>> I suppose if we're not talking about real, physical space, but about
>> some mathematical abstraction, it can have whatever properties we
>> define it to have - including the point you describe not being part of
>> the set.
>
> Well, sure. But if you're going to start screwing with the defintion of
> "unit square" rather than using the standard "set of points whose X and
> Y are both from 0 to 1", then the answer comes out of your definition
> and you don't have to ask.

Yeah, but "from 0 to 1" in which set? The integers? The rationals? The 
reals?

As you say, I guess the answer depends on how you define the question. 
(Perhaps we're not even talking about Euclidian geometry, so a unit 
"square" might not even be squarish...)

> Plus, I'm not sure that physical space is actually continuous, so it's
> possible such a square doesn't have a sqrt(2),sqrt(2) point either.

Ah yes, the "which mathematical abstraction actually describes the 
*real* world" argument. :-D

I gather space actually has a slight negative curvature, so it strictly 
isn't Euclidian...

>> Right. So you have to take some matter and mash it somehow?
>
> Basically. You can do it by taking some matter and slapping it together
> so forcefully it overcomes the non-gravitational forces. I'm not sure
> that's physically possible in the universe as it exists today.

Hmm, theoretical physics for the win!

>>>> But then, wouldn't that just mean that as soon as you remove the
>>>> force, it wouldn't be a black hole any more?
>>>
>>> No. Science!
>>
>> I don't know... Normally if you remove external force, pressure
>> decreases.
>
> It's not pressure. It's gravity.

I know that the way black holes normally form is because matter is 
crushed by gravity. But you said that to make a small black hole, you'd 
need an external force to crush it instead.

Also, I'm still not getting how you can have [gravitational] black holes 
of different sizes. Presumably the force required to crush matter is a 
constant, so...?

>> I was under the impression that light does not require a transmission
>> medium.
>
> It doesn't. But it does have different speeds in different media.

OK.

>> I don't understand.
>
> The speed of light in a diamond is less than half the speed of light in
> free space/vacuum.

Really? I didn't think it was such a big difference! o_O

> Hence, it's possible to move through a diamond at
> faster than the speed of light in a diamond.

Ah, I see.

> http://en.wikipedia.org/wiki/Cherenkov_radiation
>
> It's what gives the weird blue light coming out of nuclear reactors
> underwater - particles from the reactor hitting the water at speeds
> faster than the speed of light through water.

The real world is utterly fricking weird!

-- 
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*

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