## The Golden Ratio and Other Metal Ratios

The Golden Ratio, φ=(1+sqrt(5))/2 arises in connection with the
Fibonacci Sequence.

The Fibonacci sequence is defined as: F(n+2) = F(n+1) + F(n), with F(0) = 0 and F(1)
= 1. To solve the recurrence explicitly, one observes that a solution which satisfies
the recurrence has to satisfy the characteristic equation x^{n+2} =
x^{n+1} + x^{n}, or the equation: x^{2} = x + 1, from which we
get: x_{1,2} = φ, 1-φ.

The general solution to the recurrence, then, will be F(n) = a*φ^{n} +
b*(1-φ)^{n}, where a and b are constants. a and b have to satisfy the
initial terms as F(0) = 0 and F(1) = 1, from which we get the Binet expression, which
solves the Fibonacci recurrence explicitly:

Interesting variations come about when
we consider the recurrence F(n+2) = p*F(n+1) + q*F(n), with F(0) = 0 and F(1) = 1
and p,q in {1,2,3,4,...}. Following exactly similar tactics, this recurrence can
be solved explicitly as:
where now R is a particular "metal ratio".
For p=q=1, we get R=φ, the Golden Ratio. For p=3 and q=1, we get the *Bronze
Ratio*, R = 3.302775638. Let's see some of these metal ratios with Maple and
the corresponding sequences which are generated by the modified recurrence F(n+2)
= p*F(n+1) + q*F(n), with F(0) = 0 and F(1) = 1:
> Rm:=(p,q)->(p+sqrt(p^2+4*q))/2;

> for p from 1 to 3 do

> for q from 1 to 3 do

> print(p,q,evalf(Rm(p,q)));

> od;

> od;

p |
q |
R |
Sloane Sequence |
Continued Fraction |

1 |
1 |
φ=1.618033988 (Golden)^{[1]} |
A000045 |
[1,1,1,1,1,1] |

1 |
2 |
2 |
A001045 |
[2] |

1 |
3 |
2.302775638 |
A006130 |
[2,3,3,3,3,3] |

2 |
1 |
1+sqrt(2)=2.414213562 (Silver)^{[1]} |
A000129 |
[2,2,2,2,2,2] |

2 |
2 |
2.732050808 |
A002605 |
[2,1,2,1,2,1] |

2 |
3 |
3 |
A015518 |
[3] |

3 |
1 |
(3+sqrt(13))/2=3.302775638 (Bronze)^{[1]} |
A006190 |
[3,3,3,3,3,3] |

3 |
2 |
3.561552813 |
A007482 |
[3,1,1,3,1,1] |

3 |
3 |
3.791287848 |
A030195 |
[3,1,3,1,3,1] |

Construction

Metal Ratios can be constructed geometrically: The solutions to the quadratic
x^{2}-p*x-q=0, are the points where the circle whose diameter is determined
by the points A = (0,1) and B = (-p,q), intersect the x-axis. Indeed, for the Bronze
Ratio, we get the following figure using EucliDraw:

Construction of Bronze Ratio

EucliDraw reports: OC ~ 3.3028, while DC ~ 3.6055, hence OD ~ 0.3027. And indeed,
the Bronze Ratio is equal to (3 + sqrt(13))/2 ~ 3.302775638, and hence OC and DO are
the two roots of the quadratic x^{2}-3*x-1=0.

**Notes**

- To see the spectra of these ratios, consult Google
Scholar and the Spectra of the Scientists.