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

Approximation of the bronze spiral

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:

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

Bronze rectangle

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.

$3+{\cfrac {1}{3+{\cfrac {1}{3+{\cfrac {1}{3+\ddots }}}}}}=\beta$

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:

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

See also

• Metallic mean
• Golden Ratio
• Silver ratio
• Ratio