Metallic Mean Calculator
Enter a single base value and instantly see the golden ratio, silver ratio, and other metallic means as longer and shorter sides.
Input Form
Enter the base length you want to use.
Calculated proportions
Each row shows the metallic mean μₙ and the corresponding longer and shorter sides derived from your base value.
| Visual & name | Ratio μₙ | Longer side | Shorter side |
|---|---|---|---|
Golden ration = 1 | 1.6 | - | - |
Silver ration = 2 | 2.4 | - | - |
Bronze ration = 3 | 3.3 | - | - |
Copper ration = 4 | 4.2 | - | - |
Nickel ration = 5 | 5.2 | - | - |
Iron ration = 6 | 6.2 | - | - |
Enter a base value to see all metallic mean proportions.
About metallic means
For a positive integer n, the metallic mean μₙ is defined as (n + √(n² + 4)) ÷ 2. It extends the golden ratio into an infinite family of irrational numbers.
Formula: μₙ = (n + √(n² + 4)) / 2. The conjugate root is (n − √(n² + 4)) / 2, and the continued fraction is [n; n, n, …].
Metallic means appear in design systems, recursive processes, architecture, and mathematical art.
Reference metallic means
These are the most frequently cited metallic means. The table lists each order with its common name.
| Visual & name | Ratio μₙ |
|---|---|
Golden ration = 1 | 1.6 |
Silver ration = 2 | 2.4 |
Bronze ration = 3 | 3.3 |
Copper ration = 4 | 4.2 |
Nickel ration = 5 | 5.2 |
Iron ration = 6 | 6.2 |