where does fibonacci sequence occur in nature

using recording tools such as tables and branch diagrams (featured in the PowerPoints, and of your own creation in response to students’ needs) to support students’ search for patterns.This could include creating a family tree of direct relatives with class members physically acting out scenarios using people or materials.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.The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to differentiate include: The next largest bulb to the right is Period-8. Rotating to the left, you come to the next largest bulb, Period-5. The next largest bulb, which is the Period-3 bulb. Rotate to the left to the head, which is the Period 2 bulb. Starting with the largest, the main body is the Period-1 bulb. What is the angle in the Spiralizer between Period-5 and Period-8 that creates a Period-13 pattern? Īs you can see in the Mandelbrot Set below, the progression of periodicities in the Mandelbrot Set follows an interesting Pattern. 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 can verify this by counting the arms in the orbit diagram on the right. The next largest bulb in between the Period-5 and Period-8 bulbs, which is the Period-13 bulb. Click near the Period-8 bulb to zoom in, but make sure you can still see the Period-3 and Period-5 bulbs. It's the largest bulb in between these bulbs.

In the Mandelbrot Set at the top left, find the Period-8 bulb between the Period 3 and Period-5 bulbs. What is the angle in the Spiralizer between Period-3 and Period-5 that creates a Period-8 pattern? You should see many different higher order periodicities. (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.Įxplore the range of angles in the Spiralizer between 120 and 144. Note that it is distinct from the Period-5 pentagon pattern at 72 and 288 degrees. Try to find the Period-5 star pattern in the Spiralizer. You should see a 5-pointed star pattern in the orbit plotted on the right.Ĭlicking 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 aĥ-pointed star spiraling inward to a fixed point. The next biggest bulb to the left of the Period-3 bulb is the Period-5 bulb.Ĭlick inside the main body of the Mandelbrot Set just below the period 5 bulb. To zoom in a little closer to the edge of the Mandelbrot Set if you'd like. Now click in the Mandelbrot Set just below the Period-3 bulb (refer to the applet below if you've forgotten where it is.) -Click inside the left panel Set the angle in the spiralizer to 120 degrees and check the 'Connect Dots' button. Play with the Mandelbrot Set and the Spiralizer applets above to get a sense of how the periodic behavior occurs. Inward (this means the starting point is inside the Mandelbrot Set).Īpplets courtesy of Yevgeny Demidov and Andrei Buium. Sometimes the starting point causes the scale of the point to shrink as it orbits, and so it spirals Outward to infinity (this means the starting point is outside the Mandelbrot Set). Sometimes the starting point causes the orbit to expand, and then the point spirals Various periodic patterns, which shows up in the different periods of the bulbs of the Mandelbrot Set. Thus the point rotates through space and depending on the starting angle, it may orbit in Is to double the angle of the point to the origin, and to square the distance. And the way you square a complex number in (polar coordinates) Z n+1 = Z n 2 + C involves squaring a point in the 2-D complex plane. This is because the equation that creates the Mandelbrot Set, Involves rotating and expanding, just like the process graphically illustrated in the Spiralizer.

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

Where does fibonacci sequence occur in nature