Fibonacci Fractals

Fibonacci Numbers and the Mandelbrot Set

The Mandelbrot Set does not occur in nature. However, the mathematical patterns that produce the Mandelbrot Set do occur in a number of natural systems. We're going to explore the connections between the periodicities in the Mandelbrot Set and the periodicities we find with the Spiralizer. We saw in the last section that combining rotating and expanding in the Spiralizer created interesting periodic patterns, some of which resemble natural patterns in plants.

Now we'll see that the same sort of thing is happening to points in the Mandelbrot Set when they're iterated through the 2-Dimensional complex plane. The iteration process involves rotating and expanding, just like the process graphically illustrated in the Spiralizer. This is because the equation that creates the Mandelbrot Set, Z_n+1 = Z_n² + C involves squaring a point in the 2-D complex plane. And the way you square a complex number in (polar coordinates) is to double the angle of the point to the origin, and to square the distance. Thus the point rotates through space and depending on the starting angle, it may orbit in various periodic patterns, which shows up in the different periods of the bulbs of the Mandelbrot Set. Sometimes the starting point causes the orbit to expand, and then the point spirals outward to infinity (this means the starting point is outside the Mandelbrot Set). Sometimes the starting point causes the scale of the point to shrink as it orbits, and so it spirals inward (this means the starting point is inside the Mandelbrot Set).

Play with the Mandelbrot Set and the Spiralizer applets above to get a sense of how the periodic behavior occurs.

Set the angle in the spiralizer to 120 degrees and check the "Connect Dots" button. Now click in the Mandelbrot Set just below the Period-3 bulb (refer to the applet below if you've forgotten where it is.) <Alt>-Click inside the left panel to zoom in a little closer to the edge of the Mandelbrot Set if you'd like. The next biggest bulb to the left of the Period-3 bulb is the Period-5 bulb. Click inside the main body of the Mandelbrot Set just below the period 5 bulb. You should see a 5-pointed star pattern in the orbit plotted on the right. Clicking inside the bulb creates a stable Period-5 orbit, while clicking just inside the main body, but right next to the Period-5 bulb the orbits form a 5-pointed star spiraling inward to a fixed point.

Try to find the Period-5 star pattern in the Spiralizer. Note that it is distinct from the Period-5 pentagon pattern at 72 and 288 degrees. (You need to make sure to connect the dots to be able to see the difference.) The star pattern can be found at 144 and 216 degrees. Explore the range of angles in the Spiralizer between 120 and 144. You should see many different higher order periodicities. Find the angle in this range that creates a Period-8 pattern.

Questions:
What is the angle in the Spiralizer between Period-3 and Period-5 that creates a Period-8 pattern? [ ]

In the Mandelbrot Set at the top left, find the Period-8 bulb between the Period 3 and Period-5 bulbs. It's the largest bulb in between these bulbs. Note the Period-8 pattern in the orbit. <Alt>-Click near the Period-8 bulb to zoom in, but make sure you can still see the Period-3 and Period-5 bulbs. Click below the next largest bulb in between the Period-5 and Period-8 bulbs, which is the Period-13 bulb. You can verify this by counting the arms in the orbit diagram on the right.

Now try to find the angle in the Spiralizer in this same range between the Period 5-star (144 degrees) and the Period-8 star. You'll need to explore cerefully and use the arrow keys to adjust by .1 degrees to get it exactly.

What is the angle in the Spiralizer between Period-5 and Period-8 that creates a Period-13 pattern? [ ]

As you can see in the Mandelbrot Set below, the progression of periodicities in the Mandelbrot Set follows an interesting Pattern. Follow the bulbs in decreasing scale. Starting with the largest, the main body is the Period-1 bulb. Rotate to the left to the head, which is the Period 2 bulb. Rotate to the right and find the next largest bulb, which is the Period-3 bulb. Rotating to the left, you come to the next largest bulb, Period-5. The next largest bulb to the right is Period-8. The next largest bulb to the left is Period-13, and so on forever, and the next periodicity is the sum of the periodicities of the two bulbs before it.