Silver Rectangles and the Ways of Kings
The article explains the mathematical properties of silver rectangles, which are similar to golden rectangles but based on a different ratio. Whereas a golden rectangle can be subdivided into a square and a smaller golden rectangle, a silver rectangle can be divided into two similar silver rectangles of different sizes.
Background
- The **golden rectangle** (ratio ~1.618) is famous in art and architecture because removing a square leaves a smaller rectangle with the same proportions — a property called "self-similarity." The ratio is the golden ratio φ (phi).
- Silver rectangles follow a related but less-known pattern. The **silver ratio** (δ_s ≈ 2.414) is one of the "metallic ratios," a family that generalizes the golden ratio. For silver rectangles, removing two largest-possible squares leaves a similar rectangle, rather than one square as with gold.
- John D. Cook is a mathematician and blogger who writes accessible posts about math, computing, and their connections to the real world. This post is a short, technical-but-clear exploration of these geometric properties, aimed at readers with basic algebra.