This is the second post in a series by Shunsuke Katayama, in which he has conversations with a version of himself who decided to not pursue mathematics. Read the first post here.
Naught: Great, so now I have an idea about what the research aspect of Math is.
Prime: Brilliant. I didn’t know what research in Mathematics was before I started doing it either, so you’re already ahead of where I was last December!
Naught: So tell me more about yourself. We had that whole discussion on geometry and getting to a party in the middle of nowhere, but how does it relate to your research?
Prime: Well, I am investigating a quadrilateral living on a 2-dimensional plane, which are unique up to what we call “Affine Special Linear Group (ASL2Z)”
Naught: Wow, there’s a lot of big words there!!
Prime: I know, calm down. We’ll digest them together.
Naught: All right, let’s see if I can follow you.
Prime: So from the basics: a quadrilateral is anything with four sides.
Naught: I wasn’t born yesterday, you know.
Prime: Haha, sorry. So quadrilaterals living on a 2-dimensional plane.
Naught: So far so good.
Prime: …which are unique up to certain transformations.
Naught: All right, not so good…
Prime: No worries. Um…so let’s think about a pizza.
Naught: A pizza?
Prime: Stick with me, it’ll make sense later. Tell me, how would you define a pizza?
Naught: Um… a flat round baked bread with stuff on top?
Prime: If something like that was the size of your hand, would you call that a pizza?
Naught: Sure.
Prime: The size of a table?
Naught: Why not?
Prime: But what if that pizza was folded in half?
Naught: Oh, trick question. Then it’s a calzone!
Prime: Okay, so we can conclude that a pizza is invariant to its size but it must maintain its roundedness. You also say that pizzas can have stuff on top, so the toppings don’t have to be unique either– they can vary from cheese to prosciutto to even, God forbid, pineapples. In other words, pizzas are unique up to transformations in size and toppings, but not to shape.
Naught: I guess you can say that.
Prime: Now we can bring this analogy over to the quadrilaterals. Let’s say you make a list of things that you can change while still keeping a pizza a pizza. Think of “ASL2Z” as a similar list; instead of a list of pizza properties, it’s a list of transformations that you can do to quadrilaterals in this geometry and still call them the same or unique. It’s not very important what is actually on the list for the purposes of our discussion.
Naught: Let me process that.
Prime: Take your time. It may be hard because you’re used to thinking in Euclidean geometry where a smaller rectangle is not the same as a bigger rectangle. But if you think about it, you can rotate, reflect, and move rectangles and still have the same rectangle! These transformations are, similarly, on a list called “Euclidean group” and denoted AOnR, in case you’re interested.
Naught: Right…
Prime: But, say you’re only concerned about the area of the rectangle, and you want the area to be 10. If so, you can have rectangles of sides {1, 10} or {2, 5} and call them the “same”.
Naught: Oh, so you’re less restrictive in the definition of uniqueness in your geometry.
Prime: That’s one way to put it. In fact, in my geometry, the area will be preserved under these transformations. So going back to the analogy of the party, say the party we were invited to was at a huge mansion with a large garden. Although it’ll have one address, the mansion could be in a range of latitudes and longitudes, but all indicating the same place!
Naught: Oh, cool, cool. I think I’m starting to unravel the world of your geometry.
Prime: Don’t worry, more is coming. You’re going to have to destroy your notion of length and angle, as they are meaningless in this geometry, for next time.
The Abstract 