Many people have heard of the '''Golden ratio''', phi ('''φ'''). Phi is just one of a series
of related ratios that are referred to as the "'''Metallic ratios'''".
The '''Golden ratio''' was discovered and named by ancient civilizations as it was
thought to be the most pure and beautiful (like Gold). The '''Silver ratio''' was was
also known to the early Greeks, though was not named so until later as a nod to
the '''Golden ratio''' to which it is closely related. The series has been extended to
encompass all of the related ratios and was given the general name '''Metallic ratios''' (or ''Metallic means'').
''Somewhat incongruously as the original Golden ratio referred to the adjective "golden" rather than the metal "gold".''
'''Metallic ratios''' are the real roots of the general form equation:
x2 - bx - 1 = 0
where the integer '''b''' determines which specific one it is.
Using the quadratic equation:
( -b ± √(b2 - 4ac) ) / 2a = x
Substitute in (from the top equation) '''1''' for '''a''', '''-1''' for '''c''', and recognising that -b is negated we get:
( b ± √(b2 + 4) ) ) / 2 = x
We only want the real root:
( b + √(b2 + 4) ) ) / 2 = x
When we set '''b''' to '''1''', we get an irrational number: the '''Golden ratio'''.
( 1 + √(12 + 4) ) / 2 = (1 + √5) / 2 = ~1.618033989...
With '''b''' set to '''2''', we get a different irrational number: the '''Silver ratio'''.
( 2 + √(22 + 4) ) / 2 = (2 + √8) / 2 = ~2.414213562...
When the ratio '''b''' is '''3''', it is commonly referred to as the '''Bronze''' ratio, '''4''' and '''5'''
are sometimes called the '''Copper''' and '''Nickel''' ratios, though they aren't as
standard. After that there isn't really any attempt at standardized names. They
are given names here on this page, but consider the names fanciful rather than
canonical.
Note that technically, '''b''' can be '''0''' for a "smaller" ratio than the '''Golden ratio'''.
We will refer to it here as the '''Platinum ratio''', though it is kind-of a
degenerate case.
'''Metallic ratios''' where '''b''' > '''0''' are also defined by the irrational continued fractions:
[b;b,b,b,b,b,b,b,b,b,b,b,b,b,b,b,b...]
So, The first ten '''Metallic ratios''' are:
:::::: {| class="wikitable" style="text-align: center;"
|+ Metallic ratios
!Name!!'''b'''!!Equation!!Value!!Continued fraction!!OEIS link
|-
|Platinum||0||(0 + √4) / 2|| 1||-||-
|-
|Golden||1||(1 + √5) / 2|| 1.618033988749895...||[1;1,1,1,1,1,1,1,1,1,1...]||[[OEIS:A001622]]
|-
|Silver||2||(2 + √8) / 2|| 2.414213562373095...||[2;2,2,2,2,2,2,2,2,2,2...]||[[OEIS:A014176]]
|-
|Bronze||3||(3 + √13) / 2|| 3.302775637731995...||[3;3,3,3,3,3,3,3,3,3,3...]||[[OEIS:A098316]]
|-
|Copper||4||(4 + √20) / 2|| 4.23606797749979...||[4;4,4,4,4,4,4,4,4,4,4...]||[[OEIS:A098317]]
|-
|Nickel||5||(5 + √29) / 2|| 5.192582403567252...||[5;5,5,5,5,5,5,5,5,5,5...]||[[OEIS:A098318]]
|-
|Aluminum||6||(6 + √40) / 2|| 6.16227766016838...||[6;6,6,6,6,6,6,6,6,6,6...]||[[OEIS:A176398]]
|-
|Iron||7||(7 + √53) / 2|| 7.140054944640259...||[7;7,7,7,7,7,7,7,7,7,7...]||[[OEIS:A176439]]
|-
|Tin||8||(8 + √68) / 2|| 8.123105625617661...||[8;8,8,8,8,8,8,8,8,8,8...]||[[OEIS:A176458]]
|-
|Lead||9||(9 + √85) / 2|| 9.109772228646444...||[9;9,9,9,9,9,9,9,9,9,9...]||[[OEIS:A176522]]
|}
There are other ways to find the '''Metallic ratios'''; one, (the focus of this task)
is through '''successive approximations of Lucas sequences'''.
A traditional '''Lucas sequence''' is of the form:
x''n'' = P * x''n-1'' - Q * x''n-2''
and starts with the first 2 values '''0, 1'''.
For our purposes in this task, to find the metallic ratios we'll use the form:
x''n'' = b * x''n-1'' + x''n-2''
( '''P''' is set to '''b''' and '''Q''' is set to '''-1'''. ) To avoid "divide by zero" issues we'll start the sequence with the first two terms '''1, 1'''. The initial starting value has very little effect on the final ratio or convergence rate. ''Perhaps it would be more accurate to call it a Lucas-like sequence.''
At any rate, when '''b = 1''' we get:
x''n'' = x''n-1'' + x''n-2''
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144...
more commonly known as the Fibonacci sequence.
When '''b = 2''':
x''n'' = 2 * x''n-1'' + x''n-2''
1, 1, 3, 7, 17, 41, 99, 239, 577, 1393...
And so on.
To find the ratio by successive approximations, divide the ('''n+1''')th term by the
'''n'''th. As '''n''' grows larger, the ratio will approach the '''b''' metallic ratio.
For '''b = 1''' (Fibonacci sequence):
1/1 = 1
2/1 = 2
3/2 = 1.5
5/3 = 1.666667
8/5 = 1.6
13/8 = 1.625
21/13 = 1.615385
34/21 = 1.619048
55/34 = 1.617647
89/55 = 1.618182
etc.
It converges, but pretty slowly. In fact, the '''Golden ratio''' has the slowest
possible convergence for any irrational number.
;Task
For each of the first '''10 Metallic ratios'''; '''b''' = '''0''' through '''9''':
* Generate the corresponding "Lucas" sequence.
* Show here, on this page, at least the first '''15''' elements of the "Lucas" sequence.
* Using successive approximations, calculate the value of the ratio accurate to '''32''' decimal places.
* Show the '''value''' of the '''approximation''' at the required accuracy.
* Show the '''value''' of '''n''' when the approximation reaches the required accuracy (How many iterations did it take?).
Optional, stretch goal - Show the '''value''' and number of iterations '''n''', to approximate the '''Golden ratio''' to '''256''' decimal places.
You may assume that the approximation has been reached when the next iteration does not cause the value (to the desired places) to change.
;See also
* [[wp:Metallic_mean|Wikipedia: Metallic mean]]
* [[wp:Lucas_sequence|Wikipedia: Lucas sequence]]