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Golden Rectangle Calculator
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The previous online golden rectangle calculator is a tool to calculate the golden section also known as golden number or divine proportion. It is found in many forms of nature, art, architecture, and other fields.
The golden ratio calculator is designed to help you find relationships based on the golden number represented by the Greek letter phi (φ), where Phi ≈ 1.618, especially useful in the field of design or photography.
How to calculate the golden ratio?
The golden ratio arises when you divide a line into two parts, in such a way that the relationship between the longer part (a) and the shorter part (b) is equal to the relationship between the entire line (a + b) and the longer part (a). This geometric relationship is expressed as: (a + b) / a = a / b = φ (where φ is the golden number, approximately 1.618).
So, with 1.618 (Golden Ratio or Ratio of Gold)… How is this sequence of numbers applied in art, architecture, or design? Well, the composition of the so-called golden rectangle is used because it facilitates its use. To better understand:
For example, if you have a line that measures 10 units in length, and you want to divide this line into two parts, “a” and “b”, in such a way that they meet the golden ratio, we should follow these steps:
- Definition of the golden ratio:
- The mathematical formula would be: (a + b) / a = a / b = φ (approximately 1.618)
- We know that a + b = 10 (the total length of the line).
- Establish an equation:
- We use the equation a / b = φ to express “a” in terms of “b”: a = φ * b.
- Substitute in the main equation:
- Now, we substitute “a” in the equation (a + b) / a = φ: (φ * b + b) / (φ * b) = φ.
- Solve for “b”:
- Simplifying the equation, we get: b = 10 / (φ + 1).
- Using the approximate value of φ (1.618), we calculate: b ≈ 3.82 units.
- Calculate “a”:
- Now that we have “b”, we can calculate “a”: a = 10 – b ≈ 6.18 units.
- Verify the ratio:
- Finally, we verify that the ratio is met where: (a + b) / a = 10 / 6.18 ≈ 1.618 and for a / b = 6.18 / 3.82 ≈ 1.618.
Based on the dimensions obtained from the previous example, we will form the golden rectangle
The spiral that appears in the golden rectangle is called the golden spiral. This spiral is constructed by drawing arcs within successive squares that are formed by dividing the rectangle into golden sections.
Now, we need to understand two important points closely related to the value of the golden number:
- Why does the formula for the golden number sometimes appear as: φ = (1 + √5) / 2?
The formula φ = (1 + √5) / 2 is the exact mathematical form to calculate the golden number, which is a quadratic equation. It provides the exact value – for example, for a mathematician -, unlike decimal approximations from a geometric perspective.
- What is the relationship between the Fibonacci sequence and the golden number?
The Fibonacci sequence (discovered by Leonardo of Pisa) is an infinite sequence of numbers that starts with 0 and 1, and each subsequent number is the sum of the two preceding ones. That is: 0,1,1,2,3,5,8,13,21,34,55,89,… For example:
- The first number is 0.
- The second number is 1.
- The third is 0+1=1.
- The fourth is 1+1=2.
- The fifth is 1+2=3, and so on.
If you take two consecutive numbers from the Fibonacci sequence and divide the larger by the smaller, you get a value that increasingly approximates the golden number (ϕ) as you progress in the sequence:
| Larger number | Smaller number | Ratio (larger / smaller) | Golden number approximation (φ = 1.618…) |
|---|---|---|---|
| 1 | 1 | 1 / 1 = 1.000 | Far from the golden number |
| 2 | 1 | 2 / 1 = 2.000 | Gets a little closer |
| 3 | 2 | 3 / 2 = 1.500 | Closer |
| 5 | 3 | 5 / 3 = 1.666… | Getting closer |
| 8 | 5 | 8 / 5 = 1.600 | Very close |
| 13 | 8 | 13 / 8 = 1.625 | Even closer |
| 21 | 13 | 21 / 13 = 1.615… | Almost identical to the golden number! |
| 34 | 21 | 34 / 21 = 1.619… | Practically equal to the golden number |
This demonstrates the deep connection between the Fibonacci sequence and the golden ratio
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Frequently asked questions about calculating the golden number – FAQ
Why is the golden number considered aesthetically pleasing?
It is thought that proportions based on the golden number are harmonious and attractive to the human eye.
What does φ mean in the golden ratio?
The φ (phi) is the symbol of the golden ratio and is worth 1.618. It is like “pi” (π) for circles, but this number describes “ideal” proportions in rectangles, spirals, or faces.
How do I know if something has the golden ratio?
Measure two parts of an object (like a long and a short one). Divide the longer length by the shorter one. If the result is close to 1.618, it has the golden ratio. For example, 8 cm / 5 cm = 1.6, which is close.
Why do they say the golden ratio is perfect?
They say it is perfect because it appears in beautiful things like flowers, shells, or famous paintings, and our eyes like that proportion. Mathematically, it is unique because if you square 1.618 or invert it, it remains connected to itself.
How do I make a golden rectangle?
Draw a square (say 5 cm per side). Then, extend one side by multiplying 5 by 1.618 (about 8 cm). Connect the points, and you will have a rectangle of 5 cm by 8 cm, which follows the golden ratio.
What is the golden ratio with exact numbers?
It is not an exact number like 2 or 3, but irrational. It is (1 + √5) ÷ 2, which gives 1.61803398… and continues infinitely. That’s why we use 1.618 as an approximation.
Can I calculate the golden ratio without a calculator?
Yes, but it is approximate. Use simple numbers like 5 and 8 (which are close to 1.618 when you divide them: 8 ÷ 5 = 1.6). Or do it with a ruler and compass to draw a golden rectangle and measure it.
What does the golden ratio have to do with Fibonacci?
The Fibonacci sequence (1, 1, 2, 3, 5, 8, 13…) approaches the golden ratio. If you divide a number in the sequence by the previous one (like 13 ÷ 8), you get something close to 1.618. The larger the numbers, the closer it is.
How do I use the golden ratio in a drawing?
Divide your canvas into two parts where the larger is 1.618 times the smaller. For example, if the total is 10 cm, the long part would be 6.18 cm and the short part 3.82 cm. Place important objects on those lines to make it look balanced.
How to calculate the golden ratio in Excel?
To calculate the golden ratio in Excel, you can use the formula directly: Type =(1+SQRT(5))/2 in a cell to get the value of φ. To apply φ in calculations, use cell references. For example: If you have a value in cell A1, multiply by φ by typing =A1*(1+SQRT(5))/2.
