersects every line you might draw. Lines intersect themselves. Parallel lines sometime intersect, such as the longitude markings on a globe meeting at the poles.

So the opening line of my life story would intersect itself because I exist in a Non-Euclidean space. (And I just checked: the spelling I used the in tweet is less common, but also considered correct. Whew!)

Homework: Define a new system of geometry in which the shortest distance between two points is plenty long enough, and really it’s how you use it that matters, and not how big it is, and who are you to judge anyway?

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