Draft:Bronze Ratio
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The bronze ratio is a ratio in mathematics where 3 times a larger quantity plus the smaller quantity, divided by the larger quantity, is equal to the larger quantity divided by the smaller quantity. Its value is equal to 3+√13/2
and is the third metallic mean, approximately equal to 3.30277563773...[1][2] The mean/ratio is usually denoted with the symbol β, or with the symbol μ but it usually varies and there is no standard symbol
We can denote the relation of this ratio algebraically as:
Using the continued fraction that all metallic means follow of, [n; n, n, n, ...]: the bronze ratio can be expressed as:
Calculation[edit]
When we multiply and re-arrange the equation from above, we get
Using the quadratic equation on this gives us:
We also can use the 3-bonacci sequence to slowly approach the bronze ratio:[3][4] 1/0, 3/1, 10/3, 33/10, 109/33, etc.
Properties[edit]
Some properties of the bronze ratio are that 1/β = √13-3/2 and that any power of β is equal to 3 times the previous power plus the second previous power. which can be represented as:
We also can express it trigonometrically as:[5]
See also[edit]