The Golden Ratio definition, or Golden Mean or Golden Section, is a ratio expressed by the decimal value 1.61803... It is an irrational number, like {eq}\pi {/eq} or e, meaning that it never terminates and never repeats. Like pi, it is expressed with a Greek letter phi - {eq}\phi {/eq}. The Golden Ratio is a proportional concept that can be used to describe the relative lengths of two line segments.
These line segments obey the Golden Ratio
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In this figure, we can see line segment AB and line segment BC. The ratio of the length of AB to AC is {eq}\phi {/eq}, the Golden Ratio. However, the ratio of segment BC to the whole segment AC is also {eq}\phi {/eq}.
Another interesting geometric construction involving the Golden Ratio is the Golden Rectangle.
A golden rectangle
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Begin with a rectangle whose sides obey the Golden Ratio. This is called a Golden Rectangle. Then, add a line segment separating the rectangle into a perfect square and another rectangle, and that new rectangle is another Golden Rectangle!
Successive golden rectangles with a Golden Spiral inscribed
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This process can repeat, and a curve drawn through the successive squares is called a Golden Spiral, or a spiral whose logarithmic growth factor is (you guessed it) {eq}\phi {/eq}.
Consider now another geometric object called a regular pentagon. If we construct a regular pentagon with side length a, and draw diagonals within that pentagon, then the length of any diagonal will be {eq}a\cdot \phi {/eq}. If that's not amazing enough, notice that if we draw every possible diagonal, we create a smaller regular pentagon inside the big one. The lengths of these sides are {eq}\frac{a}{\phi^2} {/eq}. However, another of the more interesting things about the Golden Ratio is that {eq}\phi^2 = \phi + 1 {/eq}, so {eq}\phi^2 = 2.61833... {/eq}, making this problem a little more easily solved. Finally, notice the triangle shaded in yellow within the diagonals. This is called a Golden Triangle, and it is a special isosceles triangle that has the unique property that if one of its base angles is bisected, the resulting smaller triangle is similar to the larger one.
Another fascinating place where the Golden Ratio appears is the Fibonacci Sequence. This is a mathematical sequence that begins with 1, 1, ... and then, each subsequent entry in the sequence is the sum of the previous two entries. So, the first two entries are 1, 1, the next is 2, followed by 2+1 = 3, then 3+2 = 5, and so on. The Fibonacci Sequence looks like this:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144...
So, what does this have to do with the Golden Ratio? Well, if we take ratios of adjacent numbers in the Fibonacci Sequence, as the sequence gets larger, those ratios approach {eq}\phi {/eq}!
Let's talk about how to calculate the Golden Ratio. There are several methods by which we can see how to find the Golden Ratio, and we will look at a few examples here.
Golden Ratio Formula
There are several ways to state the Golden Ratio formula. Recall the line segment example from above. In this example, the Golden Ratio can be found by solving the proportion {eq}\frac{AB}{BC} = \frac{BC}{AC} {/eq}. The Golden Ratio may also be found directly by evaluating {eq}\frac{1}{2}(1+\sqrt{5}) {/eq}.
Perhaps the most interesting way to calculate the Golden Ratio is by repeated division.
Begin by dividing 1 by 2. Then, add 1. This gives a result q. Then, proceed by dividing 1 by q and add 1 to that result. Repeat this process. Let's take a look:
| New value | Divide by new value | Add 1 |
| 2 | 1/2 = 0.5 | 0.5+1 = 1.5 |
| 1.5 | 1/1.5 = 0.666... | 0.666+1 = 1.666... |
| 1.666... | 1/1.666... = 0.6 | 0.6+1 = 1.6 |
| 1.6 | 1/1.6 = 0.625 | 0.625+1 = 1.625 |
| 1.625 | 1/1.625 = 0.6153... | 0.6153+1 = 1.6153... |
| 1.6153... | 1/1.6153... = 0.619... | 0.619+1 = 1.619... |
So, repeating this process many times, each result gets closer and closer to {eq}\phi {/eq}! This leads to a self-contained formula for {eq}\phi {/eq}:
$$\phi = 1 + \frac{1}{\phi} $$
That the Golden Ratio can be defined in terms of itself is another unique property of this fascinating number.
Golden Ratio Number
As mentioned, the Golden Ratio number is irrational and is equal to about 1.61803..., but much like we use {eq}\pi {/eq} to denote the number 3.14159... we use the Greek letter {eq}\phi {/eq}, or phi, to denote the Golden Ratio. It is an infinite, non-repeating decimal that cannot be expressed as a fraction.
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FAQs
Golden ratio is a special number and is approximately equal to 1.618. Golden ratio is represented using the symbol “ϕ”. Golden ratio formula is ϕ = 1 + (1/ϕ).
How do you explain golden ratio to students? ›
The golden ratio, also known as the golden number, golden proportion or the divine proportion, is a ratio between two numbers that equals approximately 1.618. Usually written as the Greek letter phi, it is strongly associated with the Fibonacci sequence, a series of numbers wherein each number is added to the last.
What is the golden ratio simply explained? ›
Putting it as simply as we can (eek!), the Golden Ratio (also known as the Golden Section, Golden Mean, Divine Proportion or Greek letter Phi) exists when a line is divided into two parts and the longer part (a) divided by the smaller part (b) is equal to the sum of (a) + (b) divided by (a), which both equal 1.618.
What is the golden ratio in simplest form? ›
golden ratio, in mathematics, the irrational number (1 + Square root of√5)/2, often denoted by the Greek letter ϕ or τ, which is approximately equal to 1.618.
How to calculate your golden ratio? ›
Thus, the following equation establishes the relationship for the calculation of golden ratio: ϕ = a/b = (a + b)/a = 1.61803398875... where a and b are the dimensions of two quantities and a is the larger among the two.
How do you demonstrate the golden ratio? ›
You take a line and divide it into two parts – a long part (a) and a short part (b). The entire length (a + b) divided by (a) is equal to (a) divided by (b). And both of those numbers equal 1.618. So, (a + b) divided by (a) equals 1.618, and (a) divided by (b) also equals 1.618.
What is the golden ratio of God? ›
This ratio - 1.618 - is an approximation of its true value of [1+√5)/2]. This ratio has served mankind in three ways: it provides beauty, function, and reveals how wise, good, and powerful the Creator is.
What is the golden ratio in art for dummies? ›
What is the Golden Ratio? The golden ratio (also known as the golden section, and golden mean) is the ratio 1:0.62. Use it to divide lines and rectangles in an aesthetically pleasing way. In the above square A is 0.62 of the rectangle.
How do you apply golden ratio in your life? ›
Here are a few ways you can use it in your everyday life:- Use it as a guide when creating visual compositions, whether you're designing a website or arranging a vase of flowers. The golden ratio is said to be aesthetically pleasing, so following its proportions can help create an attractive design.
What is a real life example of a ratio? ›
Recipes are a good of examples of using ratios in real life. For the lemonade, 1 cup sugar to 5 cups water so if I had 2 cups of sugar I would need 10 cups of water. The ratio here is 2 jars to 5 dollars or 2:5.
The Math: Golden Ratio and Fibonacci Sequence
Two numbers are in the golden ratio if: their ratio is the same as the ratio of the numbers added together to the larger of the two numbers, or put more clearly: a/b = (a+b)/a. For example if our numbers are a= 8, b= 5 then we have 8/5 = (8+5)/8, or 1.6 = 1.625.
What is an example of the golden ratio in a house? ›
The golden ratio for choosing furniture
The sofas each represent around 60% of the space, the smaller coffee table is around 40%. So, if you are choosing a sofa, you will want to look for a coffee table that's around two thirds its length for a balanced feel.
What does the golden ratio symbolize? ›
Furthermore, the golden ratio is also associated with the idea of divine proportions or cosmic harmony. Many artists and philosophers throughout history have attributed spiritual or metaphysical significance to the golden ratio, seeing it as a representation of divine or universal order.
How do you calculate the ratio? ›
Ratios compare two numbers, usually by dividing them. If you are comparing one data point (A) to another data point (B), your formula would be A/B. This means you are dividing information A by information B. For example, if A is five and B is 10, your ratio will be 5/10.
What is the Fibonacci sequence formula? ›
The Fibonacci sequence formula deals with the Fibonacci sequence, finding its missing terms. The Fibonacci formula is given as, Fn = Fn-1 + Fn-2, where n > 1. It is used to generate a term of the sequence by adding its previous two terms.
What is the formula for ratio? ›
Ratios compare two numbers, usually by dividing them. If you are comparing one data point (A) to another data point (B), your formula would be A/B. This means you are dividing information A by information B. For example, if A is five and B is 10, your ratio will be 5/10.
What is the formula for the golden triangle? ›
The golden triangle is an isosceles triangle in which the ratio of a and b from the figure above, in other words, the ratio of the hypotenuse and base is equal to the golden ratio. a/b = phi or φ. It is also known as the sublime triangle.
How do you find the golden ratio in Fibonacci? ›
The Golden Ratio is a relationship between two numbers that are next to each other in the Fibonacci sequence. When you divide the larger one by the smaller one, the answer is something close to Phi. The further you go along the Fibonacci Sequence, the closer the answers get to Phi.
