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 reset=2004 type=formula formulafile=basicer.frm formulaname=FinDivBrot-2 function=recip passes=1 center-mag=+0.004236797608/-0.000422139251/2104673\ /1/85/0 params=1.99/-0.95/1000/0 float=y maxiter=2000 inside=255 logmap=9 colors=000Xzz_zzbzzezzczzazz`zzZzzXzzYzzqdUnbRk`Oh\ ZMeXKbVI_THXRHVPITMJQLLOLMQLKRLIUMGWOE_QCbSAeT8hU6\ kU4nU2oU5qU8rUBrUEsUHtUKuUNuUQqUSmAUiBWeBYcC_cDacE\ ccEecFgcGicHkcHmzzzzznzzcxyUvxOuwNswNrvMpuLouLpsKq\ pJrnJskIthItfHucGv`GwZFxWFyTEySEp_ZoZZnZZnZZmZZmZZ\ lYZkYZkYZjYZjYZeVW`SUXPScMQcKccHccEccBcc9cf3uh6qiB\ njGkkLhmOenSboWZp_WraTsdQtgPujUvlZogchchacmVcrOcvH\ czBczCczCczCczCczCczCczDczDczDczDczDczDcz5uz7sz8qz\ 9ozAmzCkzDizEgzFezGdzHezHfzHgzHhzHizHjzHkzHlzImzIn\ zIozIpzIqzIrzIszItzGqzFnzEmzDmzBmzAmz9mz8mz7mz5mz4\ mz3mz2mz1mz9mzHmzPhzWqzVrzVrzVrzVrzVrzVrzVrzVszVsz\ VszVszVszVszVszSqzQozOmzMmzKmzImzFmzDmzBzz9zz7zz5z\ z3zzHzzVzzhzzvzlwzkwzjwziwzhwzgwzgwzfwzewzdwzcwzcw\ zbwzawz`wz_wzzkzz`zzbzzczzezzfzzhzzizzjzzlzzmzzozz\ pzzqzzkzzfzz`zzWzzQzzLzzGzzIzzKzzMzzOzzQzzRzz3zz4z\ z8zzBzzEzzHzzKzzNzzRzzH0P } 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=========================================