Metallic mean
![If one removes the largest possible square from the end of a gold rectangle one is left with a gold rectangle. If one removes two from a silver, one is left with a silver. If one removes three from a bronze, one is left with a bronze.](/Images/godic/202501/06/Gold,_silver,_and_bronze_rectangles.svg0504.png")
![A golden triangle. The ratio a:b is equivalent to the golden ratio φ. In a silver triangle this would be equivalent to δS.](/Images/godic/202501/06/Golden_Triangle.svg0504.png")
The more general simple continued fraction expressions
are known as the metallic means or silver means (also ratios or constants) of the successive natural numbers. The golden ratio (1.618...) is the metallic mean between 1 and 2, while the silver ratio (2.414...) is the metallic mean between 2 and 3. The term "bronze ratio" (3.303...), or terms using other names of metals (copper, nickel), are occasionally used to name subsequent metallic means. The values of the first ten metallic means are shown at right. Notice that each metallic mean is a root of the simple quadratic equation