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Topic: Matheology § 160
Replies: 1   Last Post: Nov 24, 2012 3:51 PM

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Posts: 8,833
Registered: 1/6/11
Re: Matheology � 160
Posted: Nov 24, 2012 3:51 PM
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In article
WM <mueckenh@rz.fh-augsburg.de> wrote:

> Matheology § 160
> {{Another application of set theory?}} One of the remarkable
> observations made by the Voyager 2 probe was of the extremely fine
> structure of the Saturn ring system. [...] The Voyager 1 and 2
> provided startling images that the rings themselves are composed of
> thousands of thinner ringlets each of which has a clear boundary
> separating it from its neighbours.
> This structure of rings built of finer rings has some of the
> properties of a Cantor set. The classical Cantor set is constructed by
> taking a line one unit long, and erasing its central third. This
> process is repeated on the remaining line segments, until only a
> banded line of points remains. {{Materialized points are certainly not
> available in the Saturn ring system.}}
> [H. Takayasu: "Fractals in the physical sciences", Manchester
> University Press (1990) p. 36]
> http://books.google.de/books?id=NRYNAQAAIAAJ&pg=PA180&lpg=PA180&dq=Takayasu:+%
> 22Fractals+in+the+physical+sciences%22&source=bl&ots=-_jQrSNVTs&sig=ttuEDGX_63
> 81_T2AdLEBT8HIT20&hl=de&sa=X&ei=Yb6GT-iFDcvUsgao6ITPBg&sqi=2&ved=0CDMQ6AEwAQ#v
> =onepage&q=Takayasu%3A%20%22Fractals%20in%20the%20physical%20sciences%22&f=fal
> se
> Mandelbrot conjectures that radial cross-sections of Saturn's rings
> are fat Cantor sets. For supporting evidence, click each picture for
> an enlargement in a new window.
> http://classes.yale.edu/fractals/labs/paperfoldinglab/fatcantorset.html
> http://www.youtube.com/watch?v=Ztgqa_5vumI
> Regards, WM

While nature is full of fractals (in which a pattern on one scale is
repeated on apparently ever smaller scales) like the Cantor set
construction, in nature the number of possible smaller scales on which
such a pattern can repeat is always finite, unlike a purely mathematical
Cantor set, whether fat or not!

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