The Golden Ratio (Phi) and the Fibonacci Sequence in Nature and Art

Modality: In-Person

Assignment: A Nature-Based Learning Exploration

The Golden Ratio (phi), and the Fibonacci sequence in nature.

1.  See the Golden Ratio.  On-line activity—See if you can imagine the golden ratio in the works of art of Leonardo Da Vinci in the following activity: 

2. Now, read the following overview of how the golden ratio and the Fibonacci sequence of numbers occur in nature.  

ACTIVITY:

Arrange visit (as class?) to collect artifacts, including those with sunflower or other budding flower petals, or those with coniferous trees (pine cones) to collect samples. Can you observe the Fibonacci sequence as described in the attached summary article?

3. ESTIMATE the ratio (phi), Euler’s number, by computing successive ratios of the numbers in the Fibonacci sequence:  Construct this special sequence by simply adding the previous two numbers in the sequence to get the next number.  Here is the start:

{0, 1, 1, 2, 3, ……} 

What number does the ratio of successive terms appear to approach? Continue computing ratios as far out as you can go! What do you notice about the ratios computed by divided each number in the sequence by the number preceding it? 

3. DRAW the mollusk spiral:  Using the sequence of numbers in the Fibonacci sequence, draw the famous “mollusk shell” spiral on lined graph paper by creating the box outline of the curve using successive lengths of the numbers of the Fibonacci sequence as each successive perpendicular side of the box outline. 

5. COMPUTE the exact value of (phi): Compute the value of the golden ratio exactly from the proportion defined from the ratio of the two parts that comprise a line segment, where the ratio of the long part to the short is the same as the ratio of the entire line segment to the long part. What quadratic equation do you form from this proportion?  Solve it! 

Student Work