Math Adventures/Fibonacci and the Golden Ratio

These activities introduce the Fibonacci numbers, the golden ratio, and their interrelationship. These have many applications in nature and mathematics.

Italian mathematician Fibonacci was studying the growth of a rabbit population based on idealized assumptions when he wrote the sequence 0, 1, 1, 2, 3, 5, 8, 13, …. The numbers making up this sequence are now called Fibonacci numbers.

Activities

What is the next number in the Fibonacci sequence shown above? (Answer, 21)

What is the general rule for obtaining the next number in the sequence? (Answer, the next number in the sequence is the sum of the two preceding numbers)

Write out the first 15 terms in the sequence. (Answer, the sequence begins: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377 ,610, 987, 1597, 2584, 4181, 6765 … )

Tiles and Spirals

Fibonacci Tiles

Create a series of tiles based on the Fibonacci numbers as follows:

• Begin with a piece of graph paper printed with a grid pattern.
• Select a grid square near the center of the paper. Outline that grid and label it “1”.
• Select a grid next to the first one. Outline that grid and label it “1”. These two outlined grids now form an edge of length two. Use this edge as one side of a square. Outline that square and label it “2”.
• These outlined squares now provide an edge of length three. Use this edge as one side of a square. Outline that square and label it “3”.
• Continue outlining and labeling adjacent squares of sizes 5, 8, 13, 21, and continue the sequence as long as there is space on the paper.
• The resulting diagram should look like the tile diagram shown here on the right.
• Draw an arc connecting opposite corners of each square to create the Fibonacci spiral, shown in the diagram on the right.

Fibonacci Spiral

Natural Occurrences

Yellow chamomile head showing the arrangement in 21 (blue) and 13 (aqua) spirals.

Fibonacci sequences appear in biological settings, such as branching in trees, arrangement of leaves on a stem, the fruitlets of a pineapple, the flowering of artichoke, an uncurling fern and the arrangement of a pine cone, and the family tree of honeybees.

The image on the right shows the head of a yellow chamomile with the arrangement of 21 and 13 spirals highlighted.

In botany, the word phyllotaxis describes the arrangement of leaves on a plant stem. Phyllotactic spirals form a distinctive class of patterns in nature. Many of these spirals exhibit characteristics described by Fibonacci numbers.

Study the spirals occurring in each of the following plants and count the number of leaves or petals in each .

How often do the Fibonacci numbers occur?

• Aeonium,
• Succulent plants, especially aloe polyphylla,
• Conifer cones,
• Sun Flowers, and
• Artichoke.

The Golden Ratio

Line segments in the golden ratio

A simple geometric construction, shown here on the right, displays an important quantity known as the golden ratio. This ratio has been studied since ancient times and it has a fascinating relationship to the Fibonacci numbers.

From the diagram we see the golden ratio can be expressed by solving the algebraic formula: (a + b) / a = a/b for the ratio.

Mathematicians use the Greek letter phi ($\varphi$ or $\phi$) to represent the golden ratio.

Solving the equation for $\varphi$ results in an irrational number with a value of: $\varphi ={\frac {1+{\sqrt {5}}}{2}}=1.6180339887\ldots .$

Natural Occurrences

A golden rectangle has golden ratio proportions. Many consider this proportion aesthetically pleasing and shapes approximating the golden rectangle appear often in architecture and art.

Relation to the Fibonacci Sequence

It is remarkable that sequential Fibonacci numbers approximate the golden ratio.

Carry out this activity to explore the convergence of Fibonacci numbers to the golden ratio.

1. Start with the two consecutive Fibonacci numbers 8 and 5.
2. Recall the golden ratio is approximately 1.6180339887
3. Calculate the ratio 8/5. (Answer 1.6)
4. Notice the ratio of 8/5 is a rough approximation of the golden ratio.
5. Now select the next two Fibonacci numbers, 8 and 13.
6. Calculate the ratio 13/8. (Answer 1.625)
7. Notice the ratio of 13/8 is even closer to the golden ratio. How close is it?
8. Continue to choose adjacent Fibonacci numbers, compute their ratio and compare to the golden ratio.
9. Does the ratio converge toward the golden ratio? How close does it get? Are successive ratios greater than or less than the actual golden ratio? (Answer, they alternate greater and lesser than the golden ratio.)