Re: The Gordian knot


John Jones wrote:
Rupert wrote:
I don't follow you. I thought you said your string was an infinite
string to start with. I assumed that meant a linear string with no ends.

Sorry about that, the string has two ends. There is an ellipsis on the
end, so that instead of 1,2,3,4, (...) infinity
we have
1,2,3,4, (...) infinity (...)


What do you mean by a "random coiling"?

A random coiling means that the string is wound in a random manner.
That is, the infinite string (which has two ends) is not formulaically
wound. It does not, for example, coil within one half of the string and
then, by a single thread, enter the other half and continue coiling
there; or again, the string is not wound such that it enters a
hemisphere of the ball and after travelling within that hemisphere for
the distance of the radius of the sphere, returns to the other
hemisphere; nor is it necessarily wound around the circumference.

On any reasonable precise statement of the problem, I would say they
are equal.

The number of cut ends is infinite.

Actually, if we assume the string is compact and also make the
assumption I gave earlier, that's not true.

I thought that if the two ends of
the string (before cutting) were in one cut hemisphere, this would mean
that it had two more cut ends than the other hemisphere. But then how
would I be able to express this, if infinity plus two is infinity ?

There's a canonical injection from the ends of one hemisphere to the
ends of the other hemisphere. You could point out that in one case this
is not surjective.

.

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