If a fractal is symmetrical over the x-axis, and you
take the top half as one half the bank of a lake, and
the bottom half as the other, the line dividing them
in 2 is finite and completely straight. This seems to
be the case in The Mandelbrot Set (see Mike's link:
http://www.mindgames.com/mb.phtml): the midline would
be a finite horizontal straight line dividing the
fractal picture exactly in two. Thus we can see that
in the case of a river, just because the banks might
be of infinite length (and of a fractal pattern), it
does NOT mean the midline of the river must be.
Ok, now given that the banks of a river are not
completely symmetrical, but there is a convergence
here to a finite and measurable line since the
middleline can only exist at one point per each point
you move down the river, thus not abling it to have
fractal properties.
Of course this is only for rivers where the border is
in the river. Any river with one of the banks as the
border (ie old Red River) would naturally follow a
fractal pattern and have infinite length.
--- "Brian J. Butler" <
bjbutler@...>
wrote:
> On Thursday 17 April 2003 07:58 pm, you wrote:
> OK, I give up. And I give you an F in fractal
> geometry.
> BJB
>
> > I really have not misunderstood any of the points
> made in opposition to
> > measuring a physical boundary... I have only said
> that it can be done, and
> > it is done. I don't agree with the notion that
> there is infinite length in
> > a river that flkoiws only, say, 10 miles. That's
> an absurd notion. A
> > highway might be measured by its length along a
> centerline in the median,
> > while the distance along its outside shoulders may
> be different... still,
> > the distance is not infinite! We can prove that by
> arriving at our
> > destination.
> >
> > Once a high water mark is established, if that is
> the boundary, then it can
> > be measured. I didn't say it was easy to go 1,000
> miles along the banks of
> > the Red River (I believe the OK-TX boundary is the
> center of the channel,
> > which would be much less difficult to measure),
> but it can be done.
> >
> > At a human scale, the water line does not have to
> be measured around
> > molecules, or grains of sand, to be called
> accurate. As with my highway
> > example.
> >
> > -----Original Message-----
> > From: Brian J. Butler
> [mailto:bjbutler@...]
> > Sent: Thursday, April 17, 2003 4:18 PM
> > To: BoundaryPoint@yahoogroups.com
> > Subject: Re: [BoundaryPoint] OKTX --finally!
> figures
> >
> >
> > On Thursday 17 April 2003 05:04 pm, you wrote:
> > Let me trry one more time, and then if you still
> don't get it, I will give
> > up. Suppose a boundary is defined as the high
> water mark along a river
> > bank.
> > Further suppose the river bank is irregular, which
> should not be much of a
> > stretch if you have ever looked at a river bank.
> The irregularities exist
> > at
> > many scales, from broad bends measured in miles,
> to smaller meanders
> > measured
> > in hundreds of feet, to smaller gouges measured in
> tens of feet, to rocks
> > meausured in feet, to pebbles measured in inches,
> to small pebbles, ...
> > etc.
> >
> > Clearly if you consider smaller and smaller
> irregularities, the length of
> > the
> > river bank increases without bound, as illustrated
> by the two measurements
> > given for the OK-TX boundary. Since you think the
> length is bounded, it
> > implies that you stop measuring around features
> below some size threshold.
> > My
> > question to you is: At what scale do you stop
> measuring, and why? Please
> > actually think about the question.
> > BJB
> >
> > > It would really not go on like that. If the
> boundary is defined, one
> > > follows that definition. If OK-TX has been
> defined as a certain location
> >
> > in
> >
> > > the channel of the Red River, one would follow
> that and measure it as
> > > s/he goes. If the menaderings are part of the
> definition, that would be
> > > followed.
> > >
> > > The hard number really is out there. It is not
> infinite.
> > >
> > > -----Original Message-----
> > > From: Brian J. Butler
> [mailto:bjbutler@...]
> > > Sent: Thursday, April 17, 2003 1:55 PM
> > > To: BoundaryPoint@yahoogroups.com
> > > Subject: Re: [BoundaryPoint] OKTX --finally!
> figures
> > >
> > >
> > > On Thursday 17 April 2003 12:13 pm, you wrote:
> > > And following really small meanders
> > >
> > > Red River - -1234 miles
> > > East Panhandle line -- 133.6 miles
> > > North Panhandle line -- 167 miles (minus 2.2
> miles TXNM)
> > >
> > > Total -- 1532.4 miles
> > >
> > >
> > > And following really small meanders and medium
> sizes irregularities:
> > >
> > > Red River -- 2816 miles
> > > East Panhandle line -- 133.6 miles
> > > North Panhandle line -- 167 miles (minus 2.2
> miles TXNM)
> > >
> > > Total -- 3114.4 miles
> > >
> > > Etc.
> > >
> > > > Including only larger river bends:
> > > >
> > > > Red River -- 480.0 miles
> > > > East Panhandle line -- 133.6 miles
> > > > North Panhandle line -- 167 miles (minus 2.2
> miles TXNM)
> > > >
> > > > Total -- 778.4 miles
> > > >
> > > > Following the smaller meanderings of the
> rivers:
> > > >
> > > > Red River -- 726 miles
> > > > East Panhandle line -- 133.6 miles
> > > > North Panhandle line -- 167 miles (minus 2.2
> miles TXNM)
> > > >
> > > > Total -- 1024.4 miles
> > > >
> > > > Source: Texas Almanac
> > > >
> > > > --Joe
> > > >
> > > > (Will post some pics of my visit to Copperhill
> TN/McCaysville GA soon.
> > > > The border runs through a grocery store and a
> church)
> > > >
> > > >
> > > >
> > > >
> > > >
> > > > Your use of Yahoo! Groups is subject to
> >
> > http://docs.yahoo.com/info/terms/
>
> --
> Brian J. Butler
> BJB Software, Inc.
> 508-429-1441
> bjbutler@...
> http://www.bjbsoftware.com
>
__________________________________________________
Do you Yahoo!?
The New Yahoo! Search - Faster. Easier. Bingo
http://search.yahoo.com