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 Ο†:1\varphi:1Ο†:1, where Ο†=1+52β‰ˆ1.618\varphi =\tfrac{1+\sqrt{5}}{2}\approx 1.618Ο†=21+5β€‹β€‹β‰ˆ1.618.
  • If the longer side is bbb and the shorter side is aaa, then b/aβ‰ˆ1.618b/a\approx 1.618b/aβ‰ˆ1.618.

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 k2βˆ’kβˆ’1=0k^2-k-1=0k2βˆ’kβˆ’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 12.944/8β‰ˆ1.61812.944/8\approx 1.61812.944/8β‰ˆ1.618.
  • More generally, any rectangle with long side β‰ˆ 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 β€œgolden spiral,” which can be drawn by nesting squares inside a golden rectangle.

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