Wanted: punch line for joke

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ametheus
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Wanted: punch line for joke

Postby ametheus » Tue Apr 20, 2010 8:18 pm UTC

A while ago, when I was on Math Camp¹, I had a really weird math dream. I awoke with a question in my head that sounded just like the beginning of a very funny, nerdy joke in a genre comparable to "changing a light bulb". However, no matter how often I tried, I've never been able to dream of a punch line to the joke. Maybe someone with more imagination than me can help me out, and finish it for me?

The question is:

How many topologists² can you fit inside a soda can?

Looking forward to your responses.

Thijs-Johan
Department of Mathematics, Leiden University


¹: Yes, this is a real thing: http://www.vierkantvoorwiskunde.nl/
²: A topologist is someone who specializes in the field of Topology. Note that any topologist would likely interpret the outside of the soda can as the inside, therefore the can would contain every topologist in the universe. This comment is in itself an excellent suggestion for an answer, that, in most audiences, would at least produce a titter or two. However, anyone familiar with the study of topology would think this rather obvious. I'm looking for an answer slightly more creative than this.

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Qaanol
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Re: Wanted: punch line for joke

Postby Qaanol » Tue Apr 20, 2010 8:57 pm UTC

ametheus wrote:How many topologists can you fit inside a soda can?

Spoiler:
Depends whether the cover is open.

or (variation on the same theme)
Spoiler:
None, because it has an open cover.
Last edited by Qaanol on Wed Apr 21, 2010 12:01 am UTC, edited 5 times in total.
wee free kings

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z4lis
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Re: Wanted: punch line for joke

Postby z4lis » Tue Apr 20, 2010 9:47 pm UTC

How many topologists can you fit inside a soda can?


Spoiler:
I'm not sure, but if you'd like to find out, I think I saw a few compactification theorems that might be useful.
What they (mathematicians) define as interesting depends on their particular field of study; mathematical anaylsts find pain and extreme confusion interesting, whereas geometers are interested in beauty.

letterX
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Re: Wanted: punch line for joke

Postby letterX » Tue Apr 20, 2010 10:55 pm UTC

Spoiler:
Well, if you cut up and rotate Banach and Tarski in just the right way?

Both of them.

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Quaternia
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Re: Wanted: punch line for joke

Postby Quaternia » Tue Apr 20, 2010 11:37 pm UTC

Spoiler:
With a little (Dehn) surgery I'm sure we can fit a few...
Yakk wrote:hey look, the algorithm is a FSM. Thus, by his noodly appendage, QED

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Re: Wanted: punch line for joke

Postby skeptical scientist » Wed Apr 21, 2010 5:36 am UTC

I came up with one similar to Qaanol's, but Z4lis was the only one who made me chuckle.
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Yesila
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Re: Wanted: punch line for joke

Postby Yesila » Wed Apr 21, 2010 5:56 am UTC

avoiding the math punchlines I'll just go with the:

Are we talking 12 oz soda or those wimpy 8oz cans?

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Yakk
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Re: Wanted: punch line for joke

Postby Yakk » Wed Apr 21, 2010 4:44 pm UTC

If we where to open the can, there would be no inside or outside, so clearly we must deal with a closed can. To determine the topological properties of the can, however, we can imagine opening the can by rotating it by pi/2. When we do so, bubbles form in a continuous process inside as they grow from the infinitesimal. Without loss of generality, we'll presume that the bubbles form an open cover of the can -- and, as a finite quantity of bubbles leaves the can, there must be a finite sub-cover. Thus the can is compact. As compacted cans are recycled, this implies that the fundamental group of the 4-can is trivial. If we then show that the can can can can can be path-connected, and we presume the axiom of choice, we can then show that the can can can can can be deformed into the surface of the can can can can can if we can go that far. And as the following diagram commutes:

Code: Select all

 
            ...         ...           ...                ...
             |           |             |                  |
      s_0    v  s_1      v   s_2       v     s_3          v       s_4
  ... -->   Can  -->  Can Can --> Can Can Can -->  Can Can Can Can --> ...
             | d_1       | d_2         | d_3              | d_4
             v           v             v                  v
  ... <--  Glug  <--  Gluglug <-- Glugluglug  <--  Gluglugluglug   <-- ...
      c_00   |   c_01    |    c_02     |      c_03        |        c_04
             v           v             v                  v
            ...         ...           ...                ...

the answer is 6.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

greengiant
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Re: Wanted: punch line for joke

Postby greengiant » Wed Apr 21, 2010 7:25 pm UTC

How about
Spoiler:
Pah. Topologists in a soda can's easy. The real trick's getting Felix Klein into a bottle.

Or
Spoiler:
Plenty as long as you act quickly. After all, the inside of a soda can's a shrinking space. - Been a while since I did any topology but I think any compact space is a shrinking space so the inside of a (sealed) soda would work

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Govalant
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Re: Wanted: punch line for joke

Postby Govalant » Wed Apr 21, 2010 8:33 pm UTC

Maybe you all thought of this and discarded it. But how about..

Spoiler:
All of them!
Now these points of data make a beautiful line.

How's things?
-Entropy is winning.

ametheus
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Re: Wanted: punch line for joke

Postby ametheus » Tue May 11, 2010 11:20 am UTC

Yes, I did; it's in my original post.

I'd like to thank everyone for their replies, I think I'm going with Yakk's answer if that's okay with him.

Cheers,

Thijs

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Yakk
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Re: Wanted: punch line for joke

Postby Yakk » Tue May 11, 2010 8:09 pm UTC

Sure? But do clean it up. I got a bit silly at the end.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

ametheus
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Re: Wanted: punch line for joke

Postby ametheus » Tue May 11, 2010 8:16 pm UTC

I like silly; it's sort of the point.

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4=5
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Re: Wanted: punch line for joke

Postby 4=5 » Fri May 14, 2010 7:23 am UTC

Yakk wrote:Sure? But do clean it up. I got a bit silly at the end.

I'm having trouble parsing "the can can can can can". Could you explain what the first 4 cans mean?

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Yakk
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Re: Wanted: punch line for joke

Postby Yakk » Fri May 14, 2010 3:42 pm UTC

I took the term 4-can (the 4 dimensional can), and expanded it to a can can can can (4 cans). I then used the word can immediately afterwards to make it sillier.

To fully ruin the joke, such as it is:
Spoiler:
* If we where to open the can, there would be no inside or outside, so clearly we must deal with a closed can.
This is a pun on the various meanings of open and closed in cans and in topology
* To determine the topological properties of the can, however, we can imagine opening the can by rotating it by pi/2.
Again, open pun, plus rotating by pi/2 (which changes no topological properties usually), referring to opening the tab at the top.
* When we do so, bubbles form in a continuous process inside as they grow from the infinitesimal.
A pun on continuous, and a visual joke relating bubbles to open sets.
* Without loss of generality, we'll presume that the bubbles form an open cover of the can
WLog joke, and bubbles being open sets forming an open cover setting up:
* -- and, as a finite quantity of bubbles leaves the can, there must be a finite sub-cover.
A pretty standard low-level topological proof, with a misunderstanding of what "finite sub-cover" means, which concludes with:
* Thus the can is compact.
This sets up the next pun
* As compacted cans are recycled, this implies that the fundamental group of the 4-can is trivial.
Here we make a pun on compact and cans. Then I made a pun on recycled and fundamental groups. The 4-can is pulled out of thin air, so I can make a pointed sequence of bad puns:
* If we then show that the can can can can can be path-connected, and we presume the axiom of choice, we can then show that the can can can can can be deformed into the surface of the can can can can can if we can go that far
Which is a sequence of puns about the Poincaré conjecture, plus can/can puns. Doing the can can while reading the above would be going to far, if I can make that claim. The "if we presume the axiom of choice" is both (I think) actually required by the conjecture, and a pretty common requirement for a proof in topology.
* And as the following diagram commutes:
This is an algebraic topology/category theory joke, where you have "the following diagram commutes" followed by a really strong statement that (at first glance) often appears like it came out of left field.

The diagram itself contains a pun or two. The d_r arrows are drink arrows, the s_n arrows are tin-increasing arrows, and the C_0n arrows are puns on C02 (ie, carbonation going down as you glug away at the drink).

The form of the diagram is pretty similar to various category theory/algebraic topology diagrams that actually do prove (or describe requirements for) surprisingly strong statements.

* the answer is 6.
6 has no meaning in this context. 0 would probably have been more amusing, as would 4. "the answer is clear" would also have been better.

I think I caught all of my silliness above?
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.

ametheus
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Re: Wanted: punch line for joke

Postby ametheus » Fri May 14, 2010 7:54 pm UTC

Really? I thought you meant the Canonical map (which is often abbreviated with "can"), and had applied it four times on the can, thus resulting in can[can[can[can[can]]]]. The quotient groups of two of such entities are quite commonly standard nonabelian groups such as D_4.
This way, your diagram actually almost makes sense. (Apart from it being absolute nonsense, of course.)


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