Re: Peano's space-filling curve

From: John Morgan (john.morgan_at_REMOVECAPSataraxie.fr)
Date: 06/07/04 

Date: Mon, 7 Jun 2004 14:42:44 +0200

David C. Ullrich <ullrich@math.okstate.edu> wrote in message
news:l354c0974tb47j6gtbn56lk09gmdopqfgk@4ax.com...
> On Sat, 5 Jun 2004 13:50:34 +0200, "John Morgan"
> <john.morgan@REMOVECAPSataraxie.fr> wrote:

> >For example, in [0,1]^2 the Peano curve will
> > pass through (1/3, 1/3) but not (1/3, 2/3).
>
> Yes it does pass through that point.

I have mentioned two points above. If you mean (1/3, 1/3)
it's just what I said. But if you mean (1/3,2/3) you will
need to explain to me why. The geometry screams out that it
does not. Indeed every recursion of the Peano construct
"creates" further points with rational coordinates that
will never be crossed. Does the algorithm defining the
drawing of the curve change somehow as one passes from the
finite to the infinite?

> The answers have been given many times, very clearly.
> Why you're unable to glean them is hard to say.

Simply, because your generalisation "very clearly" actually
applies to the limited perspective that you and your fellow
mathies have. I'm not going to re-iterate any of what I have
said before about this. It's in Google.

> ><snip>

> >This must have an inverse f^-1:, a bijection, and I
> >wonder why this inverse won't map [0,1] to [1,0]^2.
>
> That map does have an inverse, and the inverse does
> map [0,1] onto [0,1]^2.

So if there is a bijection from[0,1] to [0,1]^2 why did
nobody say so at the beginning of all of this. I was told
there was a surjection, but no mention of a bijection.

> > Is it something to do with the next bit concerning
> > identities and which I don't yet fully understand
> > though I'm working on it.
>
> This is funny. A few posts ago Grubb said he hoped
> he'd cleared something up, I speculated that he
> hadn't, and you said yes he had.
> But it's very clear from this that it hasn't.

Once more it seems things are "very clear" to you. What Dan
Grubb said was he hoped he'd cleared "some things up", which
is not quite the same as "cleared something up". You may be
the king pin mathy on this group but that actually seems to
work to the disadvantage of both of us. You might try
leaving some of the people whose posts I reply to, to answer
for themselves.

> Two facts:
>
> (i) There _is_ a bijection from I^2 onto I. Its inverse
> _is_a bijection from I onto I^2. Such a function
> _cannot_ be continuous.
>
> (ii) There also _is_ a continuous function mapping I onto
> I^2. It is not 1-1, hence not a bijection.

You continue to assume that I can instantly see the truth
behind these bald statements, despite the fact that I tell
you I am finding it difficult to do so. I interpret the
above as follows: (i) assures us that line and square have
the same cardinality, while (ii) guarantees that the Peano
curve, that conforms to (ii), fills the square because of
(i). If that's wrong, then put me right.
>
< snip>
>
> In particular you should note that while what he said
> is exactly correct, it in no way contradicts what others
> have told you about the irrelevance of AC to the truth
> of (i) and (ii) above - "this fact" is a different fact.

I don't recall I said anything about "this fact" either
way. Actually, I have only just registered the fact that
there are bijections both ways due to being "told"
previously that there was a surjection from [1,0] to
[1,0]^2. And before you say "yes that's true" I had
initially assumed that the fact I was being offered a
surjection implied that there was no bijection. Please, all
of you, try to get your act together. This might help me
arrange mine likewise.

Cheers

John

Relevant Pages

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