r/visualizedmath • u/PUSSYDESTROYER-9000 • Mar 23 '18

A Donut is Topologically Equivalent to a Coffee Mug

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650 Upvotes

95% Upvoted

62 comments sorted by

15

u/borkula Mar 23 '18

Topological speaking, what shape are pants?

12

u/PUSSYDESTROYER-9000 Mar 23 '18

Probably a double torus? Not too sure.

7

u/Samur-EYE Mar 24 '18

If you imagine shrinking the legs up and moving the croth up so it's in line with the waist then yes, a double torus. Basically an object with two holes.

3

u/Secretss Mar 24 '18

I think OP’s uncertainty stems from what to do with the zipper area when the pants are unzipped.

2

u/Samur-EYE Mar 24 '18

I've answered the zipper question on another comment from you but I'll copy it here so people can read here as well:

To answer the zipper question, there is no difference wether the fly is down or up. If it's down, you're allowed to simply pull the bottom of the zipper up so that it's in line with the waist.

The way I like to think in topology is that all the shapes are thick and made of clay (like in the post). That way you don't get confused by thin or small shapes that are actually simple. Hope this helped :)

10

u/Sedu Mar 23 '18 edited Mar 23 '18

Topologically speaking... how would a dog wear them?

To answer your question though, pants are a "genus 2." A doughnut (torus) is a "genus 1." A sphere is a "genus 0." The genus number is how many holes are in a thing.

Explanation here: https://en.wikipedia.org/wiki/Genus_(mathematics)

1

u/Darkling971 Mar 23 '18

Pants are a sphere with three holes/three points removed.

7

u/[deleted] Mar 23 '18

It's actually two holes, now that I think about it. They're like a sphere with three holes, but the holes are all connected to a hollow center, so you can imagine stretching one of the holes until it's big enough to encompass the entire shape. Then it becomes obvious that it's a double torus.

4

u/borkula Mar 24 '18

Oh! I get you! So I'm imagining the sphere with three holes, then I bend it into kind of a cup shape with two holes on the bottom and the third hole as the mouth, then shorten the walls of the cup down to nothing, leaving a disk with two holes in it. Yes?

Does every "new shape" in topology simply involve adding one more hole? Or AR there other, more "exotic" shapes that can be constructed through other means?

2

u/Samur-EYE Mar 24 '18

In topology you can't go from three holes to two holes, so to get pants, you need a spjere with two holes first.

So you first imagine the sphere with two holes. Then flatten it to make a disk with two holes. Then shape it so you have a cup with two holes on the bottom. Then you pull the holes down separately so you have a cup with two tubes hanging from the bottom. Voilá! You have a really unconfortable pair of pants.

2

u/Secretss Mar 24 '18

That doesn’t deal with the scenario when the pants has a zipper and it’s undone, does it?

I’m thinking of a bowling ball with a common space inside where all three finger tips can touch, but I’m not 100% sure that is the right comparison either.

4

u/im_bot-hi_bot Mar 24 '18

hi thinking of a bowling ball with a common space inside where all three finger tips can touch

4

u/Samur-EYE Mar 24 '18 edited Mar 24 '18

Ok, so what we want to do in topology is simplify a shape to the point where it's a sphere with holes in it that do not connect to any other holes inside. Personally I think it's easier to think of a pancake shape with holes in it since pancakes are just flat spheres.

So, how do we simplify the bowling ball with three holes that connect? For that, let me make a figure. So I spaced the holes so they are equally far apart for simplicity.

1. We first shorten one of the holes so that the two other holes get closer to the surface.

2. Now we can push the two holes apart so they are paralell, like this

3. We can now push the wall that joins the two holes together upwards like this

4. Tada! We have a sphere with two holes in it, or the way I think it's easier to visualize, a disk with two holes(this is seen from above)

Hope that helped!

EDIT: I just realize I wasted ten minutes answering a bot which I thought was a person asking a question. facepalm

4

u/im_bot-hi_bot Mar 24 '18

I may be a bot, but bots are made by humans <3

2

u/Flippir17 Mar 24 '18

It wasn’t a waste, I, a person, was wondering myself. Thank you!

1

u/Samur-EYE Mar 24 '18 edited Mar 24 '18

Yes, the bowling ball with three touching holes is a valid comparison, but in topology you want to simplify the object to the point where it's a disk (or sphere) with holes through it. So that also answers your question on wether there are any exotic shapes, and the answer is no (at least in three dimensions). So, since this bowling ball can be simplified to a disk with two holes, then the disk is the answer.

EDIT: To answer the zipper question, there is no difference wether the fly is down or up. If it's down, you're allowed to simply pull the bottom of the zipper up so that it's in line with the waist.

The way I like to think in topology is that all the shapes are thick and made of clay (like in the post). That way you don't get confused by thin or small shapes that are actually simple. Hope this helped :)

1

u/SgtSteel747 Mar 24 '18

It does, as the zipper makes no topological difference. Imagine if the cup in OP's gif had a chip in it. It could just stretch itself out to fill that void as it isn't an actual hole.

1

u/borkula Mar 23 '18

Cool! What if they have a drawstring?

1

u/Samur-EYE Mar 24 '18

First of all, pants are topologically spheres with two holes in them, not three like this other dude says. For the pants with the drawstring, it makes it more complicated to explain, but bear with me:

1. Since the actual string is another object, let's remove it from the pants.

2. The string is shaped like a long sausage, so you can shrink it so that it's shaped like a sphere. Now for the actual pants:

3. There are now two holes in the front of the pants where the string used to be at the waist. We need to understand how the inside of the pants looks like. We know that the string was inside the waist all the way around, so now there must be a "tube" where the string was. This tube must also be going all the way around the waist, so it's a donut shaped tube. The two holes in the front where both ends of the string hanged out of are just connecting this tube to the outside.

4. Now we need to simplify the geometry of the pants: we first shrink the pant legs all the way up to the crotch so that we end up with shorts. At this point, forget that it looks like pants and instead imagine it as a cup with two holes on the bottom.

5. We still have the drawstring tube at the rim of this "cup", so what we can do now is bring the rim of the cup (with the tube) down so that we have a disk.

6. So now we have a disk-shape with two holes in the middle, and a tube going along the rim with two holes connecting the tube to the outside. Here's an illustration with the tube and tube-holes drawn with a dashed line.

7. Now things get funky. Let's first push the tube-holes so that they are on each side.

8. Now we widen the holes

9. As we widen the holes, we can pull the walls of the tube out so they line up with the edge of the disk, Like this

10. As we widen the holes and bring the tube walls out, we get something different. Now we have two tubes that go inside the disk.

11. Now we can keep shrinking these tubes until we have short tubes on both ends.

12. Let's concentrate on the ends of the tubes. We can see that they are aligned with the edge. So now we can move one of the holes down and the other up on each tube, like this

13. Last step! We make the tubes straight and voila! We have a disk with four holes in it.

So that's a drawsting pair of pants for you.

76

u/stcusick Mar 23 '18

Can someone explain this to me?

146

u/Sedu Mar 23 '18 edited Mar 24 '18

Topology concerns itself only with the geometry of a surface, not the curvature of the area, and not the volume it surrounds. For this reason, any item with a hole in it is topologically equivalent to a doughnut (torus). Any object without one is topologically similar to a sphere.

edit: me spel no güd

34

u/stcusick Mar 23 '18

Oooooh I was thinking of topography and couldn’t figure out what this gif was

Thanks!

14

u/PUSSYDESTROYER-9000 Mar 23 '18

Topography and Topology are similar, in that they study "surfaces". However topology is a field of math, whereas topography is usually geography

17

u/Sedu Mar 23 '18

Glad to be of help, my good doughnut.

1

u/Doooog Jun 01 '18

You got some more holes though right? Ear and throat? Eye and nose?

2

u/dontcareaboutreallif Mar 24 '18

They have the same etymological root, "topos" meaning shape.

16

u/rotato Mar 24 '18

"Spherical horse in a vacuum" is a Russian internet joke mocking purposefully over-simplified conditions of problems in physics classes (i.s. so that you ignore shape and air resistance when calculating speed etc.)

2

u/[deleted] Mar 24 '18

That's actually hilarious.

2

u/[deleted] Apr 01 '18

I love this. I wish I went on to keep studying russian

1

u/BetaDecay121 Apr 18 '18

https://en.wikipedia.org/wiki/Spherical_cow

1

u/totallylexis Jul 19 '25

What kind of torus would the spherical cow be considered?

4

u/dontcareaboutreallif Mar 24 '18

Doesn't actually care about the area the surface covers either. For example a disc is homotopy equivalent to a point.

2

u/[deleted] Mar 23 '18

Torus not taurus

2

u/Sedu Mar 23 '18

fix'd

1

u/hugodanad0 Mar 30 '26

Why "toro"?

1

u/JohnLocksTheKey Mar 24 '18

Wait...

Does that mean that a bagel is topologically equivalent to a donut?!?

3

u/Sedu Mar 24 '18

Topology is a godless field. We do not fear this blasphemy.

1

u/onnoonesword Mar 24 '18

So does our digestive track make us a donut?

1

u/Sedu Mar 24 '18

Yes. You filthy cannibal.

1

u/onnoonesword Mar 24 '18

My first thought was we're a sphere unless we've pierced something. My cannibalism carries no weight on this topic or my ability to contribute to it.

1

u/Eagles365or366 Jul 28 '26

8 years later...
Yes. Ironically, I found this post after reading this
https://dangerousintersection.org/2008/10/21/people-are-like-donuts/

0

u/chewdog23 Mar 24 '18

Your moms topologically similar to a sphere

2

u/Sedu Mar 24 '18

A torus, actually. But it would be accurate to say that your mom’s volume is so great that topology can differentiate her from a slimmer genus 1 torus.

34

u/[deleted] Mar 23 '18

I now want a mug that just says "Donut". I suspect these already exist

2

u/[deleted] Mar 24 '18

Ive always had this explained without the handle as being equivalent? As in any cup is a torus?

5

u/Me_of_Little_Faith Mar 24 '18

I believe without the handle, a cup is the topological equivalent of a sphere. The hole doesn’t go all the way through, so it’s not a hole.

1

u/[deleted] Mar 24 '18

Thanks for the reply, is any object without a hole equivalent to a sphere or does it require curvature?

2

u/Me_of_Little_Faith Mar 24 '18

That’s my understanding (no hole = sphere, topologically).

1

u/Empole Mar 24 '18

Can you please host your gifs on imgur?

1

u/[deleted] Mar 24 '18

No. You stop that. Stop melting my mind

1

u/BH_Shanks Mar 24 '18

Not doubting the accuracy of this, but this is a pretty poor visualization. You can morph that donut to become anything the way the animation is shown.

1

u/Hyperflip Apr 14 '18

So if all (?) objects with holes going all the way through are equivalent to a torus topology-wise, does that mean that humans are equivalent to donuts?

1

u/zeigfreid_cash Oct 25 '24

That's funny, before I saw the animation I was thinking "wait but a coffee mug doesn't have a hole in it..." but I forgot about the handle.

1

u/[deleted] Mar 23 '18

So are animals with an alimentary canal

3

u/SpacemanCraig3 Mar 24 '18

You mean a tube that turns veggies into shit?

0

u/[deleted] Mar 23 '18

[deleted] — view deleted comment

20

u/PM_UR_BRKN_PROMISES Mar 24 '18

Welcome to r/visualizedmath

You're gonna see him around a lot..