Fractal visionaries and enthusiasts:

Most everyone is familiar with the so-called 'Let's make a Deal' paradox, which is really no paradox at all.  You do not have a 50-50 chance of making the correct choice when one of the other doors is eliminated, you have a 67-percent chance of winning by switching your choice to the remaining door.  The true mystery is the fact that it is so hard to explain to others why this must be true.  (See below for my attempt to explain it.)

For a real paradox, check today's image, which is a scene in the left-over debris in the southwest part of the largest bud of the minibrot on the west branch of the filament extending from the large period-3 north bud on the main bay of the Mandelbrot set.  The minibrot is in the process of morphing from a quadratic into a near-circular order-15 one.

So far, so good.  But things go quite awry when we start counting.  Normal, well-adjusted order-15 minibrots have 14 lobes and are surrounded by rings of 15 elements, but the minibrots in today's image are anything but normal.  begin with the minibrot in the upper left corner.  A careful count reveals the expected 14 lobes.  But now count the surrounding elements, and surprisingly, you will find 16, not the proper 15.

This is only the start of the craziness.  The large minibrot in the center of the scene is apparently a merging of two order-15 minibrots, and as such should have 28 lobes.  Unfortunately, when I counted, I found 29.  It's quite a mystery where the extra lobe comes from.  I have not yet tried to count the elements surrounding this merged minibrot, but I doubt that they equal the expected 30.

The name "Paradox Country" refers to this paradoxical craziness.

The rating of 7-1/2 is all I could do for the image.  The longer I see the colors, the harsher and muddier they appear.  If it were not for the math interest, the image would rate something more like a 6.

The calculation time of just under 2 minutes will pass quickly, bringing only minor restless eye syndrome.

As I wrote at the start of this discussion, the 'Let's Make a Deal' paradox is no paradox at all.  When you choose a door, you have a 2/3 chance of having chosen a losing door.  This means there is a 2/3 chance that one of the other 2 doors is the winning door.  When one of the other doors is shown to be a loser, nothing has changed with your original choice.  There is still a 2/3 chance that it is a loser, but now the 2/3 chance that one of the other 2 doors is the winner has been combined into one door, which now has a 2/3 likelihood of being the winner.  So switch your choice to the other door and your chance of winning will jump from 33 percent to 67 percent.

The clouds continued today here at Fractal Central, while the weather remained dry.  The fractal cats (of the escape-time variety) thought the temperature of 48F +9C should have been much warmer.

The fractal people made it through the day with only minor inconveniences.  Now that the commercial work is winding down, the things that once were larger inconveniences now seem less inconvenient.  The next FOTD will conveniently be posted in 24 hours.  Until then, take care, and always choose the other door.

Jim Muth
jamth@mindspring.com
jimmuth@aol.com


START PARAMETER FILE=======================================

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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=========================================