Golden Ratio: Discover the Phi Number, Golden Proportion and Its Mathematical Value
Did you know that the Mona Lisa, the Egyptian pyramids, and even the arrangement of sunflower seeds share the same mathematical secret? This mystery is called the golden ratio, also known as the golden proportion, phi number or golden ratio. In this complete guide, you’ll discover what this fascinating number is, how to calculate it, and why it’s present in so many aspects of our lives.
GOLDEN RATIO VALUE
An irrational number that appears in mathematics, art, nature and architecture
What is the Golden Ratio?
The golden ratio (represented by the Greek letter Φ Phi) is a special irrational number with unique mathematical properties. It is also known as:
- Golden proportion: Relationship between two segments of a line
- Golden number: For its value and mathematical beauty
- Golden ratio: Proportion considered aesthetically perfect
- Divine proportion: Name given during the Renaissance
- Phi number (φ): For the Greek letter that represents it
Mathematical definition: Two quantities are in golden ratio if the ratio of their sum to the larger quantity is equal to the ratio of the larger quantity to the smaller one.
How to Calculate the Golden Ratio? 3 Easy Methods
Method 1: Quadratic Equation
If we have that Φ = a/b and (a+b)/a = a/b, then:
Multiplying by Φ: Φ² = Φ + 1
Solving the quadratic equation: Φ² – Φ – 1 = 0
Using the quadratic formula:
Taking the positive solution: Φ = (1 + √5) / 2 ≈ 1.6180339887
Method 2: Fibonacci Sequence
The Fibonacci sequence is intimately related to the golden ratio:
- Sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89…
- Rule: Each number is the sum of the two previous ones
- Property: Dividing a number by the previous one approximates Φ
13/8 = 1.625
21/13 ≈ 1.615
34/21 ≈ 1.619
55/34 ≈ 1.6176
89/55 ≈ 1.61818
Method 3: Geometric Construction
Constructing a golden rectangle:
- Draw a square with side 1
- Mark the midpoint of the base
- Draw a line from this point to an opposite corner
- Use that line as radius to extend the base
- The resulting rectangle has golden proportion
Applications of the Golden Ratio in Real Life
🌿 In Nature
- Sunflower seed spiral: Pattern in golden spiral
- Snail shells: Logarithmic spiral based on Φ
- Tree branches: Arrangement following the proportion
- Relationship in human body: Navel as golden point
- Hurricanes and galaxies: Natural spiral shapes
🏛️ In Architecture
- Egyptian pyramids: Proportions based on Φ
- Parthenon of Athens: Design according to golden ratio
- Notre Dame Cathedral: Facade with golden proportions
- Eiffel Tower: Curve follows golden spiral
- Modern architecture: Le Corbusier and the Modulor
🎨 In Art and Design
- Mona Lisa: Face and composition with Φ
- Vitruvian Man: Da Vinci and golden proportions
- The Last Supper: Geometric composition
- Graphic design: Layouts and typography
- Photography: Related rule of thirds
The Golden Circle and the Golden Spiral
What is the Golden Circle?
The golden circle is a geometric construction where the relationship between the radius and certain chords follows the golden ratio. It is used in logo design and architecture.
The Golden Spiral
The golden spiral is constructed from successive golden rectangles. It is a logarithmic spiral that grows by a factor of Φ for each quarter turn. It appears in:
- Nautilus shells
- Arrangement of seeds in sunflowers
- Spiral galaxies
- Hurricane waves
Practical Examples and Curiosities
📐 Numerical Examples
Example 1: Golden rectangle
If the shorter side measures 1, the longer side will be:
Example 2: Divided segment
Dividing a segment of length 100 in golden ratio:
Smaller part = 100 – 61.8 = 38.2
Example 3: Inverse golden rectangle
The inverse of Φ has interesting properties:
Golden Ratio in the Human Body
Leonardo da Vinci studied golden proportions in the human body:
- Total height / Height to navel ≈ Φ
- Height to navel / Height to head ≈ Φ
- Arm length / Forearm length ≈ Φ
- Hand length / Finger length ≈ Φ
- Smile width / Nose width ≈ Φ
Relationship with Other Mathematical Concepts
The Golden Ratio and Mathematics
Algebraic properties:
Φ = 1 + 1/Φ
Φ = √(1 + Φ)
Relationship with π and e:
- Φ appears in identities with π: cos(36°) = Φ/2
- Relationship with number e: limits of sequences
- Connected to trigonometry: angles of 36° and 72°
Modern Applications of the Golden Ratio
💻 Technology and Design
Web design and UX:
- Proportions in layouts and grids
- Typography sizes scaled with Φ
- Spacing between visual elements
Artificial Intelligence:
- Optimization algorithms based on Φ
- Neural networks and optimal proportions
- Data analysis and natural patterns
Finance and Economics:
- Technical market analysis (Fibonacci retracements)
- Proportions in economic growth
- Predictive models based on sequences
Conclusion: The Mathematical Beauty of the Golden Ratio
The golden ratio or golden proportion is much more than a simple mathematical value. It is a bridge between pure mathematics and the natural world, between art and science, between the abstract and the concrete.
From sunflowers in the field to Renaissance masterpieces, from ancient pyramids to modern web design, the number Φ seems to be woven into the very fabric of reality.
Ready to discover the golden ratio for yourself? Look around you: in flowers, in buildings, in art, and even in your own body. Mathematics isn’t just numbers on paper; it’s the secret language of the universe, and the golden ratio is one of its most beautiful words.