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Science Forum Index  »  Math - Numerical Analysis Forum  »  Seeking for a dyadic solenoid image
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VK
Posted: Wed Mar 07, 2007 12:24 pm
Guest
If it's wrong NG, a pointer to the right one would be much
appreciated.

I'm seeking a graphics representation of the dyadic solenoid,
something like the 3-adic picture sample at <http://en.wikipedia.org/
wiki/Solenoid_group>

Google search didn't help so far:
<http://images.google.com/images?svnum=10&hl=en&gbv=2&q=dyadic
+solenoid&btnG=Search>

Any picture would go even if a handwriting scan, I'll make the final
image by my own on 3D editor.


If I understand correctly such solenoid encompasses all decimal
rationals representable as a finite binary sequence, so it is kind of
"projection of binary world as seen by a decimal observer". If so then
is it possible to have a binary rational which is not representable as
a finite decimal sequence? What would be the graphical representation
of the "projection of decimal world as seen by a binary observer"?
Sorry if I said something really stupid. I'm a linguist by my primary
education (math linguistics related "survival minimum" in math many
years ago on bachelor studies).
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Zbigniew Karno
Posted: Mon Mar 12, 2007 4:37 pm
Guest
On 7 Mar, 17:24, "VK" <schools_r...@yahoo.com> wrote:
Quote:
If it's wrong NG, a pointer to the right one would be much
appreciated.

I'm seeking a graphics representation of the dyadic solenoid,
something like the 3-adic picture sample at <http://en.wikipedia.org/
wiki/Solenoid_group

Google search didn't help so far:
http://images.google.com/images?svnum=10&hl=en&gbv=2&q=dyadic
+solenoid&btnG=Search

Any picture would go even if a handwriting scan, I'll make the final
image by my own on 3D editor.

If I understand correctly such solenoid encompasses all decimal
rationals representable as a finite binary sequence, so it is kind of
"projection of binary world as seen by a decimal observer". If so then
is it possible to have a binary rational which is not representable as
a finite decimal sequence? What would be the graphical representation
of the "projection of decimal world as seen by a binary observer"?
Sorry if I said something really stupid. I'm a linguist by my primary
education (math linguistics related "survival minimum" in math many
years ago on bachelor studies).


You can find some interesting description of the dyadic solenoid
as the intersection of a decreasing sequence of so called "glued
chains" (see Fig. 3 and 4 - only two stages of such construction)
in the paper "On pointed 1-movability and related notions" (Fund.
Math. 114 (1981) No.1, pp. 29-52) by Jozef Krasinkiewich available
at link

http://matwbn.icm.edu.pl/ksiazki/fm/fm114/fm11414.pdf

I would like to note also that at link

http://www.karlin.mff.cuni.cz/~pyrih/e/e2001v2/c/ect/node52.html

is a very suggestive description by Andrzej Gutek of the dyadic
solenoid as the quotient (identification) space of the Cantor
band C x I (where C is the dyadic Cantor set and I is the unit
interval).

Regards,
Z. Karno
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