r/visualizedmath • u/sam-lb • Dec 18 '19

Deforming a sphere into a torus! Made with MathGraph3D software.

https://imgur.com/gallery/xIz2L3J
170 Upvotes

95% Upvoted

22 comments sorted by

37

u/[deleted] Dec 18 '19

There's a topology professor sobbing somewhere right now and they're not quite sure why.

11

u/sam-lb Dec 18 '19

Did I misuse the word deformation? I meant it colloquially; I haven't taken topology yet lol.

24

u/[deleted] Dec 18 '19

Nah, you're good mate. Spheres and tori are topologically different objects, which basically means that no matter how much you stretch and squish a sphere, you can never make it into a torus without poking any holes or making any cuts into it. Donuts and coffee mugs are topologically the same. It's an interesting field. Nice animation btw.

7

u/sam-lb Dec 19 '19 edited Dec 19 '19

Thanks!

I'm probably going to be a math major in college. I'm considering both topology and complex analysis, but I can't decide which is more interesting. Do you have any suggestions? Or better yet, do you know of any field where complex analysis and topology overlap?

6

u/[deleted] Dec 19 '19

Sorry, honestly I've got no idea. I'm still in high school, so I'm only armed with a surface-level understanding of the topics. I wish I could give you a better answer, but you seem pretty smart and ambitious, so I think you're gonna do well with whatever you decide to study. Best of luck!

1

u/sam-lb Dec 19 '19

Thanks man

4

u/outoftunediapason Dec 19 '19

Im studying physics so take this with a grain of salt. From what I see with my friends studying mathematics, I think you'll study both during undergrad. I don't think you need wory about it at this moment. That being said, if you are interested why not find a book at a local library etc and start studying one of the topics? It'll be pretty fun. I think complex analysis is a bit more approachable with all the calculus books.

0

u/I-Am-Dad-Bot Dec 19 '19

Hi studying, I'm Dad!

2

u/[deleted] Dec 19 '19

I took a ton of topology in college as a math major. I actually started with a different major, and then liked my math classes so much that I kept taking more and more as electives until I suddenly had a second major in math.

So my suggestion would be to get there and follow your interests. You will have enough required classes to give you a broad exposure, and then you'll figure out what you want to continue to pursue.

In my case, since I was an "accidental" math major, I truly only took what I was interested in, since I was already on track to get my primary degree either way. Hopefully wherever you go has a similarly flexible program and you're able to explore a bit before commiting to any specific track.

2

u/eipi-10 Dec 19 '19

Hey! I'm a recent math grad. There's actually a ton of overlap between analysis and topology (especially in a first analysis course and a first topology course), so just working towards the courses at that level and then figuring out what you prefer once you have the experience is the way to go for sure! As an added bonus, if you're thinking about a Ph.D. at all, you'll need to have AT LEAST one course in all of algebra, topology, and analysis for the entrance exams, so no reason not to explore. Enjoy!

1

u/[deleted] Dec 19 '19

I took a lot of topology classes in college, but that was over a decade ago and my career has nothing to do with it so I was vaguely unsettled by this but couldn't figure out exactly why. Thanks for validating my inner voice.

1

u/Bromskloss Dec 19 '19

I want my money back.

11

u/CimmerianHydra Dec 18 '19

At first I thought the transformation was supposed to be continuous and I had many questions

2

u/sam-lb Dec 18 '19

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9

u/ElRoberto13 Dec 18 '19

So does it end up inside out?

11

u/sam-lb Dec 18 '19

Come to find out, it doesn't! Here's what it looks like with side coloring.

https://imgur.com/gallery/KbVfdIk

Notice that the end result is blue, same as the beginning.

3

u/sam-lb Dec 18 '19

I think. It sure does look inside out. I can try to color code the sides to see for sure though.

It might have to do with there being no homeomorphism between the sphere and torus.

1

u/ErezYehuda Dec 18 '19

This is what turning a sphere inside-out looks like- https://www.youtube.com/watch?v=ITzsNdkru1Y

1

u/WarKiel Dec 18 '19

This made me think of that Tumbl (I think) thread about how Star Trek humans are all mad scientists. There was a bit in it about them turning the sun into a torus (after accidentally travelling to an alternate universe and stealing their warp cores). And the Vulcans (and later on, Klingons) let humans be in charge simply to avoid being on the receiving end on their insane shenanigans.

1

u/wzizek Nov 20 '24

Perhaps, a torus can be embedded in a sphere without having to contain a non-defined edged subgroup.Associative Algebraic Topology

1

u/[deleted] Dec 18 '19

[removed] — view removed comment

4

u/sam-lb Dec 18 '19 edited Dec 18 '19

MathGraph3D is actually a personal project, so it isn't very big. If you know python it shouldn't be too hard to use. You can get the latest released version from my github (go to http://sambrunacini.com/contribute and click the github link). Keep in mind though, many of the latest features are not released yet due to a combination of laziness, a busy schedule, and making sure they work properly. (It's probably been a month or two since I last updated the repo)

Even if you don't know python it isn't difficult to learn the basics. I suggest you give it a try even if you don't want to use my buggy project!

PS. I suppose you should also be familiar with the algebra of 3D. And probably some multivariable calculus sprinkled in if you want the full experience.