Ideas that are compelling, yet unhealthy.A Math Problem
Imagine you had an infinitely long, infinitely thin string with two ends, you take the tangle of string and put it inside a box, while leaving the two ends on the outside.
The Problem:
Define an infinitely long path between two points that stays within a finite volume. The path canot intersect itself. The location of each point on the path must be a function of the distance along it from either of the two points.
The Problem:
Define an infinitely long path between two points that stays within a finite volume. The path canot intersect itself. The location of each point on the path must be a function of the distance along it from either of the two points.
posted by indrax at 1:24 PM
6 Comments:
If it's infinite, how can it have ends?
if the string is infinitely thing it has no volume and can thus cover an infinite distance without intersecting itself.
It can have ends, and have them a finite distance apart, because it is infinitely tangled. If you start from either end, no matter how far you go, you will always find yourself somewhere in the tangle.
One tricky thing is also proving that the two ends are connected at some point in 'the middle'. This splits the problem in to two of the same problem again, which suggests some fractal solutions, But I don't think fractal solutions can satisfty the requirement of the position being a function of the distance.
This is a really great thought experiment, however i'd have to refer back to the second comment. I think you could perhaps remove the box from the experiment and still be left with something as equally compelling to prove. In fact, you'd most likely be referring to the works of Mr. Hawking himself. Regardless, you've got me thinking :)
No, the first commenter was correct. Something of infinite length -- by very definition -- cannot have an end.
Not true anonymous, For example,The perimiter of the Koch Snowflake is infinite, and any segment of it would also be infinite in length. The Snowflake would work for my prolem, but it is not a function, so there is no way to find the X-Y cordinate of the point 50 units along the snowflake.
fec:
I'm not sure how it would work without the box. I'd love to hear more.
Post a Comment
<< Home