Wikipedia:Reference desk/Archives/Mathematics/2012 February 18

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February 18

Topological relationship between a sphere and a cone?

slice of sphere and imaginary paraboloid. Dmcq (talk) 14:56, 19 February 2012 (UTC)

Specifically, for a story I'm writing, I'm curious if there's any special mathematical relationship between the surface of the sphere of Earth and the hollow stepped cone of Dante's Hell. Is there some clever topological way one can be folded to produce the other? Please to remember, I'm a lay questioner - keep it simple...

Adambrowne666 (talk) 07:58, 18 February 2012 (UTC)

Well, for the simplest topological relationship, homeomorphism, sharp edges and spikes don't matter, so you can smooth them out (our article has a picture of a coffee cup being folded and stretched to make a doughnut). You could easily stretch a closed stepped cone to make a sphere (you could even flip it, so the inside of the cone becomes the outside of the sphere, as long as you're talking about mathematical abstractions rather than solid rock). The problem is that the pit is, if I recall correctly, open at the top - certainly it is in pictures like this: File:Stradano Inferno Map Lower.jpg. You could flatten out the steps of the pit, and stretch it around into a kind of ball shape, but you'd still have a hole that you couldn't close. I don't think there's a way to fold that hole away (or to be more rigorous, I don't think there's a way to map every point on a solid sphere onto a punctured ball continuously), since you fundamentally change the properties of the shape (a ball with a hole in it doesn't have a clearly defined interior - a sphere does). Smurrayinchester 11:52, 18 February 2012 (UTC)

Thanks, Smurray - that's interesting - so you'd end up with a sphere with a hole; makes me think of the 19th Century speculation on the idea of there being an entry to the hollow Earth somewhere around the Arctic. I must admit, I'm being sort of plagiaristic here, referring to the great old Christopher Priest novel The Inverted World, where the protagonists' altered perceptions causes them to see the Earth and Sun etc as pseudospheres. Thanks again Adambrowne666 (talk) 13:14, 18 February 2012 (UTC)

If you allow an imaginary radius as you go up past the north pole you get a paraboloid going off to infinity there from $r={\sqrt {R^{2}-z^{2}}}$. Dmcq (talk) 15:37, 18 February 2012 (UTC)

Thanks, Dmcq, that's a sort of magnificent answer, but I can't see it - I see from your profile you're interested in the visualisation of mathematics - maybe you could take pity on me and try and depict it to someone mathematically shortsighted? (BTW, I'm fascinated by your link to EPI - gonna look into that further.) Adambrowne666 (talk) 05:07, 19 February 2012 (UTC)

How about this then? If Chariots of the Gods can make money there must be something in trying to sell Dante's hell at the poles with this. ;-) Dmcq (talk) 14:56, 19 February 2012 (UTC)

Ultracool! Thanks heaps. I can certainly use that.

Strange voting system

I'm not sure if I read about this somewhere or made it up. Does the following system have a name? Every eligible voter is also a potential candidate. All votes cast for a particular person X get "paid forward" to whoever X votes for. Anyone wishing to be elected votes for themself and receives all votes cast directly for them as well as any that were paid forward from other voters. Votes are paid forward until they arrive at a self-voter, and otherwise (if there's a loop of votes) they never get counted. Sound familiar to anyone? Staecker (talk) 21:37, 18 February 2012 (UTC)

Delegated voting, also known as liquid democracy, is close, although it doesn't quite involve the same pooling of votes that goes on here (people nominate themselves as candidates, rather than voting for themselves, though apart from preventing loops that is just a semantic quibble really). Smurrayinchester 21:58, 18 February 2012 (UTC)

Thanks! That's close enough to probably be what I was thinking of. Staecker (talk) 22:34, 18 February 2012 (UTC)

3y-2x=5+9y-2x

How to graph this equation? I just curious. — Preceding unsigned comment added by 65.92.151.169 (talk) 23:59, 18 February 2012 (UTC)

First solve for X or Y (in this case, only Y is possible, and even that will make for a rather dull graph). StuRat (talk) 00:04, 19 February 2012 (UTC)

Add 2x to both sides to get 3y=5+9y. Subtract 3y from each side and subtract 5 from each side to get -5=6y. Divide both sides by 6 and switch the sides to get y=-5/6. The graph will be a straight horizontal line at -5/6.--Mattmatt1987 (talk) 17:34, 19 February 2012 (UTC)

I was hoping they would do their own homework, after I told them what they needed to do. StuRat (talk) 22:26, 19 February 2012 (UTC)