Golden Ratio Formula & Fibonacci Sequence
1. The Golden Ratio (φ - Phi)
Definition
Two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities.
If \(a > b > 0\), then:
\[\frac{a}{b} = \frac{a + b}{a} = \varphi\]where \(\varphi\) (phi) is the golden ratio.
Golden Ratio Value
The golden ratio is an irrational number:
\[\varphi = \frac{1 + \sqrt{5}}{2} \approx 1.618033988749...\]Alternative Formulas for Golden Ratio
- \(\varphi = 1 + \frac{1}{\varphi}\)
- \(\varphi^2 = \varphi + 1\)
- \(\varphi = 2 \sin(54°)\)
- \(\frac{1}{\varphi} = \varphi - 1 \approx 0.618033988749...\)
Key Property: The golden ratio is the only positive number that becomes its own reciprocal when 1 is subtracted: \(\frac{1}{\varphi} = \varphi - 1\)
2. Deriving the Golden Ratio
Starting from:
\[\frac{a}{b} = \frac{a + b}{a}\]Step 1: Let \(\varphi = \frac{a}{b}\), so \(a = \varphi b\)
Step 2: Substitute into the equation:
\[\varphi = \frac{\varphi b + b}{\varphi b} = \frac{b(\varphi + 1)}{\varphi b} = \frac{\varphi + 1}{\varphi}\]Step 3: Multiply both sides by \(\varphi\):
\[\varphi^2 = \varphi + 1\]Step 4: Rearrange to standard form:
\[\varphi^2 - \varphi - 1 = 0\]Step 5: Apply the quadratic formula:
\[\varphi = \frac{1 \pm \sqrt{1 + 4}}{2} = \frac{1 \pm \sqrt{5}}{2}\]Step 6: Since \(\varphi > 0\), take the positive solution:
\[\varphi = \frac{1 + \sqrt{5}}{2} \approx 1.618033989\]3. The Fibonacci Sequence
Definition
The Fibonacci sequence is a series where each number is the sum of the two preceding numbers:
First 20 Fibonacci Numbers
| \(n\) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|---|
| \(F_n\) | 0 | 1 | 1 | 2 | 3 | 5 | 8 | 13 | 21 | 34 |
| \(n\) | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 |
| \(F_n\) | 55 | 89 | 144 | 233 | 377 | 610 | 987 | 1597 | 2584 | 4181 |
4. The Golden Ratio and Fibonacci Connection
Fundamental Relationship: The ratio of consecutive Fibonacci numbers approaches the golden ratio as \(n\) increases:
\[\lim_{n \to \infty} \frac{F_{n+1}}{F_n} = \varphi\]Ratio of Consecutive Fibonacci Numbers
| Fibonacci Number \(F_n\) | Next Number \(F_{n+1}\) | Ratio \(\frac{F_{n+1}}{F_n}\) | Decimal Value |
|---|---|---|---|
| 2 | 3 | 3/2 | 1.5 |
| 3 | 5 | 5/3 | 1.666666... |
| 5 | 8 | 8/5 | 1.6 |
| 8 | 13 | 13/8 | 1.625 |
| 13 | 21 | 21/13 | 1.615384... |
| 21 | 34 | 34/21 | 1.619047... |
| 89 | 144 | 144/89 | 1.617977... |
| 144 | 233 | 233/144 | 1.618055... |
| 233 | 377 | 377/233 | 1.618025... |
| 987 | 1597 | 1597/987 | 1.618034... |
Notice how the ratio converges to \(\varphi \approx 1.618033989\)
5. Binet's Formula
Closed-Form Formula for Fibonacci Numbers
Binet's formula allows us to calculate any Fibonacci number directly without recursion:
Or equivalently:
\[F_n = \frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^n - \left(\frac{1-\sqrt{5}}{2}\right)^n\right]\]Given: Find \(F_7\)
Step 1: Use Binet's formula with \(n = 7\):
\[F_7 = \frac{(1.618034)^7 - (1-1.618034)^7}{\sqrt{5}}\]Step 2: Calculate the powers:
\[\varphi^7 = (1.618034)^7 \approx 29.034\] \[(1-\varphi)^7 = (-0.618034)^7 \approx -0.034\]Step 3: Substitute:
\[F_7 = \frac{29.034 - (-0.034)}{\sqrt{5}} = \frac{29.068}{2.236} \approx 13\]6. Properties Connecting Golden Ratio and Fibonacci
Important Properties
- Approximation Property: For large \(n\), \(F_n \approx \frac{\varphi^n}{\sqrt{5}}\)
- Next Term Estimate: \(F_{n+1} \approx \varphi \cdot F_n\) (for large \(n\))
- Cassini's Identity: \(F_{n+1} \cdot F_{n-1} - F_n^2 = (-1)^n\)
- Sum Formula: \(\sum_{i=0}^{n} F_i = F_{n+2} - 1\)
- Golden Ratio in Fibonacci: \(\varphi = \frac{F_{n+1}}{F_n}\) as \(n \to \infty\)
Fibonacci Identities Using Golden Ratio
- \(\varphi^n = F_n \varphi + F_{n-1}\)
- \(\varphi^n = \frac{1}{2}(L_n + F_n\sqrt{5})\) where \(L_n\) are Lucas numbers
- \(F_{2n} = F_n(2F_{n+1} - F_n)\)
- \(F_{n+m} = F_n F_{m+1} + F_{n-1} F_m\)
7. Applications in Nature and Art
Natural Occurrences
- Flower Petals: Many flowers have petals in Fibonacci numbers (3, 5, 8, 13, 21, 34)
- Spiral Patterns: Sunflower seeds, pinecones, and pineapples show Fibonacci spirals
- Tree Branching: Branch patterns often follow Fibonacci ratios
- Shell Spirals: Nautilus shells approximate golden ratio spirals
- Galaxy Spirals: Spiral galaxies often exhibit golden ratio proportions
- Human Body: Proportions of fingers, face, and body approximate golden ratio
Applications in Art and Architecture
- Classical Architecture: Parthenon in Greece uses golden ratio proportions
- Renaissance Art: Leonardo da Vinci's works incorporate golden ratio
- Modern Design: Logo design and layout composition
- Photography: Rule of thirds approximates golden ratio
- Music: Musical scales and composition structure
8. The Golden Rectangle and Spiral
Golden Rectangle
A rectangle whose side lengths are in the golden ratio:
If you remove a square from a golden rectangle, the remaining rectangle is also a golden rectangle.
Fibonacci Spiral
Created by drawing quarter-circle arcs connecting opposite corners of squares with Fibonacci number side lengths (1, 1, 2, 3, 5, 8, 13, 21...). This approximates the golden spiral, which is a logarithmic spiral with growth factor φ.
9. Worked Examples
Problem: Verify that \(\varphi^2 = \varphi + 1\)
Solution:
\[\varphi = \frac{1 + \sqrt{5}}{2}\] \[\varphi^2 = \left(\frac{1 + \sqrt{5}}{2}\right)^2 = \frac{(1 + \sqrt{5})^2}{4} = \frac{1 + 2\sqrt{5} + 5}{4} = \frac{6 + 2\sqrt{5}}{4} = \frac{3 + \sqrt{5}}{2}\] \[\varphi + 1 = \frac{1 + \sqrt{5}}{2} + 1 = \frac{1 + \sqrt{5} + 2}{2} = \frac{3 + \sqrt{5}}{2}\]Problem: We know \(F_{12} = 144\). Estimate \(F_{20}\) using the golden ratio.
Solution: For large \(n\), \(F_{n+k} \approx F_n \times \varphi^k\)
Here, \(k = 20 - 12 = 8\)
\[F_{20} \approx F_{12} \times \varphi^8 = 144 \times (1.618)^8\] \[F_{20} \approx 144 \times 46.979 = 6,764.98 \approx 6,765\]Key Insight: The golden ratio and Fibonacci sequence are deeply interconnected. As Fibonacci numbers grow larger, their ratio converges precisely to the golden ratio, making them fundamental tools in mathematics, nature, and design.