Golden Section

Overview

Engineering reference data for Golden Section in miscellaneous.

Key Formulas

Unit Conversion

$y = x \cdot k$

Multiply by conversion factor.

Linear Interpolation

$y = y_1 + \frac{(x - x_1)(y_2 - y_1)}{x_2 - x_1}$

Estimate between two known points.

Percentage

$p = \frac{\text{part}}{\text{whole}} \times 100\%$

Part as fraction of whole.

Variables

Geometric Properties

The Golden Section creates a specific proportional relationship when dividing a line segment. Given a line segment AC, a point B divides it such that the ratio of the whole segment (AC) to the larger part (BC) is equal to the ratio of the larger part (BC) to the smaller part (AB). This relationship defines the unique value φ.

$$\frac{AC}{BC} = \frac{BC}{AB} = \phi$$

This proportional relationship is the foundation for constructing the Golden Rectangle, where the sides are in the ratio 1 : φ.

Connection to the Fibonacci Sequence

The Golden Ratio φ is intrinsically linked to the Fibonacci sequence (0, 1, 1, 2, 3, 5, 8, 13, ...), where each number is the sum of the two preceding ones. As the sequence progresses, the ratio between consecutive Fibonacci numbers converges to φ.

$$\lim_{n \to \infty} \frac{F_{n+1}}{F_n} = \phi \approx 1.6180339887...$$

For example:

• 3 / 2 = 1.5
• 5 / 3 ≈ 1.666...
• 8 / 5 = 1.6
• 13 / 8 = 1.625

This convergence provides a practical way to approximate φ using integer ratios.

References

• Original Source