I Try To Understand the Coffee-Cup-Donut

I got frustrated this morning about all the ways in which I have utterly failed to learn math, which led to a conversation about a mathematical concept I was introduced to in college and could. not. understand. I took one abstract math class on a whim but had to quit after a week or so because I didn’t understand the professor when she said “A coffee cup and a donut are the same shape” which is a common “joke” introduction to Topology. The conversation that follows is me learning what in the hell that woman was talking about by talking with my ever-patient girlfriend:

A: A fun exercise is to think about the alphabet. Just the capital letters and to group the letters based on which can and can’t be morphed into each other.
A: REMEMBER. No cutting or gluing. So, for example, A and R can be grouped together
but B can’t go in that group because you’d have to “glue” the bottom part together.
Me: But B could turn into A!
A: Not without cutting something
Me: No, you could smush the bottom up, just like the coffee cup. This is why I don’t understand, I think. The rules seem so arbitrary.
Me: You can smush any shape into any other shape!
A: So, I think eliminating holes counts as gluing. You cannot create or eliminate a space.
Me: OMG then the coffee cup can’t turn into a donut!
A: Yeah it can.
Me: 😡
A: Hold on! No frustration.
Me: You are eliminating a space!
A: Ah, that is not a space.
Me: OMG. 😡
A: Ok, so on a very basic level, think of a solid 2d circle. It lives in 2d space. It divides up that space in a very basic way. There is the space outside the circle, and the circle. That’s it. Now imagine cutting a hole out of that circle. Now that “flat donut”  is dividing up the space in a different way.
Me: Wait. Cutting a hole out of that circle?
A: Yeah, you have a flat solid circle.
Me: Oh, ok, filled in.
A: Right
Me: And then you make it a donut. Flat donut.
A: A flat donut.
Me: Bialy.
A: We’re just talking 2d right now.
Me: Flat bagel.
A: Yeah. Ok so now you have the flat bagel.
Me: Bialy
A: The space INSIDE the flat bagel…bialy?
Me: A bialy is a flat bagel.
A: Ah. I got it. Ok. We are both learning things. Ok so you have the bialy, the space inside the bialy, and the space outside the bialy.
Me: Right.
A: You are allowed to change the SHAPE of any of those things but you CANNOT create new spaces.
Me: Ok, so you could make the bialy into a crueller.
A: Yeah, totally.
Me: Or a cinnamon twist. But you could not make it into a carrot.
A: Right.
Me: Because it would no longer have a hole.
A: You could create a GIANT dent in it if you want, that’s fine, but all the spaces must remain intact.
Me: So you could make it into a hairtie.


About Logan Howlett

Small and fierce.
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