;     Date: Mon, 07 Jun 2010 22:31:44 -0400
;     From: Jim Muth <jamth@mindspring.com>
;  Subject: [Fractint] FOTD 08-06-10 (One Froggy Fractal [7])
;       Id: <1.5.4.16.20100607223302.2a5f056a@pop.mindspring.com>
; ---------
; 
; FOTD -- June 08, 2010 (Rating 7)
; 
; Fractal visionaries and enthusiasts:
; 
; The Klein bottle comment at the end of a recent FOTD discussion 
; was totally in jest, as are most of my closing comments, with a 
; few exceptions.
; 
; Since the true Klein bottle is an abstraction, it would hold 
; zero gallons of presumably 3-D water, which itself would be an 
; abstraction in 4-D space.  The familiar 3-D Klein-bottle model 
; could hold water, but the water would not be enclosed.  It would 
; merely be a puddle sloshing around in a basin of the single-
; sided surface.
; 
; The Klein bottle is actually mis-named.  It is not at all a 
; bottle with an inside that could be sealed.  It is actually a 
; two-dimensional surface with no edges or breaks, which twists in 
; 4-D space in such a manner that its apparent two sides are 
; actually different parts of the one and only side.
; 
; A way to almost picture the true Klein bottle is to imagine a 
; cylinder like a length of garden hose bent into a circle, with 
; the ends joined to form a donut (torus).  All we need to do is 
; find a way to gradually turn the hose inside-out without making 
; cuts, so that halfway around the loop the inside of the hose 
; finds itself on the outside.  It's impossible in 3-D space but 
; simple in 4-D space, where the 2-D surface may rotate in place 
; on itself.
; 
; To see an analog of the problem, take a broad, short rubber 
; band, cut it, twist it, and rejoin the edges so that it forms a 
; flexible Mobius strip.  Then squash the strip between two panes 
; of glass in an effort to flatten it into two dimensions.  It 
; will become apparent that a true Mobius strip must intersect 
; itself when it is squeezed into its model in 2-D space, just as 
; a 4-D Klein bottle must intersect itself when squeezed into its 
; model in 3-D space.
; 
; There is also a 'figure-8' model of the Klein bottle, which is 
; less well known but actually a bit closer to the real thing.
; 
; The FOTD is about fractal geometry and not topolgy.  So let's 
; get on to today's 7-rated image.
; 
; I named today's image "One Froggy Fractal" because of the vague 
; frog-like shape near the center.  The whole image is a close up 
; of the area where the two prominent minibrots in yesterday's 
; image almost merge.  In addition to reminding me of a frog, the 
; image bears a striking resemblance to a dwarf galaxy such as 
; what one might see in a deep sky photograph taken by the space 
; telescope.
; 
; The calculation time of over 10 minutes might seem a bit slow.  
; The way to save time is to visit the FOTD web site at:
; 
;        <http://www.Nahee.com/FOTD/>
; 
; and view the finished image there.
; 
; Cooler weather moved into Fractal Central on Monday, much to the 
; fractal cats' appreciation.  The partly cloudy skies, lower 
; humidity and temperature of 73F 23C made things pleasant for all 
; concerned.  My day was acceptable, as was FL's day also.  The 
; fractal cats were happy that no intruding tomcats tried to break 
; in.  The next FOTD will be posted in 24 hours.  Until then, take 
; care, and what results in 5-D space when a Klein bottle is given 
; an extra twist in 4-D space?
; 
; 
; Jim Muth
; jamth@mindspring.com
; jimmuth@aol.com
; 
; 
; START PARAMETER FILE=======================================

One_Froggy_Fractal { ; time=0:10:43.29-SF5 on P4-2000
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frm:FinDivBrot-2   { ; Jim Muth
z=(0,0), c=pixel, a=-(real(p1)-2),
esc=(real(p2)+16), b=imag(p1):
z=(b)*(z*z*fn1(z^(a)+b))+c
|z| < esc }

; END PARAMETER FILE=========================================
; 
; 
; 
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