Remote Space Learning - The Fibonacci Sequence

Lecture 02: The Fibonacci Sequence and Golden Ratio

The Fibonacci sequence is a series of numbers in which each term is the sum of the two preceding ones, starting from 0 and 1. This sequence grows progressively as follows: 0, 1, 1, 2, 3, 5, 8, 13, and so on. The Fibonacci sequence is closely related to the Golden Ratio, an irrational number approximately equal to 1.618. As the sequence progresses, the ratio of successive Fibonacci numbers converges toward the Golden Ratio. This unique ratio appears widely in nature, art, and architecture, creating visually pleasing proportions in structures like the Parthenon and patterns like spirals in shells. Together, the Fibonacci sequence and the Golden Ratio reveal an intrinsic harmony within natural forms and human design.

Leonardo Fibonacci, born around 1170 in Pisa, Italy—famously known for the Leaning Tower—was the son of Guglielmo and Alessandra Bonacci, a merchant family. His father's surname, Bonacci, translates to "calm" in Italian, likely reflecting Guglielmo's reputedly kind and straightforward nature. Fibonacci's name, often thought to mean "son of Bonacci" (figliu di Bonacci), honors this family background.

Fibonacci is celebrated as one of the most influential mathematicians of the medieval period. His major contributions include his renowned works Liber Abaci (Book of Calculation) and Liber Quadratorum (Book of Squares), which laid foundational concepts in number theory and popularized Hindu-Arabic numerals in Europe. Liber Abaci is particularly famous for introducing the Fibonacci sequence, a series that reveals profound mathematical properties and has applications in various natural and scientific phenomena. Through these works, Fibonacci left an enduring legacy in mathematics and beyond.

In Liber Abaci, Fibonacci introduces the famous "Rabbit Problem" with the following scenario:

"A man places a pair of rabbits in an enclosure surrounded by walls. If each month every pair of rabbits produces a new pair, which in turn becomes productive from the second month onward, how many pairs of rabbits will there be at the end of one year?"

The Golden Ratio, symbolized by the Greek letter ϕ (phi) and approximately equal to 1.618, is a unique mathematical constant that has captivated thinkers for centuries due to its properties and aesthetic appeal. The ratio is defined by a simple yet profound relationship: when a line segment is divided into two parts, a and b, where a>b, the ratio of the whole segment to the larger part is the same as the ratio of the larger part to the smaller part.

The Fibonacci sequence begins with 0 and 1, and each subsequent term is the sum of the two preceding terms. This process creates a sequence that grows as follows:

F(0) = 0
F(1) = 1
F(2) = F(1) +F(0) = 1 + 0 = 1
F(3) = F(2) + F(1) = 1 + 1 = 2
F(4) = F(3) + F(2) = 2 + 1 = 3
F(5) = F(4) + F(3) = 3 + 2 = 5
F(6) = F(5) + F(4) = 5 + 3 = 8
F(7) = F(6) + F(5) = 8 + 5 = 13
F(8) = F(7) + F(6) = 13 + 8 = 21
F(9) = F(8) + F(7) = 21 + 13 = 34

The first 10 terms of the Fibonacci sequence are: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34.

The Fibonacci sequence is notable for its connection to the Golden Ratio (approximately 1.618). As we divide each term by its previous term, the ratio approaches the Golden Ratio:

F(2) / F(1) =1 / 1 = 1
F(3) / F(2) = 2 / 1 = 2
F(4) / F(3) = 3 / 2 = 1.5
F(5) / F(4) = 5 / 3 ≈ 1.6667
F(6) / F(5) = 8 / 5 = 1.6
F(7) / F(6) = 13 / 8 = 1.625
F(8) / F(7) = 21 / 13 ≈ 1.6154
F(9) / F(8) = 34 / 21 ≈ 1.6190

As the sequence progresses, this ratio converges towards 1.618, the value of the Golden Ratio.

Fibonacci Sequence Problems

Calculating Fibonacci Terms
Calculate the 12th term in the Fibonacci sequence. Start with F(0) = 0, F(1) = 1, where each term is the sum of the two preceding terms.
Sum of Fibonacci Terms
Find the sum of the first 10 terms of the Fibonacci sequence. List the terms as you add them up.
Identifying Fibonacci Ratios
In the Fibonacci sequence, calculate the ratio F(8) / F(7) and F(9) / F(8). How close are these values to the Golden Ratio (1.618)?
Fibonacci in Nature
A flower has 13 petals, and each petal follows a spiral pattern. If this pattern follows the Fibonacci sequence, how many petals would the next flower in the sequence have?
Fibonacci Numbers in Real Life
A plant’s leaves grow in a pattern that follows the Fibonacci sequence. If the first two leaves appear in the first month, and each month new leaves grow according to the sequence, how many leaves will the plant have by the end of the 6th month?

Golden Ratio Problems

Calculating the Golden Ratio in a Rectangle
A rectangle has a length of 34 cm and a width of 21 cm, which are consecutive Fibonacci numbers. Verify if this rectangle approximates the Golden Ratio by calculating the ratio length / width.
Golden Ratio in Proportions
The height of a painting is 24 inches, and the width is designed to follow the Golden Ratio. What should the width of the painting be (rounded to the nearest inch)?
Golden Ratio Spiral
Using squares of Fibonacci dimensions (1x1, 2x2, 3x3, 5x5, etc.), draw a spiral by connecting the corners of each square with an arc. How many squares (up to what dimension) do you need to reach a total length of 34 units?
Golden Ratio in Human Body
If the length of a person’s forearm is 21 cm and their hand length is 13 cm, calculate the ratio of forearm to hand length. Is this ratio close to the Golden Ratio?
Finding the Golden Ratio in a Division Problem
Suppose you divide a 100 cm line segment into two parts, a and b, where a>b. If the ratio a/b is approximately 1.618, find the approximate lengths of a and b.

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