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Math Lesson 4.2.5 - Applications of Ratios in Practice - The Golden Ratio and Fibonacci Numbers

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Welcome to our Math lesson on Applications of Ratios in Practice - The Golden Ratio and Fibonacci Numbers, this is the fifth lesson of our suite of math lessons covering the topic of Rates. Applications of Ratios and Rates in Practice, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Applications of Ratios in Practice - The Golden Ratio and Fibonacci Numbers

Besides the applications of ratio discussed in the previous tutorial, there is another important and special application of ratio in practice, known as the golden ratio. Thus, if we have two numbers a and b (where a is the biggest) which meet the condition

a + b/a = a/b = φ

we say that a and b form a golden ratio, expressed by the letter φ.

The golden ratio is a constant number, whose value is

φ = 1 + √5/2 = 1.6180339887

Since √5 is irrational, the digits after the decimal point continue on forever without repeating.

Let's use the properties of fractions to find a possible relationship between the numbers a and b involved in a golden ratio. Thus, we can write

a + b/a = a/b
a/a + b/a = a/b
1 + b/a = a/b
1 = a/b - b/a
1 = a2 - b2/a × b
a2 - b2 = a × b

We will explain in the algebra section that

a2 - b2 = (a - b) × (a + b)

Therefore, substituting in the previous equation, we obtain

(a + b) × (a - b) = a × b

Hence, the numbers forming a golden ratio must meet the condition "sum × difference = product".

Example 5

A two-storey building forms a golden ratio when the ground floor (the highest from the two) is 5.2 m high. What is the height of the first floor?

Solution 5

We have a = 5.2 m. Since

a + b/a = a/b ≈ 1.618

we obtain for the height of the first floor:

5.2/b ≈ 1.618
b = 5.2/1.618
≈ 3.21 m

Obviously, it is very difficult to guess which numbers meet the condition to be in a golden ratio. Most numbers qualified for this are not integers, which makes finding the values of a and b an impossible task. This issue was addressed by Fibonacci - an Italian mathematician, who discovered a special sequence of numbers, where the next number is obtained by the sum of the previous two. The first numbers of Fibonacci sequence is

Fibonacci sequence = {1,2,3,5,8,13,21,34,55,89}

For example, 1 + 2 = 3; 2 + 3 = 5; 3 + 5 = 8 and so on.

But, what is the relationship between the golden ratio and Fibonacci numbers? Well, the more the numbers in the Fibonacci sequence increase, the closer they go to the value of the golden ratio. Thus,

1 + 2/2 = 3/2 = 1.5
2 + 3/3 = 5/3 = 1.666
3 + 5/5 = 8/5 = 1.6
5 + 8/8 = 13/8 = 1.625
8 + 13/13 = 21/13 = 1.6153
13 + 21/21 = 34/21 = 1.6190
21 + 34/34 = 55/34 = 1.6176
etc.

As you see, the values get closer and closer to 1.6180339887, which is the true value of the Fibonacci sequence.

Items that form a golden ratio between them produce a very fascinating view that has been used since antiquity in art and construction. This is because the building structures that form a golden ratio are stronger and more stable. We can find the golden ratio in a number of things, including:

  • In plants: You can find the golden ratio in the spiral arrangement of leaves (called a phyllotaxis) on some plants, or in the golden spiral pattern of pinecones, cauliflower, pineapples, and the arrangement of seeds in sunflowers.
  • In art: In many cases, artists have been inspired by the aesthetics of the golden ratio and incorporated it into their works.
  • In architecture: The Parthenon in Greece incorporates the golden ratio in many of its design elements. In the twentieth century, Swiss architect Le Corbusier used the golden ratio in his Modulor system for the scale of architectural proportion. The United Nations Secretariat Building in New York City was designed using the golden ratio: the size and shape of the windows, columns, and some sections of the building are based on the golden ratio.

Example 6

A piece of furniture is 233 cm high. How wide must it be to form the nicest view? (Width must be shorter than height.) Use the Fibonacci series to find the missing dimension.

Solution 6

The nicest view is obtained when height and width form a golden ratio. From the Fibonacci series, we have 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233,

Therefore, the width must be 144 cm as

H + W/H = 233 cm + 144 cm/233 cm
= 377 cm/233 cm
= 1.61802

This value is very close to that of the golden ratio, so we obtain the nicest view of the furniture by using the above dimensions.

More Rates. Applications of Ratios and Rates in Practice Lessons and Learning Resources

Ratio and Proportion Learning Material
Tutorial IDMath Tutorial TitleTutorialVideo
Tutorial
Revision
Notes
Revision
Questions
4.2Rates. Applications of Ratios and Rates in Practice
Lesson IDMath Lesson TitleLessonVideo
Lesson
4.2.1What Are Rates? Clarifying Misconceptions
4.2.2Two Associated Unit Rates
4.2.3Rate of a Quantity Change
4.2.4Applications of Rates in Practice
4.2.5Applications of Ratios in Practice - The Golden Ratio and Fibonacci Numbers

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