Golden Rectangle

A rectangle in the golden ratio, and its self-similarity

Intermediate (high-school to early-undergraduate level)

Goal of this page

Understand the definition of the golden rectangle and its relation to the golden ratio $\varphi$, and grasp its self-similarity (removing a square leaves a similar rectangle) and the golden spiral.

1. What is a golden rectangle

A golden rectangle is a rectangle whose side ratio is the golden ratio $\varphi:1$.

Golden ratio:

$$\varphi = \dfrac{1+\sqrt5}{2} \approx 1.618$$

When the long side $L$ and short side $W$ satisfy $\dfrac{L}{W}=\varphi$, the rectangle is called a golden rectangle.

Figure 1: A golden rectangle (long : short = $\varphi$ : 1). Blue is the square on the short side; orange is the remaining rectangle.

2. Self-similarity

Removing a square whose side equals the short side of a golden rectangle leaves a rectangle that is again a golden rectangle. This corresponds to the defining relation of the golden ratio,

$$\varphi = 1 + \dfrac{1}{\varphi}, \qquad \text{i.e.}\quad \varphi^2 = \varphi + 1$$

The square-removal can be repeated indefinitely, so similar golden rectangles nest without end.

Figure 2: Repeating the square-removal nests similar golden rectangles infinitely. The coloured frames are the nested golden rectangles; the black dot is the limit point.

3. The golden spiral and applications

Golden spiral: a special case of the logarithmic spiral $r = a\,\varphi^{2\theta/\pi}$, whose radius is multiplied by $\varphi$ every $90^\circ$. Joining quarter circles in the nested squares gives a curve that approximates it (Figure 3); note that this quarter-circle curve is itself not a logarithmic spiral, only an approximation.

Figure 3: A curve made by joining quarter circles in the nested squares (orange) and the true golden spiral (blue). The golden spiral is a logarithmic spiral whose radius grows by $\varphi$ every $90^\circ$; the quarter-circle construction closely approximates it.

Fibonacci sequence: the ratio $\dfrac{F_{n+1}}{F_n}$ of consecutive Fibonacci numbers converges to $\varphi$. Hence a rectangle whose sides are consecutive Fibonacci numbers (a Fibonacci rectangle) has an aspect ratio approaching $\varphi$, so its shape approaches the golden rectangle.

Frequently asked questions

Q1. What is a golden rectangle?

It is a rectangle whose side ratio is the golden ratio φ=(1+√5)/2≈1.618; the long side and short side are in the ratio φ:1.

Q2. What happens when you remove a square?

Removing a square whose side equals the short side of a golden rectangle leaves another golden rectangle (self-similarity). This corresponds to the property φ²=φ+1 of the golden ratio, and the operation can be repeated indefinitely.

Q3. What is the golden spiral?

The golden spiral is a logarithmic spiral whose radius is multiplied by the golden ratio φ (≈1.618) every 90° turn. Joining quarter circles in the nested squares gives a curve that closely approximates this golden spiral. It arises naturally from the self-similar structure of the golden rectangle.

References and related topics

Golden ratio (Number theory: Introduction) — definition and properties of φ
Fibonacci numbers (Sequences: Introduction) — the ratio converges to φ
Regular polygons (Geometry: Introduction) — the regular pentagon and the golden ratio

References

Livio, M. (2002). The Golden Ratio, Broadway Books.
Wikipedia: Golden rectangle