Note: Analogous to the golden ratio, the bronze ratio is the positive solution to the equation x2=3x+1, which equals 23+32+4 and has the continued fraction expansion 3+3+3+…11. The approximation evaluated in this problem is accurate to three decimal places (3.303). Assuming the continued fraction expansion has a limiting value x0, it can be seen that x0=3+x01, which is equivalent to x02=3x0+1.
The bronze ratio, 23+32+4, is analogous to the golden ratio, 21+12+4. The golden ratio is associated with
1+1+1+⋯11
Similarly, the silver ratio is 22+22+4, associated with
2+2+2+⋯11.
These metallic ratios are also related to generalizations of the Fibonacci sequence, arise as lengths of diagonals in regular polygons, and have many other geometric and algebraic interpretations as well.