Draft:Bronze Ratio

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... 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:


 * $$ \frac{3x + y}{x} = \frac{x}{y} \equiv \beta $$

Using the continued fraction that all metallic means follow of, [n; n, n, n, ...]: the bronze ratio can be expressed as:


 * Gold, silver, and bronze rectangles vertical.png$$ 3 + \cfrac{1}{3 + \cfrac{1}{3 + \cfrac{1}{3 + \ddots}}} =\beta $$

Calculation
When we multiply and re-arrange the equation from above, we get


 * $${\beta}^2 - 3\beta - 1 = 0.$$

Using the quadratic equation on this gives us:


 * $$ \beta = \frac{3+\sqrt{13}}{2} $$

We also can use the 3-bonacci sequence to slowly approach the bronze ratio: $1⁄0$, $3⁄1$, $10⁄3$, $33⁄10$, $109⁄33$, etc.

Properties
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:


 * $$ \beta^n = 3\beta^{n-1} + \beta^{n-2} $$

We also can express it trigonometrically as:


 * $$8\cos\frac{\pi}{13}\cos\frac{3\pi}{13}\cos\frac{4\pi}{13} = \beta $$