what is a golden rectangle

A golden rectangle is a special rectangle whose side lengths are in the golden ratio, meaning the long side divided by the short side is about 1.618 to 1.

What is a golden rectangle?

• A golden rectangle has side lengths in the ratio φ:1\varphi:1φ:1, where φ=1+52≈1.618\varphi =\tfrac{1+\sqrt{5}}{2}\approx 1.618φ=21+5​​≈1.618.

• If the longer side is bbb and the shorter side is aaa, then b/a≈1.618b/a\approx 1.618b/a≈1.618.

Key defining property

• If you cut a square off a golden rectangle, the remaining smaller rectangle is similar to the original, meaning it has the same proportions.

• This self-similarity leads to the equation k2−k−1=0k^2-k-1=0k2−k−1=0 for the ratio k=b/ak=b/ak=b/a, whose positive solution is the golden ratio.

Simple numeric example

• Take a rectangle whose short side is 8 units; a golden rectangle would have a long side of about 12.944 units, since 12.944/8≈1.61812.944/8\approx 1.61812.944/8≈1.618.

• More generally, any rectangle with long side ≈ 1.618 times the short side is a good practical approximation to a golden rectangle.

Where it appears

• Golden rectangles are often associated with art and architecture, where these proportions are considered especially aesthetically pleasing.

• They are also related to the Fibonacci sequence and the spiral often called the “golden spiral,” which can be drawn by nesting squares inside a golden rectangle.

Information gathered from public forums or data available on the internet and portrayed here.