Thought Experiments in Mathematics: Gabriel's Horn
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Thought Experiments in Mathematics: Gabriel's Horn
1–10 of 43 posts
Re: Thought Experiments in Mathematics: Gabriel's Horn
#2This answer on Math SE illustrates the concept of finite volume-infinite area quite nicely (http://math.stackexchange.com/a/14632/14643).
Note that the converse is proven only for surfaces of revolution and for differentiable functions. Finding a pathological counter example that violates these assumptions would be interesting.
Re: Thought Experiments in Mathematics: Gabriel's Horn
#3The link is an image of a print-formatted copy of the Wikipedia article https://en.wikipedia.org/wiki/Gabriel%27s_Horn which is more easily read directly on many devices.
Re: Thought Experiments in Mathematics: Gabriel's Horn
#4The most interesting thing about it is that at one point people thought it was a paradox.
Re: Thought Experiments in Mathematics: Gabriel's Horn
#5This answer on Math SE illustrates the concept of finite volume-infinite area quite nicely ( http://math.stackexchange.com/a/14632/14643 ). Note that the converse is proven only for surfaces of revolution and for differentiable functions. Finding a pathological counter example that violates these assumptions would be interesting.
My intuition is that the converse is impossible in general. There is an axis along which an object with infinite volume is infinite, and taking cross sections along this axis you should be able to prove that the object has infinite area.
Re: Thought Experiments in Mathematics: Gabriel's Horn
#6This answer on Math SE illustrates the concept of finite volume-infinite area quite nicely ( http://math.stackexchange.com/a/14632/14643 ). Note that the converse is proven only for surfaces of revolution and for differentiable functions. Finding a pathological counter example that violates these assumptions would be interesting.
The converse is impossible in any context where the isoperimetric inequality is applicable.
Re: Thought Experiments in Mathematics: Gabriel's Horn
#7This answer on Math SE illustrates the concept of finite volume-infinite area quite nicely ( http://math.stackexchange.com/a/14632/14643 ). Note that the converse is proven only for surfaces of revolution and for differentiable functions. Finding a pathological counter example that violates these assumptions would be interesting.
Section 7.1.3 of the article has a proof that such a counterexample cannot exist.
Of course, this being mathematics, changing the rules can produce a counterexample. A simple way of doing that is by declaring the inside of the shape the outside and vice versa.
I expect one could construct a counterexample around a black hole, too.
Re: Thought Experiments in Mathematics: Gabriel's Horn
#8I'm a big fan of the Alexander horned sphere, which is even weirder. The interior is simply connected (like the interior of a sphere) but the exterior isn't.
Re: Thought Experiments in Mathematics: Gabriel's Horn
#9The most interesting thing about it is that at one point people thought it was a paradox.
It took lots of very intelligent mathematicians to realize that infinities are weird.
Even today, it takes quite some schooling to come to terms with some of the _easy_ examples.
For example, I expect that easily over 95% of university graduates disagree with the statement "There are as many rational numbers as there are prime numbers.".
I even fear that is true for the easier to believe "0.9999999... = 1"
Re: Thought Experiments in Mathematics: Gabriel's Horn
#10The most interesting thing about it is that at one point people thought it was a paradox.
It took lots of very intelligent mathematicians to realize that infinities are weird. Even today, it takes quite some schooling to come to terms with some of the _easy_ examples. For example, I expect that easily over 95% of university graduates disagree with the statement "There are as many rational numbers as there are prime numbers." . I even fear that is true for the easier to believe "0.9999999... = 1"
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