Many people are familiar with the Golden Ratio (φ), but it is just one member of a broader family known as the Metallic Ratios. These ratios describe a recursive relationship between the sides of a rectangle. Given a rectangle with side lengths A and B (B > A), the Metallic Ratios satisfy the equation:
A / B = (B – n × A) / A
where n is a fixed integer.
For the Golden Ratio (n = 1):
A / B = (B – A) / A
which leads to:
φ² – φ – 1 = 0
solving this gives:
φ = (1 + √5) / 2 ≈ 1.618…
For the Silver Ratio (n = 2):
A / B = (B – 2A) / A
which leads to:
ψ² – 2ψ – 1 = 0
solving this gives:
ψ = 1 + √2 ≈ 2.414…
Different values of n define other Metallic Ratios. The table below presents some of them.
Curious minds, don’t stop here—explore the fascinating properties of the other Metallic Ratios and see what patterns and surprises you can uncover!
