06/29/2009, 08:35 AM
(06/28/2009, 02:31 PM)bo198214 Wrote:(06/27/2009, 11:50 PM)Kouznetsov Wrote: You have no need to deal with so exotic objects like sexp, in order to reproduce the same paradox. Consider the logatirhmic spiral x=log(t)cos(pt), y=log(t)sin(pt) .
For posiive p and real t, this spiral winds infinitely around sero.
"If we slowly continuously transform" this spiral to the straight line (just reduce t to zero), then you may wander, how the infinite number of turns of this spiral arond zero becomes finite (no windings at all).
Well there must be a critical point where it changes from infinite winding to finite winding.
we have a continuous transform h:[0,1]->region of CC, with h(0)=T and h(1)=S.
Where is this point t in the transform of T into S (can it visibly recognized?) that sexp(h(t)) is infintely winding while sexp(h(t')) is only finitely winding for t'<t.
Henryk, you reveal the misprint in my post. Sorry. I just corrected there
"(just reduce t to zero)" to "(just reduce p to zero)".
Now t parametrizes the set (curve),
and p is parameter we play in order to modify the "speed" of winding. Only at p=0, there is no winding, although at small p, it is difficult to count many turns, we need strong zooming in.
Do you see any pradox about the logarithmic spiral?