Learning Your Way AroundThe Spiraling RuleWhen you explore any of the valleys in between the main body and the circular bulbs, you will find spirals.
The four images below show a sequence of zooms into the main valley that divides the 'head' and the 'body' of the Mandelbrot Set. This area is often
called the Seahorse Valley.
The deeper we venture down the crevice of the Seahorse Valley, the more tightly twisted the spirals become. We can try to zoom into the
spot where the head and the body join, at the horizontal X-axis, but we can never get there. The following four images show a variety of
spirals that come from successively deeper down the Seahorse Valley, and which twist tighter and tighter. Where the Seahorse Valley meets the X-axis,
the spirals twist infinitely tightly! This Spiraling Rule is universal - it applies to all the valleys underneath all the bulbs on the Mandelbrot Set. Activity: Explore the valley between the 'Period 3' bulb, above, and the main body. The spirals to the left of the junction point where the bulb joins the body twist in one direction, whereas the spirals on the right of this point twist in the other direction. Verify this by zooming into spirals in the valleys on both sides of the junction point. Verify that the same twisting rule applies in the valleys of the Period-3 bulb, i.e. the spirals are more twisted the deeper you go down the valley. Why do think we call the Period-3 bulb by that name? We'll learn more about the periods of the various bulbs in a few lessons, but first, we'll learn about Bifurcation... |
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